4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).nb

(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 7.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[    304557,       5965]
NotebookOptionsPosition[    298869,       5813]
NotebookOutlinePosition[    300687,       5868]
CellTagsIndexPosition[    300644,       5865]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[Cell[TextData[{
  StyleBox["Rules for integrands of the form ",
   FontFamily->"Arial"],
  Cell[BoxData[
   RowBox[{
    SuperscriptBox[
     RowBox[{
      StyleBox["(",
       FontSize->16], 
      RowBox[{
       StyleBox["a",
        FontSize->16], 
       StyleBox["+",
        FontSize->16], 
       RowBox[{
        StyleBox["b",
         FontSize->16], 
        StyleBox[" ",
         FontSize->16], 
        RowBox[{
         StyleBox["Tan",
          FontSize->16], 
         StyleBox["[",
          FontSize->16], 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
    RowBox[{"(", 
     RowBox[{
      StyleBox["A",
       FontSize->16], 
      StyleBox["+",
       FontSize->16], 
      StyleBox[
       RowBox[{"B", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}],
       FontSize->16], 
      StyleBox["+",
       FontSize->16], 
      RowBox[{
       StyleBox["C",
        FontSize->16], 
       StyleBox[" ",
        FontSize->16], 
       SuperscriptBox[
        StyleBox[
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}],
         FontSize->16], "2"]}]}], 
     StyleBox[")",
      FontSize->16]}]}]],
   CellDingbat->None,
   CellChangeTimes->{
    3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
     3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
     3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
     3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
     3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
     3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
     3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
     3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
     3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
     3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
     3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
     3.5193325694253187`*^9, {3.5193415004828687`*^9, 
     3.5193415113404875`*^9}, {3.5193513965602303`*^9, 
     3.519351397420232*^9}, {3.5194037839416766`*^9, 3.5194037847117205`*^9}, 
     3.52105258275843*^9, 3.5210531741726685`*^9, {3.521057461672199*^9, 
     3.5210574760242243`*^9}, {3.521059881564049*^9, 3.521059886088057*^9}, {
     3.5213902158895197`*^9, 3.521390217543123*^9}, {3.521390827878595*^9, 
     3.521390829376197*^9}, {3.5214688792561245`*^9, 3.521468879443325*^9}, {
     3.523320201951167*^9, 3.523320202247567*^9}, {3.5288596764894824`*^9, 
     3.5288596768094826`*^9}, {3.529196266260353*^9, 
     3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
     3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
     3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
     3.529285215443486*^9}, {3.5295115356999426`*^9, 
     3.5295115433443794`*^9}, {3.529521412127842*^9, 3.529521414647986*^9}, {
     3.529529429975619*^9, 3.529529432955224*^9}, 3.5340074894011116`*^9, 
     3.5415537711400824`*^9, {3.5415538409034047`*^9, 3.541553841106205*^9}, {
     3.5415564120531206`*^9, 3.5415564206955357`*^9}, 
     3.5415565488341613`*^9, {3.5415605631516123`*^9, 
     3.5415605860836525`*^9}, {3.541561323247347*^9, 3.5415613234501476`*^9}, 
     3.5416162115728135`*^9, {3.5416506679001875`*^9, 3.541650687980216*^9}, 
     3.560533948494043*^9, 3.560538006319724*^9, {3.5605392374714475`*^9, 
     3.5605392382814484`*^9}, 3.5606250606115856`*^9, 3.5617573869745255`*^9, 
     3.561757856045182*^9, {3.561761339582559*^9, 3.5617613636025925`*^9}, {
     3.5617652519980364`*^9, 3.561765252188037*^9}, {3.561767460571129*^9, 
     3.5617674608411293`*^9}, 3.5621081762414055`*^9, {3.5621084468550806`*^9,
      3.5621084519718895`*^9}, 3.563340740121106*^9},
   FontWeight->"Bold"]
 }], "None"]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.4965234353062396`*^9, {3.5193214825316973`*^9, 3.5193214852617016`*^9}, {
   3.523316399894187*^9, 3.5233164032341914`*^9}, {3.523318350558202*^9, 
   3.5233183516382036`*^9}, {3.536542725780803*^9, 3.536542725780803*^9}, {
   3.5367764878952584`*^9, 3.5367764886452594`*^9}, {3.544322815209103*^9, 
   3.5443228192963104`*^9}, {3.5450662624913826`*^9, 3.545066292141424*^9}, {
   3.545066402271579*^9, 3.5450664054415827`*^9}, {3.5450834884295015`*^9, 
   3.5450834895895033`*^9}, {3.5454961263762493`*^9, 
   3.5454961321462574`*^9}, {3.5454963863166127`*^9, 
   3.5454964580867133`*^9}, {3.545497046127537*^9, 3.545497046127537*^9}, {
   3.5454971130676303`*^9, 3.545497136257663*^9}, 3.545610399977621*^9, {
   3.546040778592962*^9, 3.546040778592962*^9}, {3.5461052065706367`*^9, 
   3.5461052249474688`*^9}, {3.546191339879622*^9, 3.5461913430596266`*^9}, {
   3.5462141824067917`*^9, 3.5462142145768366`*^9}, {3.5462145455473003`*^9, 
   3.5462145601073203`*^9}, {3.5463149785974817`*^9, 
   3.5463149794195285`*^9}, {3.5488738437683954`*^9, 
   3.5488738699084315`*^9}, {3.5488890647645535`*^9, 3.548889080804576*^9}, {
   3.5505951583099127`*^9, 3.5505951620351257`*^9}, {3.5505952613798075`*^9, 
   3.5505952630869055`*^9}, {3.5535321133435373`*^9, 
   3.5535321173537664`*^9}, {3.5536175341411576`*^9, 
   3.5536175352411594`*^9}, {3.55396599671187*^9, 3.55396599862998*^9}, {
   3.560190530310161*^9, 3.560190628512334*^9}, {3.5602162723838596`*^9, 
   3.5602162793138695`*^9}, {3.5606249190913877`*^9, 3.560624924321395*^9}, {
   3.5617663419995623`*^9, 3.5617663705796027`*^9}, {3.5617666456099873`*^9, 
   3.561766656350003*^9}, {3.562107691486154*^9, 3.5621077390350375`*^9}, {
   3.5621080165751247`*^9, 3.5621080185719285`*^9}, {3.5621081511877613`*^9, 
   3.5621081511877613`*^9}, {3.5633407304210925`*^9, 3.563340730891093*^9}, {
   3.575852587746504*^9, 3.575852593256511*^9}, {3.575852659996605*^9, 
   3.575852659996605*^9}, {3.575852743186721*^9, 3.575852743186721*^9}, {
   3.57585284254686*^9, 3.575852895956935*^9}, {3.575853026617118*^9, 
   3.575853038617135*^9}, {3.575853153907296*^9, 3.575853185757341*^9}, {
   3.575853290447487*^9, 3.575853290447487*^9}, {3.5758553923824325`*^9, 
   3.5758553923824325`*^9}, {3.5758554525725164`*^9, 
   3.5758554525725164`*^9}, {3.5758555750526876`*^9, 
   3.5758555750526876`*^9}, {3.5758556596228065`*^9, 
   3.5758556596228065`*^9}, {3.59711750603603*^9, 3.5971175097360353`*^9}, {
   3.599344008976363*^9, 3.599344026266387*^9}, {3.600822083799305*^9, 
   3.6008220901193137`*^9}, {3.6008222754495735`*^9, 3.600822286709589*^9}, {
   3.600822346629673*^9, 3.6008223539596834`*^9}, {3.607268100733368*^9, 
   3.607268110573382*^9}},
 TextAlignment->Center,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["1:",
   FontFamily->"Arial",
   FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
    RowBox[{"(", 
     RowBox[{"A", "+", 
      RowBox[{"A", " ", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
    RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.4965117206885777`*^9, 3.5192471175756674`*^9, {3.5192587973599553`*^9, 
   3.5192588081399703`*^9}, {3.519321495105319*^9, 3.5193214991925263`*^9}, 
   3.5193247857072587`*^9, {3.5194051897010813`*^9, 3.5194051902791147`*^9}, {
   3.5194055551249824`*^9, 3.5194055708408813`*^9}, {3.519405711316916*^9, 
   3.51940572904193*^9}, {3.519410772192028*^9, 3.5194107776364374`*^9}, {
   3.519413782484516*^9, 3.519413782827717*^9}, {3.519767588664174*^9, 
   3.519767611128214*^9}, {3.5197937941314034`*^9, 3.519793794568204*^9}, {
   3.5197947626591043`*^9, 3.519794764749508*^9}, {3.519794865354085*^9, 
   3.519794874355301*^9}, {3.523218063086026*^9, 3.523218070356036*^9}, {
   3.523316514684347*^9, 3.523316514684347*^9}, {3.5233165475343933`*^9, 
   3.5233165475343933`*^9}, {3.5233184331871357`*^9, 
   3.5233184334271364`*^9}, {3.5273599103435545`*^9, 3.527359910608755*^9}, {
   3.5326424241029196`*^9, 3.53264242435292*^9}, {3.5326584126048727`*^9, 
   3.532658412794873*^9}, {3.532726454021428*^9, 3.5327264542814283`*^9}, {
   3.5327554047263184`*^9, 3.532755404993334*^9}, {3.532798666112293*^9, 
   3.532798676322307*^9}, {3.5328778343810363`*^9, 3.5328778343810363`*^9}, 
   3.5330758294771795`*^9, {3.5330827570868783`*^9, 3.533082757336879*^9}, {
   3.534975097668671*^9, 3.534975097888672*^9}, {3.536863459221404*^9, 
   3.536863459439804*^9}, {3.540264711953947*^9, 3.5402647253239655`*^9}, {
   3.540264783504047*^9, 3.540264783504047*^9}, {3.540502117527218*^9, 
   3.540502117997218*^9}, {3.540502551737826*^9, 3.540502551937826*^9}, {
   3.5449015667548094`*^9, 3.5449015920048447`*^9}, {3.544901714065016*^9, 
   3.544901727955035*^9}, 3.544911958524931*^9, {3.5449122781717815`*^9, 
   3.5449122781717815`*^9}, {3.5455187047402697`*^9, 
   3.5455187052402706`*^9}, {3.545518898070541*^9, 3.545518911130559*^9}, {
   3.5455202868524847`*^9, 3.5455202868524847`*^9}, {3.545520317832528*^9, 
   3.545520339272558*^9}, {3.545520578812894*^9, 3.545520601212925*^9}, {
   3.545520817443228*^9, 3.545520820373232*^9}, {3.545578383065389*^9, 
   3.545578398615411*^9}, {3.5461099987254543`*^9, 3.54611000814787*^9}, {
   3.546110085383606*^9, 3.546110115616459*^9}, {3.546113143676578*^9, 
   3.5461131599162064`*^9}, {3.546113861589839*^9, 3.546113902539911*^9}, {
   3.546114166008774*^9, 3.546114166008774*^9}, {3.5461157697071905`*^9, 
   3.546115770081591*^9}, {3.546881285463888*^9, 3.546881338233962*^9}, 
   3.5468817383245225`*^9, {3.546881986034869*^9, 3.546881986034869*^9}, {
   3.5468892568770494`*^9, 3.546889259477053*^9}, {3.5470661850482807`*^9, 
   3.5470661850482807`*^9}, {3.5470663601085253`*^9, 
   3.5470663608385267`*^9}, {3.547079095010357*^9, 3.547079106460373*^9}, {
   3.5484359059573984`*^9, 3.548435906657399*^9}, {3.5484359766974974`*^9, 
   3.5484359766974974`*^9}, {3.5647716159718266`*^9, 
   3.5647716160018272`*^9}, {3.5648664240101037`*^9, 3.564866438130123*^9}, {
   3.5651351161390033`*^9, 3.5651351163390036`*^9}, {3.5804089094175644`*^9, 
   3.5804089108475666`*^9}, {3.5804898643695683`*^9, 3.580489864659569*^9}, {
   3.5804921727253036`*^9, 3.5804921728453035`*^9}, {3.580493765467533*^9, 
   3.5804937656875334`*^9}, 3.5971897182598896`*^9, {3.597191023646764*^9, 
   3.5971910240017643`*^9}, {3.5972453914603157`*^9, 
   3.5972453949603205`*^9}, {3.597245713090766*^9, 3.5972457172507715`*^9}, {
   3.601148546589867*^9, 3.601148572299903*^9}, {3.601262326634014*^9, 
   3.601262346654043*^9}, 3.6012625421043234`*^9, 3.6120369670942698`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by substitution", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.4964417379912624`*^9, 3.4964417471112747`*^9}, {
   3.496441910111503*^9, 3.4964419275315275`*^9}, 3.496451213764926*^9, 
   3.496528833243797*^9, {3.4975777080419827`*^9, 3.4975777244120054`*^9}, {
   3.4975780869725127`*^9, 3.4975780962225256`*^9}, {3.4976627907155895`*^9, 
   3.497662798718403*^9}, {3.5076647229436293`*^9, 3.507664727563636*^9}, 
   3.514253181887664*^9, {3.5455040345073204`*^9, 3.5455040366873236`*^9}, {
   3.545504158357494*^9, 3.545504246227617*^9}, {3.545505035248721*^9, 
   3.5455051376488647`*^9}, {3.5456005712019787`*^9, 
   3.5456005712019787`*^9}, {3.5480100113402867`*^9, 3.548010012360288*^9}, {
   3.548010302450694*^9, 3.548010302870695*^9}, {3.5630304726923504`*^9, 
   3.5630305459812794`*^9}, {3.5630307316528053`*^9, 
   3.5630307335092087`*^9}, {3.5630327435025606`*^9, 
   3.5630327618226085`*^9}, {3.5630346364023542`*^9, 3.563034660882388*^9}, {
   3.5630350834229803`*^9, 3.563035093942995*^9}, {3.563036967865618*^9, 
   3.563036986495644*^9}, {3.5630374367662745`*^9, 3.563037445046286*^9}, {
   3.563038586937885*^9, 3.5630385936278944`*^9}, {3.5630388430582438`*^9, 
   3.5630388526682568`*^9}, {3.5630392002887435`*^9, 
   3.5630392327687893`*^9}, {3.5630392880588665`*^9, 
   3.5630392887688675`*^9}, {3.5630396749469166`*^9, 3.563039675866918*^9}, {
   3.5630397820070667`*^9, 3.563039854937169*^9}, {3.5630408041934977`*^9, 
   3.56304081968352*^9}, {3.563040869123589*^9, 3.563040895703626*^9}, {
   3.5630410000337725`*^9, 3.563041038373826*^9}, {3.563041076383879*^9, 
   3.5630410772538805`*^9}, {3.563041161243998*^9, 3.563041193284043*^9}, {
   3.5630413434642534`*^9, 3.563041344414254*^9}, {3.5630445357587223`*^9, 
   3.5630445601687565`*^9}, {3.5630445952288055`*^9, 
   3.5630445952288055`*^9}, {3.563044995139365*^9, 3.5630450259994087`*^9}, {
   3.5630450931295023`*^9, 3.5630451258995485`*^9}, {3.5630465995342903`*^9, 
   3.563046638464345*^9}, {3.563046678854401*^9, 3.563046678854401*^9}, {
   3.5630467805945435`*^9, 3.5630467813945446`*^9}, {3.563046962064798*^9, 
   3.56304698510483*^9}, {3.5630475237855844`*^9, 3.5630475460756154`*^9}, {
   3.563047627255729*^9, 3.5630476379957438`*^9}, {3.563048032996297*^9, 
   3.5630480816963654`*^9}, {3.5630482232065635`*^9, 3.5630482502566013`*^9}, 
   3.564780769678449*^9, 3.564780801828494*^9, {3.5647809441386933`*^9, 
   3.5647809441386933`*^9}, {3.564804839566764*^9, 3.5648048555411916`*^9}, {
   3.5648456887553034`*^9, 3.5648456967053146`*^9}, {3.564866466950164*^9, 
   3.5648664684801655`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{"F", "[", 
     RowBox[{"b", " ", 
      RowBox[{"Tan", "[", 
       RowBox[{"e", "+", 
        RowBox[{"f", " ", "x"}]}], "]"}]}], "]"}], 
    RowBox[{"(", 
     RowBox[{"A", "+", 
      RowBox[{"A", " ", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}]}], "\[Equal]", 
   RowBox[{
    FractionBox["A", 
     RowBox[{"b", " ", "f"}]], 
    RowBox[{"Subst", "[", 
     RowBox[{
      RowBox[{"F", "[", "x", "]"}], ",", "x", ",", 
      RowBox[{"b", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}]}], "]"}], 
    RowBox[{
     SubscriptBox["\[PartialD]", "x"], 
     RowBox[{"(", 
      RowBox[{"b", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}], ")"}]}]}]}]],
  CellChangeTimes->{{3.564804887614848*^9, 3.564804965942586*^9}}]
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.4964417379912624`*^9, 3.4964417471112747`*^9}, {
   3.496441910111503*^9, 3.4964419275315275`*^9}, 3.496451213764926*^9, 
   3.496528833243797*^9, {3.4975777080419827`*^9, 3.4975777244120054`*^9}, {
   3.4975778737222147`*^9, 3.4975778945722437`*^9}, {3.4976396822792068`*^9, 
   3.49763968986082*^9}, {3.497639771433363*^9, 3.49763978104298*^9}, {
   3.4976638984111347`*^9, 3.4976639153215647`*^9}, {3.4976645939383564`*^9, 
   3.497664598805565*^9}, {3.4976646480080514`*^9, 3.4976646992229414`*^9}, {
   3.4976647966763124`*^9, 3.497664832415975*^9}, 3.497664903037299*^9, {
   3.4986779242986293`*^9, 3.498677947778662*^9}, {3.499179738750054*^9, 
   3.499179740902858*^9}, {3.499213316063504*^9, 3.4992133345435305`*^9}, {
   3.4992133873636045`*^9, 3.4992133892036066`*^9}, {3.4992162641476316`*^9, 
   3.499216265707634*^9}, {3.4993685410928354`*^9, 3.4993685488628464`*^9}, {
   3.50031970240469*^9, 3.5003197069046965`*^9}, {3.5004320341975737`*^9, 
   3.5004320353096375`*^9}, {3.5026784535032854`*^9, 3.502678461573297*^9}, {
   3.5027657904804587`*^9, 3.5027657904804587`*^9}, {3.5027672190193686`*^9, 
   3.5027672190193686`*^9}, {3.5035496199411335`*^9, 
   3.5035496486311736`*^9}, {3.504028649276944*^9, 3.5040286502597466`*^9}, 
   3.50403187501501*^9, 3.5043107662485094`*^9, {3.5047476342426376`*^9, 
   3.504747635372639*^9}, {3.509473798840955*^9, 3.509473798840955*^9}, {
   3.509934658346775*^9, 3.5099346733567963`*^9}, 3.50993851024617*^9, 
   3.509982467717354*^9, {3.51016913209204*^9, 3.51016913209204*^9}, {
   3.5101701908549356`*^9, 3.5101701908549356`*^9}, {3.5104308159080315`*^9, 
   3.5104308165080323`*^9}, {3.5104512814432535`*^9, 
   3.5104512826132545`*^9}, {3.51207648865281*^9, 3.5120764894028115`*^9}, 
   3.512076647043032*^9, 3.514852964035932*^9, {3.5148630622076383`*^9, 
   3.5148630622076383`*^9}, {3.5170044468353167`*^9, 3.517004462981345*^9}, {
   3.546877072757991*^9, 3.546877072757991*^9}, {3.546877116468052*^9, 
   3.546877116468052*^9}, {3.5468800248321238`*^9, 3.5468800248321238`*^9}, {
   3.5468905744888945`*^9, 3.546890582888906*^9}, {3.549237858267603*^9, 
   3.549237858267603*^9}, {3.5647810126487894`*^9, 3.5647810126487894`*^9}, {
   3.5647827296111927`*^9, 3.5647827296111927`*^9}, {3.564804872732422*^9, 
   3.5648048735904236`*^9}, {3.5648049823070145`*^9, 
   3.5648049909962296`*^9}, {3.564805057389946*^9, 3.5648051481353054`*^9}, {
   3.5971897999200068`*^9, 3.5971897999200068`*^9}, {3.5971899198101788`*^9, 
   3.597189949867722*^9}, {3.6011485467998676`*^9, 3.6011485468198676`*^9}, {
   3.6012619517984767`*^9, 3.6012619706035037`*^9}, {3.601262021773577*^9, 
   3.6012620375235996`*^9}, {3.693418030656555*^9, 3.6934180306575556`*^9}}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.4964417379912624`*^9, 3.4964417471112747`*^9}, {
   3.496441910111503*^9, 3.4964419275315275`*^9}, 3.496451213764926*^9, 
   3.496528833243797*^9, {3.4975777080419827`*^9, 3.4975777244120054`*^9}, {
   3.4975778737222147`*^9, 3.4975778945722437`*^9}, {3.4976396822792068`*^9, 
   3.49763968986082*^9}, {3.497639771433363*^9, 3.49763978104298*^9}, {
   3.4976638984111347`*^9, 3.4976639153215647`*^9}, {3.4976645939383564`*^9, 
   3.497664598805565*^9}, {3.4976646480080514`*^9, 3.4976646992229414`*^9}, {
   3.4976647966763124`*^9, 3.497664832415975*^9}, 3.497664903037299*^9, {
   3.4986779242986293`*^9, 3.498677947778662*^9}, {3.499179738750054*^9, 
   3.499179740902858*^9}, {3.499213316063504*^9, 3.4992133345435305`*^9}, {
   3.499358051967739*^9, 3.4993580717017736`*^9}, {3.4993638312082148`*^9, 
   3.499363839028226*^9}, {3.499368533262825*^9, 3.499368534762827*^9}, {
   3.500319714354707*^9, 3.500319719674714*^9}, {3.500321207536797*^9, 
   3.5003212188668127`*^9}, {3.501358653573163*^9, 3.5013586559531665`*^9}, {
   3.502678432393256*^9, 3.5026784351732597`*^9}, {3.502766589293062*^9, 
   3.502766589293062*^9}, 3.5027671790052986`*^9, 3.5027674748646183`*^9, {
   3.5038569856949825`*^9, 3.503856992059794*^9}, {3.504031357905302*^9, 
   3.5040313587633038`*^9}, 3.5040317449575815`*^9, {3.504311260541732*^9, 
   3.504311260711732*^9}, {3.504750498076647*^9, 3.50475050021665*^9}, {
   3.5092408217803926`*^9, 3.509240824151596*^9}, {3.510164009908225*^9, 
   3.510164009908225*^9}, {3.5117214107451816`*^9, 3.5117214107451816`*^9}, {
   3.5120768665233393`*^9, 3.5120768665233393`*^9}, {3.5121850888428593`*^9, 
   3.5121850938972683`*^9}, 3.546877116548052*^9, {3.5468800973122253`*^9, 
   3.5468800973122253`*^9}, {3.547066202678305*^9, 3.547066202678305*^9}, {
   3.5470791133403826`*^9, 3.5470791133403826`*^9}, {3.548435986417511*^9, 
   3.548435986417511*^9}, {3.564866488150193*^9, 3.564866488150193*^9}, 
   3.6012623755290847`*^9}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
   RowBox[{"(", 
    RowBox[{"A", "+", 
     RowBox[{"A", " ", 
      SuperscriptBox[
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox["A", 
      RowBox[{"b", " ", "f"}]]}]}], " ", 
   RowBox[{"Subst", "[", 
    RowBox[{
     RowBox[{"\[Integral]", 
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"a", "+", "x"}], ")"}], "m"], 
       RowBox[{"\[DifferentialD]", "x"}]}]}], ",", "x", ",", 
     RowBox[{"b", " ", 
      RowBox[{"Tan", "[", 
       RowBox[{"e", "+", 
        RowBox[{"f", " ", "x"}]}], "]"}]}]}], "]"}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.496441851501421*^9, 
   3.496441872881451*^9}, {3.496528916553913*^9, 3.4965289272239285`*^9}, {
   3.497576165747217*^9, 3.4975761801372375`*^9}, {3.497577761382057*^9, 
   3.4975777938921027`*^9}, 3.49763959300025*^9, 3.497639634480723*^9, 
   3.4976398417426867`*^9, {3.49766381031778*^9, 3.4976638350282235`*^9}, {
   3.49766435104593*^9, 3.497664390373599*^9}, {3.4976647278489914`*^9, 
   3.4976647419046164`*^9}, {3.497664869980841*^9, 3.497664887437272*^9}, {
   3.498761171082196*^9, 3.498761171082196*^9}, {3.4987645684369526`*^9, 
   3.4987646442070584`*^9}, {3.49883740188522*^9, 3.4988375207310295`*^9}, {
   3.498837939529365*^9, 3.498837958467798*^9}, 3.498875518717125*^9, {
   3.4991794250335026`*^9, 3.499179452879552*^9}, {3.4991796787211485`*^9, 
   3.499179681529153*^9}, {3.4991961982482667`*^9, 3.4991962392763386`*^9}, {
   3.499196653100665*^9, 3.4991966955639396`*^9}, {3.499197135282312*^9, 
   3.499197176918785*^9}, {3.499197690444487*^9, 3.499197694048094*^9}, {
   3.499212356332161*^9, 3.4992123865622034`*^9}, {3.499212421412252*^9, 
   3.499212423032254*^9}, {3.4992126233625345`*^9, 3.499212696332637*^9}, {
   3.499357348905704*^9, 3.4993574380910606`*^9}, {3.499404301116287*^9, 
   3.49940430323629*^9}, {3.4994048006769867`*^9, 3.4994048025069895`*^9}, {
   3.5003179811489162`*^9, 3.5003179814297166`*^9}, 3.5003181853432746`*^9, {
   3.5003185563630857`*^9, 3.500318557803087*^9}, {3.5013586287231283`*^9, 
   3.5013586400931444`*^9}, {3.5027658915998363`*^9, 3.502765896981846*^9}, {
   3.5027668222170715`*^9, 3.50276686143554*^9}, {3.505344692893856*^9, 
   3.5053446929038563`*^9}, {3.5092397382492895`*^9, 
   3.5092398121778193`*^9}, {3.51016418932094*^9, 3.5101642175257893`*^9}, {
   3.515189878545912*^9, 3.515189880095914*^9}, {3.5468767933175993`*^9, 
   3.5468767933275995`*^9}, {3.5468768261776457`*^9, 3.546876840807666*^9}, {
   3.564771616011827*^9, 3.564771616051827*^9}, {3.564866512920228*^9, 
   3.5648665875203323`*^9}, {3.597189829315049*^9, 3.5971898413850665`*^9}, {
   3.6011485468398676`*^9, 3.60114859892994*^9}, {3.6012620500436172`*^9, 
   3.60126207260865*^9}, {3.601262388299103*^9, 3.601262395039113*^9}, {
   3.6934180434162855`*^9, 3.693418092031066*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"(", 
      RowBox[{"A_", "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}]}], ",", 
    "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"A", "/", 
     RowBox[{"(", 
      RowBox[{"b", "*", "f"}], ")"}]}], "*", 
    RowBox[{"Subst", "[", 
     RowBox[{
      RowBox[{"Int", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"a", "+", "x"}], ")"}], "^", "m"}], ",", "x"}], "]"}], ",", 
      "x", ",", 
      RowBox[{"b", "*", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", "*", "x"}]}], "]"}]}]}], "]"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "C", ",", "m"}], 
       "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"EqQ", "[", 
     RowBox[{"A", ",", "C"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.4993638009881725`*^9, {3.4993661760314975`*^9, 3.4993661794315023`*^9}, 
   3.4993699436547995`*^9, 3.499369995914872*^9, {3.5003179814609165`*^9, 
   3.5003179817417173`*^9}, 3.5003186747732515`*^9, {3.5003187057132945`*^9, 
   3.5003187115933027`*^9}, {3.5003194495843363`*^9, 3.500319453664342*^9}, {
   3.5003198135348454`*^9, 3.5003198411248837`*^9}, 3.500320756016165*^9, 
   3.50052634660607*^9, {3.501043281866661*^9, 3.5010432838407745`*^9}, {
   3.501358610383103*^9, 3.5013586217531185`*^9}, {3.5026774770619183`*^9, 
   3.5026774832519274`*^9}, {3.502677757052311*^9, 3.502677762832319*^9}, {
   3.502765679626664*^9, 3.502765694945891*^9}, {3.502766916862438*^9, 
   3.5027669304188614`*^9}, 3.5027671021907635`*^9, {3.503071895268139*^9, 
   3.5030719155981674`*^9}, {3.50351615772817*^9, 3.5035161598181725`*^9}, {
   3.503856818447089*^9, 3.503856835341919*^9}, {3.5053446929138565`*^9, 
   3.5053446929338565`*^9}, {3.5092398129578204`*^9, 
   3.5092399297864256`*^9}, {3.509240858050456*^9, 3.509240876630089*^9}, {
   3.5094733331253176`*^9, 3.509473337703579*^9}, 3.509475176411747*^9, 
   3.5094752858760085`*^9, {3.509501454231578*^9, 3.5095014624839926`*^9}, {
   3.5095015947098246`*^9, 3.5095015977830305`*^9}, 3.509937960073399*^9, 
   3.5099380786935654`*^9, {3.51016691275181*^9, 3.5101669291418333`*^9}, {
   3.510167526232669*^9, 3.5101675556027107`*^9}, {3.5101684080895686`*^9, 
   3.5101684312332087`*^9}, {3.512076721843137*^9, 3.5120767220331373`*^9}, 
   3.5168207803766356`*^9, {3.517001937691708*^9, 3.517001939875712*^9}, 
   3.541093954069064*^9, {3.5410951460003576`*^9, 3.541095146920759*^9}, 
   3.541312061104577*^9, {3.5454921203191385`*^9, 3.5454921203291388`*^9}, {
   3.5468768663577013`*^9, 3.5468769234377813`*^9}, {3.5468769673278427`*^9, 
   3.546876975997855*^9}, {3.5468804677827435`*^9, 3.546880470102747*^9}, 
   3.5468809973634853`*^9, {3.5470662581683826`*^9, 3.5470662605883865`*^9}, 
   3.5470790733603263`*^9, {3.547079227800543*^9, 3.547079229430545*^9}, {
   3.548435995007523*^9, 3.548436016337553*^9}, 3.548436410738105*^9, {
   3.5500102505425825`*^9, 3.550010257016594*^9}, {3.564771616081827*^9, 
   3.564771616101827*^9}, {3.564771654511881*^9, 3.564771654531881*^9}, {
   3.564866625740386*^9, 3.564866637140402*^9}, {3.564866683610468*^9, 
   3.5648667120405073`*^9}, {3.5799787188202853`*^9, 
   3.5799787229402914`*^9}, {3.5799968380806684`*^9, 
   3.5799968404706717`*^9}, {3.5803539845087166`*^9, 3.580353988428722*^9}, {
   3.580489885019597*^9, 3.580489887759601*^9}, {3.580491263331527*^9, 
   3.5804912695015354`*^9}, {3.5804913102515926`*^9, 
   3.5804913141315985`*^9}, {3.5971899775277615`*^9, 3.5971900135478134`*^9}, 
   3.5972455825005827`*^9, {3.5972457449508104`*^9, 3.597245768790844*^9}, {
   3.6011527190564566`*^9, 3.601152719066457*^9}, 3.601257203696659*^9, {
   3.601262415599142*^9, 3.601262438474175*^9}, {3.6934181518714886`*^9, 
   3.693418163582158*^9}, 3.706139449925252*^9, 3.7154553358371506`*^9},
 Background->GrayLevel[0.85]],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{"cot", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"(", 
      RowBox[{"A_", "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"cot", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}]}], ",", 
    "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"-", "A"}], "/", 
     RowBox[{"(", 
      RowBox[{"b", "*", "f"}], ")"}]}], "*", 
    RowBox[{"Subst", "[", 
     RowBox[{
      RowBox[{"Int", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"a", "+", "x"}], ")"}], "^", "m"}], ",", "x"}], "]"}], ",", 
      "x", ",", 
      RowBox[{"b", "*", 
       RowBox[{"Cot", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", "*", "x"}]}], "]"}]}]}], "]"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "C", ",", "m"}], 
       "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"EqQ", "[", 
     RowBox[{"A", ",", "C"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.4993638009881725`*^9, {3.4993661760314975`*^9, 3.4993661794315023`*^9}, 
   3.4993699436547995`*^9, 3.499369995914872*^9, {3.5003179814609165`*^9, 
   3.5003179817417173`*^9}, 3.5003186747732515`*^9, {3.5003187057132945`*^9, 
   3.5003187115933027`*^9}, {3.5003194495843363`*^9, 3.500319453664342*^9}, {
   3.5003198135348454`*^9, 3.5003198411248837`*^9}, 3.500320756016165*^9, 
   3.50052634660607*^9, {3.501043281866661*^9, 3.5010432838407745`*^9}, {
   3.501358610383103*^9, 3.5013586217531185`*^9}, {3.5026774770619183`*^9, 
   3.5026774832519274`*^9}, {3.502677757052311*^9, 3.502677762832319*^9}, {
   3.502765679626664*^9, 3.502765694945891*^9}, {3.502766916862438*^9, 
   3.5027669304188614`*^9}, 3.5027671021907635`*^9, {3.503071895268139*^9, 
   3.5030719155981674`*^9}, {3.50351615772817*^9, 3.5035161598181725`*^9}, {
   3.503856818447089*^9, 3.503856835341919*^9}, {3.5053446929138565`*^9, 
   3.5053446929338565`*^9}, {3.5092398129578204`*^9, 
   3.5092399297864256`*^9}, {3.509240858050456*^9, 3.509240876630089*^9}, {
   3.5094733331253176`*^9, 3.509473337703579*^9}, 3.509475176411747*^9, 
   3.5094752858760085`*^9, {3.509501454231578*^9, 3.5095014624839926`*^9}, {
   3.5095015947098246`*^9, 3.5095015977830305`*^9}, 3.509937960073399*^9, 
   3.5099380786935654`*^9, {3.51016691275181*^9, 3.5101669291418333`*^9}, {
   3.510167526232669*^9, 3.5101675556027107`*^9}, {3.5101684080895686`*^9, 
   3.5101684312332087`*^9}, {3.512076721843137*^9, 3.5120767220331373`*^9}, 
   3.5168207803766356`*^9, {3.517001937691708*^9, 3.517001939875712*^9}, 
   3.541093954069064*^9, {3.5410951460003576`*^9, 3.541095146920759*^9}, 
   3.541312061104577*^9, {3.5454921203191385`*^9, 3.5454921203291388`*^9}, {
   3.5468768663577013`*^9, 3.5468769234377813`*^9}, {3.5468769673278427`*^9, 
   3.546876975997855*^9}, {3.5468804677827435`*^9, 3.546880470102747*^9}, 
   3.5468809973634853`*^9, {3.5470662581683826`*^9, 3.5470662605883865`*^9}, 
   3.5470790733603263`*^9, {3.547079227800543*^9, 3.547079229430545*^9}, {
   3.548435995007523*^9, 3.548436016337553*^9}, 3.548436410738105*^9, {
   3.5500102505425825`*^9, 3.550010257016594*^9}, {3.564771616081827*^9, 
   3.564771616101827*^9}, {3.564771654511881*^9, 3.564771654531881*^9}, {
   3.564866625740386*^9, 3.564866637140402*^9}, {3.564866683610468*^9, 
   3.5648667120405073`*^9}, {3.5799787188202853`*^9, 
   3.5799787229402914`*^9}, {3.5799968380806684`*^9, 
   3.5799968404706717`*^9}, {3.5803539845087166`*^9, 3.580353988428722*^9}, {
   3.580489885019597*^9, 3.580489887759601*^9}, {3.580491263331527*^9, 
   3.5804912695015354`*^9}, {3.5804913102515926`*^9, 
   3.5804913141315985`*^9}, {3.5971899775277615`*^9, 3.5971900135478134`*^9}, 
   3.5972455825005827`*^9, {3.5972457449508104`*^9, 3.597245768790844*^9}, {
   3.6011527190564566`*^9, 3.601152719066457*^9}, 3.601257203696659*^9, {
   3.601262415599142*^9, 3.6012624576242027`*^9}, {3.6934181743827763`*^9, 
   3.6934181791240473`*^9}, 3.706139449935252*^9, {3.7154553376952567`*^9, 
   3.7154553395403624`*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["2:",
   FontFamily->"Arial",
   FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
    RowBox[{"(", 
     RowBox[{"A", "+", 
      RowBox[{"B", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
      RowBox[{"C", " ", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
    RowBox[{"\[DifferentialD]", "x"}], "  ", 
    StyleBox["when",
     FontFamily->"Arial",
     FontWeight->"Plain"], 
    StyleBox["  ",
     FontFamily->"Arial",
     FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
     RowBox[{
      RowBox[{
       RowBox[{"A", " ", 
        SuperscriptBox["b", "2"]}], "-", 
       RowBox[{"a", " ", "b", " ", "B"}], "+", 
       RowBox[{
        SuperscriptBox["a", "2"], " ", "C"}]}], "\[Equal]", "0"}]]]], 
     "None"]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.4965117206885777`*^9, 3.5192471175756674`*^9, {3.5192587973599553`*^9, 
   3.5192588081399703`*^9}, {3.519321495105319*^9, 3.5193214991925263`*^9}, 
   3.5193247857072587`*^9, {3.5194051897010813`*^9, 3.5194051902791147`*^9}, {
   3.5194055551249824`*^9, 3.5194055708408813`*^9}, {3.519405711316916*^9, 
   3.51940572904193*^9}, {3.519410772192028*^9, 3.5194107776364374`*^9}, {
   3.519413782484516*^9, 3.519413782827717*^9}, {3.519767588664174*^9, 
   3.519767611128214*^9}, {3.5197937941314034`*^9, 3.519793794568204*^9}, {
   3.5197947626591043`*^9, 3.519794764749508*^9}, {3.519794865354085*^9, 
   3.519794874355301*^9}, {3.523218063086026*^9, 3.523218070356036*^9}, {
   3.523316514684347*^9, 3.523316514684347*^9}, {3.5233165475343933`*^9, 
   3.5233165475343933`*^9}, {3.5233184331871357`*^9, 
   3.5233184334271364`*^9}, {3.5273599103435545`*^9, 3.527359910608755*^9}, {
   3.5326424241029196`*^9, 3.53264242435292*^9}, {3.5326584126048727`*^9, 
   3.532658412794873*^9}, {3.532726454021428*^9, 3.5327264542814283`*^9}, {
   3.5327554047263184`*^9, 3.532755404993334*^9}, {3.532798666112293*^9, 
   3.532798676322307*^9}, {3.5328778343810363`*^9, 3.5328778343810363`*^9}, 
   3.5330758294771795`*^9, {3.5330827570868783`*^9, 3.533082757336879*^9}, {
   3.534975097668671*^9, 3.534975097888672*^9}, {3.536863459221404*^9, 
   3.536863459439804*^9}, {3.540264711953947*^9, 3.5402647253239655`*^9}, {
   3.540264783504047*^9, 3.540264783504047*^9}, {3.540502117527218*^9, 
   3.540502117997218*^9}, {3.540502551737826*^9, 3.540502551937826*^9}, {
   3.5449015667548094`*^9, 3.5449015920048447`*^9}, {3.544901714065016*^9, 
   3.544901727955035*^9}, 3.544911958524931*^9, {3.5449122781717815`*^9, 
   3.5449122781717815`*^9}, {3.5455187047402697`*^9, 
   3.5455187052402706`*^9}, {3.545518898070541*^9, 3.545518911130559*^9}, {
   3.5455202868524847`*^9, 3.5455202868524847`*^9}, {3.545520317832528*^9, 
   3.545520339272558*^9}, {3.545520578812894*^9, 3.545520601212925*^9}, {
   3.545520817443228*^9, 3.545520820373232*^9}, {3.545578383065389*^9, 
   3.545578398615411*^9}, {3.5461099987254543`*^9, 3.54611000814787*^9}, {
   3.546110085383606*^9, 3.546110115616459*^9}, {3.546113143676578*^9, 
   3.5461131599162064`*^9}, {3.546113861589839*^9, 3.546113902539911*^9}, {
   3.546114166008774*^9, 3.546114166008774*^9}, {3.5461157697071905`*^9, 
   3.546115770081591*^9}, {3.546881285463888*^9, 3.546881338233962*^9}, 
   3.5468817383245225`*^9, {3.546881986034869*^9, 3.546881986034869*^9}, {
   3.5468892568770494`*^9, 3.546889259477053*^9}, {3.5470661850482807`*^9, 
   3.5470661850482807`*^9}, {3.5470663601085253`*^9, 
   3.5470663608385267`*^9}, {3.547079095010357*^9, 3.547079106460373*^9}, {
   3.5484290087137384`*^9, 3.54842903151377*^9}, {3.5484291151438875`*^9, 
   3.5484291151438875`*^9}, {3.5484293453342104`*^9, 3.548429355144224*^9}, {
   3.564791286603173*^9, 3.5647912866231728`*^9}, {3.5647983663541937`*^9, 
   3.5647983708938017`*^9}, 3.5647988147925816`*^9, 3.5647988767558904`*^9, {
   3.601148547649869*^9, 3.6011485723799033`*^9}, {3.6012484003491383`*^9, 
   3.601248403559142*^9}, {3.6012625470493307`*^9, 3.601262547189331*^9}, 
   3.612036967974271*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Algebraic simplification", "Subsubsection",
 CellChangeTimes->{
  3.4964418130913677`*^9, {3.496442428087432*^9, 3.4964424330170403`*^9}, 
   3.496528894323882*^9, 3.4975761400971813`*^9, {3.4975778216221414`*^9, 
   3.497577829892153*^9}, {3.4975827089309845`*^9, 3.4975827138509912`*^9}, 
   3.497639710530856*^9, 3.4976397495153246`*^9, 3.4976637968549566`*^9, {
   3.497664121273126*^9, 3.4976641274819374`*^9}, {3.4976641823940334`*^9, 
   3.497664184390837*^9}, 3.497664250254153*^9, 3.497664775694276*^9, 
   3.4976649459685745`*^9, {3.49876308829488*^9, 3.4987630935048876`*^9}, {
   3.4987648355173264`*^9, 3.498764841027334*^9}, {3.4991797271436334`*^9, 
   3.4991797277208347`*^9}, {3.4992121758819084`*^9, 3.499212176211909*^9}, {
   3.4993573070196304`*^9, 3.4993573078152323`*^9}, 3.5013524902049417`*^9, 
   3.501621399618896*^9, 3.5016217326739454`*^9, 3.5016376139935837`*^9, 
   3.5016377276810865`*^9, {3.5017223219869995`*^9, 3.501722324888605*^9}, {
   3.501722402935542*^9, 3.5017224033411427`*^9}, 3.5017229817901583`*^9, 
   3.5017231707844906`*^9, 3.5017234546797895`*^9, {3.5017235643791823`*^9, 
   3.501723566750386*^9}, {3.5017237232810616`*^9, 3.5017237250750647`*^9}, {
   3.5027664855216804`*^9, 3.5027664896088877`*^9}}],

Cell[TextData[{
 "Basis:  If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{"A", " ", 
     SuperscriptBox["b", "2"]}], "-", 
    RowBox[{"a", " ", "b", " ", "B"}], "+", 
    RowBox[{
     SuperscriptBox["a", "2"], " ", "C"}]}], "\[Equal]", "0"}]]],
 ", then ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"A", "+", 
    RowBox[{"B", " ", "z"}], "+", 
    RowBox[{"C", " ", 
     SuperscriptBox["z", "2"]}]}], "\[Equal]", 
   RowBox[{
    FractionBox["1", 
     SuperscriptBox["b", "2"]], 
    RowBox[{"(", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "z"}]}], ")"}], 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"b", " ", "B"}], "-", 
      RowBox[{"a", " ", "C"}], "+", 
      RowBox[{"b", " ", "C", " ", "z"}]}], ")"}]}]}]], "Input",
  CellChangeTimes->{{3.5027666870896344`*^9, 3.502766694577647*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4964418130913677`*^9, {3.496442428087432*^9, 3.4964424330170403`*^9}, 
   3.496528894323882*^9, 3.4975761400971813`*^9, {3.4975778216221414`*^9, 
   3.497577829892153*^9}, {3.4975827089309845`*^9, 3.4975827138509912`*^9}, 
   3.497639710530856*^9, 3.4976397495153246`*^9, 3.4976637968549566`*^9, {
   3.497664121273126*^9, 3.4976641274819374`*^9}, {3.4976641823940334`*^9, 
   3.497664184390837*^9}, 3.497664250254153*^9, 3.497664775694276*^9, 
   3.4976649459685745`*^9, {3.49876308829488*^9, 3.4987630935048876`*^9}, {
   3.4987648355173264`*^9, 3.498764841027334*^9}, {3.4991797271436334`*^9, 
   3.4991797277208347`*^9}, {3.4992121758819084`*^9, 3.499212176211909*^9}, {
   3.4993573070196304`*^9, 3.4993573078152323`*^9}, 3.5013524902049417`*^9, 
   3.501621399618896*^9, 3.5016217326739454`*^9, 3.5016376139935837`*^9, 
   3.5016377276810865`*^9, {3.5017223219869995`*^9, 3.501722324888605*^9}, {
   3.501722402935542*^9, 3.5017224033411427`*^9}, 3.5017229817901583`*^9, 
   3.5017231707844906`*^9, 3.5017234546797895`*^9, {3.5017235643791823`*^9, 
   3.501723566750386*^9}, {3.5017237232810616`*^9, 3.5017237250750647`*^9}, {
   3.5027664855216804`*^9, 3.5027664896088877`*^9}, {3.5027665557998037`*^9, 
   3.5027665566110053`*^9}, 3.5027665996046805`*^9, {3.502766706511668*^9, 
   3.5027667557297544`*^9}, 3.502767181048902*^9, {3.502767342353185*^9, 
   3.5027673502359996`*^9}, 3.502767478140624*^9, {3.5468800917622175`*^9, 
   3.54688009376222*^9}, {3.5470791398604193`*^9, 3.547079145160427*^9}, {
   3.5470792120805206`*^9, 3.5470792120805206`*^9}, {3.6012565986025095`*^9, 
   3.601256603766118*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{"A", " ", 
     SuperscriptBox["b", "2"]}], "-", 
    RowBox[{"a", " ", "b", " ", "B"}], "+", 
    RowBox[{
     SuperscriptBox["a", "2"], " ", "C"}]}], "\[Equal]", "0"}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.4964417379912624`*^9, 3.4964417471112747`*^9}, {
   3.496441910111503*^9, 3.4964419275315275`*^9}, 3.496451213764926*^9, 
   3.496528833243797*^9, {3.4975777080419827`*^9, 3.4975777244120054`*^9}, {
   3.4975778737222147`*^9, 3.4975778945722437`*^9}, {3.4976396822792068`*^9, 
   3.49763968986082*^9}, {3.497639771433363*^9, 3.49763978104298*^9}, {
   3.4976638984111347`*^9, 3.4976639153215647`*^9}, {3.4976645939383564`*^9, 
   3.497664598805565*^9}, {3.4976646480080514`*^9, 3.4976646992229414`*^9}, {
   3.4976647966763124`*^9, 3.497664832415975*^9}, 3.497664903037299*^9, {
   3.4986779242986293`*^9, 3.498677947778662*^9}, {3.499179738750054*^9, 
   3.499179740902858*^9}, {3.499213316063504*^9, 3.4992133345435305`*^9}, {
   3.499358051967739*^9, 3.4993580717017736`*^9}, {3.4993638312082148`*^9, 
   3.499363839028226*^9}, {3.499368533262825*^9, 3.499368534762827*^9}, {
   3.500319714354707*^9, 3.500319719674714*^9}, {3.500321207536797*^9, 
   3.5003212188668127`*^9}, {3.501358653573163*^9, 3.5013586559531665`*^9}, {
   3.502678432393256*^9, 3.5026784351732597`*^9}, {3.502766589293062*^9, 
   3.502766589293062*^9}, 3.5027671790052986`*^9, 3.5027674748646183`*^9, {
   3.5038569856949825`*^9, 3.503856992059794*^9}, {3.504031357905302*^9, 
   3.5040313587633038`*^9}, 3.5040317449575815`*^9, {3.504311260541732*^9, 
   3.504311260711732*^9}, {3.504750498076647*^9, 3.50475050021665*^9}, {
   3.5092408217803926`*^9, 3.509240824151596*^9}, {3.510164009908225*^9, 
   3.510164009908225*^9}, {3.5117214107451816`*^9, 3.5117214107451816`*^9}, {
   3.5120768665233393`*^9, 3.5120768665233393`*^9}, {3.5121850888428593`*^9, 
   3.5121850938972683`*^9}, 3.546877116548052*^9, {3.5468800973122253`*^9, 
   3.5468800973122253`*^9}, {3.547066202678305*^9, 3.547066202678305*^9}, {
   3.5470791133403826`*^9, 3.5470791133403826`*^9}, {3.548429141033924*^9, 
   3.548429141033924*^9}, {3.5484293653942385`*^9, 3.5484293653942385`*^9}, {
   3.5647983782102146`*^9, 3.5647983782102146`*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
   RowBox[{"(", 
    RowBox[{"A", "+", 
     RowBox[{"B", " ", 
      RowBox[{"Tan", "[", 
       RowBox[{"e", "+", 
        RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
     RowBox[{"C", " ", 
      SuperscriptBox[
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox["1", 
      SuperscriptBox["b", "2"]]}]}], " ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], 
      RowBox[{"m", "+", "1"}]], 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"b", " ", "B"}], "-", 
       RowBox[{"a", " ", "C"}], "+", 
       RowBox[{"b", " ", "C", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.496441851501421*^9, 
   3.496441872881451*^9}, {3.496528916553913*^9, 3.4965289272239285`*^9}, {
   3.497576165747217*^9, 3.4975761801372375`*^9}, {3.497577761382057*^9, 
   3.4975777938921027`*^9}, 3.49763959300025*^9, 3.497639634480723*^9, 
   3.4976398417426867`*^9, {3.49766381031778*^9, 3.4976638350282235`*^9}, {
   3.49766435104593*^9, 3.497664390373599*^9}, {3.4976647278489914`*^9, 
   3.4976647419046164`*^9}, {3.497664869980841*^9, 3.497664887437272*^9}, {
   3.498761171082196*^9, 3.498761171082196*^9}, {3.4987645684369526`*^9, 
   3.4987646442070584`*^9}, {3.49883740188522*^9, 3.4988375207310295`*^9}, {
   3.498837939529365*^9, 3.498837958467798*^9}, 3.498875518717125*^9, {
   3.4991794250335026`*^9, 3.499179452879552*^9}, {3.4991796787211485`*^9, 
   3.499179681529153*^9}, {3.4991961982482667`*^9, 3.4991962392763386`*^9}, {
   3.499196653100665*^9, 3.4991966955639396`*^9}, {3.499197135282312*^9, 
   3.499197176918785*^9}, {3.499197690444487*^9, 3.499197694048094*^9}, {
   3.499212356332161*^9, 3.4992123865622034`*^9}, {3.499212421412252*^9, 
   3.499212423032254*^9}, {3.4992126233625345`*^9, 3.499212696332637*^9}, {
   3.499357348905704*^9, 3.4993574380910606`*^9}, {3.499404301116287*^9, 
   3.49940430323629*^9}, {3.4994048006769867`*^9, 3.4994048025069895`*^9}, {
   3.5003179811489162`*^9, 3.5003179814297166`*^9}, 3.5003181853432746`*^9, {
   3.5003185563630857`*^9, 3.500318557803087*^9}, {3.5013586287231283`*^9, 
   3.5013586400931444`*^9}, {3.5027658915998363`*^9, 3.502765896981846*^9}, {
   3.5027668222170715`*^9, 3.50276686143554*^9}, {3.505344692893856*^9, 
   3.5053446929038563`*^9}, {3.5092397382492895`*^9, 
   3.5092398121778193`*^9}, {3.51016418932094*^9, 3.5101642175257893`*^9}, {
   3.515189878545912*^9, 3.515189880095914*^9}, {3.5468767933175993`*^9, 
   3.5468767933275995`*^9}, {3.5468768261776457`*^9, 3.546876840807666*^9}, {
   3.54842903831378*^9, 3.548429042463786*^9}, 3.548429073373829*^9, {
   3.564791286633173*^9, 3.5647912866531725`*^9}, 3.564798511106848*^9, {
   3.5647985657693443`*^9, 3.5647985662373447`*^9}, {3.601148547699869*^9, 
   3.601148572389903*^9}, {3.6012565761384697`*^9, 3.601256589554493*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"(", 
      RowBox[{"A_.", "+", 
       RowBox[{"B_.", "*", 
        RowBox[{"tan", "[", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], "]"}]}], "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}]}], ",", 
    "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"1", "/", 
     RowBox[{"b", "^", "2"}]}], "*", 
    RowBox[{"Int", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", "*", 
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], "*", 
       RowBox[{"Simp", "[", 
        RowBox[{
         RowBox[{
          RowBox[{"b", "*", "B"}], "-", 
          RowBox[{"a", "*", "C"}], "+", 
          RowBox[{"b", "*", "C", "*", 
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}]}]}], ",", "x"}], "]"}]}], ",", 
      "x"}], "]"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "B", ",", "C", ",", 
        "m"}], "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"EqQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}], "-", 
       RowBox[{"a", "*", "b", "*", "B"}], "+", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}]}], ",", "0"}], 
     "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.4993638009881725`*^9, {3.4993661760314975`*^9, 3.4993661794315023`*^9}, 
   3.4993699436547995`*^9, 3.499369995914872*^9, {3.5003179814609165`*^9, 
   3.5003179817417173`*^9}, 3.5003186747732515`*^9, {3.5003187057132945`*^9, 
   3.5003187115933027`*^9}, {3.5003194495843363`*^9, 3.500319453664342*^9}, {
   3.5003198135348454`*^9, 3.5003198411248837`*^9}, 3.500320756016165*^9, 
   3.50052634660607*^9, {3.501043281866661*^9, 3.5010432838407745`*^9}, {
   3.501358610383103*^9, 3.5013586217531185`*^9}, {3.5026774770619183`*^9, 
   3.5026774832519274`*^9}, {3.502677757052311*^9, 3.502677762832319*^9}, {
   3.502765679626664*^9, 3.502765694945891*^9}, {3.502766916862438*^9, 
   3.5027669304188614`*^9}, 3.5027671021907635`*^9, {3.503071895268139*^9, 
   3.5030719155981674`*^9}, {3.50351615772817*^9, 3.5035161598181725`*^9}, {
   3.503856818447089*^9, 3.503856835341919*^9}, {3.5053446929138565`*^9, 
   3.5053446929338565`*^9}, {3.5092398129578204`*^9, 
   3.5092399297864256`*^9}, {3.509240858050456*^9, 3.509240876630089*^9}, {
   3.5094733331253176`*^9, 3.509473337703579*^9}, 3.509475176411747*^9, 
   3.5094752858760085`*^9, 3.5099379677634096`*^9, 3.5099380908635817`*^9, {
   3.510162512862395*^9, 3.5101625133303957`*^9}, {3.510164194515749*^9, 
   3.5101642104433775`*^9}, 3.5101666942815046`*^9, {3.5120759144259744`*^9, 
   3.512075914615975*^9}, {3.5159797216064987`*^9, 3.515979724586104*^9}, 
   3.5168207803566356`*^9, {3.517001932668499*^9, 3.5170019351801033`*^9}, 
   3.541093952665062*^9, {3.541095139791547*^9, 3.541095141039549*^9}, 
   3.5413120604445763`*^9, {3.545492120309139*^9, 3.5454921203191385`*^9}, {
   3.546876866337702*^9, 3.546876923417782*^9}, {3.546876967297843*^9, 
   3.5468769759678555`*^9}, {3.5468791720509295`*^9, 3.546879173100931*^9}, {
   3.5468804604227333`*^9, 3.546880463732738*^9}, 3.546880992343478*^9, 
   3.5470662550883784`*^9, 3.5470790756403294`*^9, {3.5470792239905376`*^9, 
   3.54707922567054*^9}, {3.548429051213798*^9, 3.5484290582238083`*^9}, {
   3.548429162483954*^9, 3.5484291650039577`*^9}, {3.548429377794256*^9, 
   3.5484293786542573`*^9}, {3.5647912866631727`*^9, 
   3.5647913151532125`*^9}, {3.5647983874454308`*^9, 3.564798399987853*^9}, {
   3.6011527191064568`*^9, 3.601152719116457*^9}, {3.6012566944490776`*^9, 
   3.601256736241551*^9}, {3.601256892351025*^9, 3.6012569017890415`*^9}, {
   3.6012569323494954`*^9, 3.6012569591035423`*^9}, {3.6012572135116735`*^9, 
   3.601257214021674*^9}, 3.7061394499552517`*^9, {3.7154553454447002`*^9, 
   3.7154553457667184`*^9}},
 Background->GrayLevel[0.85]],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"(", 
      RowBox[{"A_.", "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}]}], ",", 
    "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"-", "C"}], "/", 
     RowBox[{"b", "^", "2"}]}], "*", 
    RowBox[{"Int", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", "*", 
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], "*", 
       RowBox[{"(", 
        RowBox[{"a", "-", 
         RowBox[{"b", "*", 
          RowBox[{"Tan", "[", 
           RowBox[{"e", "+", 
            RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}]}], ",", "x"}], 
     "]"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "C", ",", "m"}], 
       "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"EqQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}], "+", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}]}], ",", "0"}], 
     "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.4993638009881725`*^9, {3.4993661760314975`*^9, 3.4993661794315023`*^9}, 
   3.4993699436547995`*^9, 3.499369995914872*^9, {3.5003179814609165`*^9, 
   3.5003179817417173`*^9}, 3.5003186747732515`*^9, {3.5003187057132945`*^9, 
   3.5003187115933027`*^9}, {3.5003194495843363`*^9, 3.500319453664342*^9}, {
   3.5003198135348454`*^9, 3.5003198411248837`*^9}, 3.500320756016165*^9, 
   3.50052634660607*^9, {3.501043281866661*^9, 3.5010432838407745`*^9}, {
   3.501358610383103*^9, 3.5013586217531185`*^9}, {3.5026774770619183`*^9, 
   3.5026774832519274`*^9}, {3.502677757052311*^9, 3.502677762832319*^9}, {
   3.502765679626664*^9, 3.502765694945891*^9}, {3.502766916862438*^9, 
   3.5027669304188614`*^9}, 3.5027671021907635`*^9, {3.503071895268139*^9, 
   3.5030719155981674`*^9}, {3.50351615772817*^9, 3.5035161598181725`*^9}, {
   3.503856818447089*^9, 3.503856835341919*^9}, {3.5053446929138565`*^9, 
   3.5053446929338565`*^9}, {3.5092398129578204`*^9, 
   3.5092399297864256`*^9}, {3.509240858050456*^9, 3.509240876630089*^9}, {
   3.5094733331253176`*^9, 3.509473337703579*^9}, 3.509475176411747*^9, 
   3.5094752858760085`*^9, 3.5099379677634096`*^9, 3.5099380908635817`*^9, {
   3.510162512862395*^9, 3.5101625133303957`*^9}, {3.510164194515749*^9, 
   3.5101642104433775`*^9}, 3.5101666942815046`*^9, {3.5120759144259744`*^9, 
   3.512075914615975*^9}, {3.5159797216064987`*^9, 3.515979724586104*^9}, 
   3.5168207803566356`*^9, {3.517001932668499*^9, 3.5170019351801033`*^9}, 
   3.541093952665062*^9, {3.541095139791547*^9, 3.541095141039549*^9}, 
   3.5413120604445763`*^9, {3.545492120309139*^9, 3.5454921203191385`*^9}, {
   3.546876866337702*^9, 3.546876923417782*^9}, {3.546876967297843*^9, 
   3.5468769759678555`*^9}, {3.5468791720509295`*^9, 3.546879173100931*^9}, {
   3.5468804604227333`*^9, 3.546880463732738*^9}, 3.546880992343478*^9, 
   3.5470662550883784`*^9, 3.5470790756403294`*^9, {3.5470792239905376`*^9, 
   3.54707922567054*^9}, {3.548429051213798*^9, 3.5484290582238083`*^9}, {
   3.548429162483954*^9, 3.5484291650039577`*^9}, {3.548429377794256*^9, 
   3.5484293786542573`*^9}, {3.5647912866631727`*^9, 
   3.5647913151532125`*^9}, {3.5647983874454308`*^9, 3.564798399987853*^9}, {
   3.6011527191064568`*^9, 3.601152719116457*^9}, {3.6012566944490776`*^9, 
   3.601256736241551*^9}, {3.601256892351025*^9, 3.6012569017890415`*^9}, {
   3.6012569323494954`*^9, 3.6012569591035423`*^9}, {3.6012572135116735`*^9, 
   3.601257214021674*^9}, {3.601257457622024*^9, 3.6012574829370604`*^9}, {
   3.6102216625260315`*^9, 3.6102216635460324`*^9}, 3.706139449965252*^9, 
   3.7154553504329853`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["3.",
   FontFamily->"Arial"]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
    RowBox[{"(", 
     RowBox[{"A", "+", 
      RowBox[{"B", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
      RowBox[{"C", " ", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
    RowBox[{"\[DifferentialD]", "x"}], "  ", 
    StyleBox["when",
     FontFamily->"Arial",
     FontWeight->"Plain"], 
    StyleBox["  ",
     FontFamily->"Arial",
     FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
     RowBox[{
      RowBox[{
       RowBox[{"A", " ", 
        SuperscriptBox["b", "2"]}], "-", 
       RowBox[{"a", " ", "b", " ", "B"}], "+", 
       RowBox[{
        SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}]]]], 
     "None"]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.5192470594155855`*^9, {3.519247288125906*^9, 3.5192473207459517`*^9}, {
   3.519250976620308*^9, 3.5192509795375133`*^9}, 3.5192520188581386`*^9, {
   3.5193215829334736`*^9, 3.519321595725496*^9}, {3.5194112531544743`*^9, 
   3.5194112537784753`*^9}, {3.519411537636574*^9, 3.519411568711828*^9}, {
   3.519793882583559*^9, 3.519793883878361*^9}, {3.5214731238817034`*^9, 
   3.521473124371704*^9}, {3.523316505044334*^9, 3.523316505044334*^9}, {
   3.5233184226511183`*^9, 3.523318422861119*^9}, 3.5328367822219863`*^9, {
   3.5328742732655816`*^9, 3.5328742968684235`*^9}, {3.532874332701686*^9, 
   3.532874341843302*^9}, {3.532874546156861*^9, 3.532874552443672*^9}, {
   3.5328756408887835`*^9, 3.5328756426671867`*^9}, {3.5328756840852594`*^9, 
   3.532875690340871*^9}, {3.5328775995694237`*^9, 3.532877607619038*^9}, {
   3.5330804511936502`*^9, 3.5330804511936502`*^9}, {3.5330827213068285`*^9, 
   3.5330827218568287`*^9}, {3.5368634501889877`*^9, 
   3.5368634503449883`*^9}, {3.537654628221073*^9, 3.537654628221073*^9}, {
   3.5376556650325246`*^9, 3.5376556690825305`*^9}, {3.5376558319727583`*^9, 
   3.537655832122759*^9}, {3.5403190180369616`*^9, 3.540319018277975*^9}, {
   3.540320078404611*^9, 3.5403200969116697`*^9}, {3.54497301407633*^9, 
   3.5449730147315316`*^9}, {3.5449730455259857`*^9, 3.54497310485289*^9}, 
   3.5449734027198133`*^9, {3.544973705703545*^9, 3.5449737123023567`*^9}, {
   3.5449743847103376`*^9, 3.5449743847103376`*^9}, {3.5449897190542192`*^9, 
   3.5449897371142445`*^9}, 3.54501330464754*^9, 3.5450140205483975`*^9, {
   3.5450140745556927`*^9, 3.545014074867693*^9}, {3.545014174442668*^9, 
   3.5450141765018716`*^9}, 3.5450143633121996`*^9, {3.5450664467216406`*^9, 
   3.5450664468916407`*^9}, {3.5450708578278165`*^9, 3.545070858737818*^9}, {
   3.5454157832263145`*^9, 3.5454158086387587`*^9}, {3.5454159825478644`*^9, 
   3.54541600849071*^9}, {3.5454302617354054`*^9, 3.5454302618654056`*^9}, 
   3.5454968316872363`*^9, {3.545499279040663*^9, 3.545499279340663*^9}, {
   3.5454995447410345`*^9, 3.5454995452610354`*^9}, 3.54550074137271*^9, {
   3.54550107702318*^9, 3.5455010772431803`*^9}, 3.5455014895037575`*^9, 
   3.5455017531441264`*^9, {3.5455036281367517`*^9, 3.545503628336752*^9}, 
   3.545520129082264*^9, {3.5455208855433235`*^9, 3.545520885973324*^9}, {
   3.545520964293433*^9, 3.5455210208435125`*^9}, {3.545529041549029*^9, 
   3.545529065459062*^9}, {3.5455293083994026`*^9, 3.5455293521794634`*^9}, {
   3.5455294046695375`*^9, 3.5455294056895385`*^9}, {3.5455295327397165`*^9, 
   3.5455295327397165`*^9}, {3.545577377053981*^9, 3.5455774501340833`*^9}, 
   3.545577510944168*^9, 3.546115756337967*^9, 3.5468835263890266`*^9, {
   3.5468896017375326`*^9, 3.546889622177561*^9}, {3.547236869117025*^9, 
   3.5472368693270254`*^9}, {3.564791286583173*^9, 3.564791286603173*^9}, {
   3.5647987393040485`*^9, 3.564798778725318*^9}, {3.5647988986271286`*^9, 
   3.5647989025739355`*^9}, {3.601148547779869*^9, 3.6011485723999033`*^9}, {
   3.601248407459148*^9, 3.601248410829153*^9}, {3.6012625488643336`*^9, 
   3.6012625489943333`*^9}, 3.612036968794272*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["1.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
     RowBox[{"(", 
      RowBox[{"A", "+", 
       RowBox[{"B", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
       RowBox[{"C", " ", 
        SuperscriptBox[
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"A", " ", 
          SuperscriptBox["b", "2"]}], "-", 
         RowBox[{"a", " ", "b", " ", "B"}], "+", 
         RowBox[{
          SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
       "\[And]", " ", 
       RowBox[{"m", "\[LessEqual]", 
        RowBox[{"-", "1"}]}]}]]]], "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.5192470594155855`*^9, {3.519247288125906*^9, 3.5192473207459517`*^9}, {
   3.519250976620308*^9, 3.5192509795375133`*^9}, 3.5192520188581386`*^9, {
   3.5193215829334736`*^9, 3.519321595725496*^9}, {3.5194112531544743`*^9, 
   3.5194112537784753`*^9}, {3.519411537636574*^9, 3.519411568711828*^9}, {
   3.519793882583559*^9, 3.519793883878361*^9}, {3.5214731238817034`*^9, 
   3.521473124371704*^9}, {3.523316505044334*^9, 3.523316505044334*^9}, {
   3.5233184226511183`*^9, 3.523318422861119*^9}, 3.5328367822219863`*^9, {
   3.5328742732655816`*^9, 3.5328742968684235`*^9}, {3.532874332701686*^9, 
   3.532874341843302*^9}, {3.532874546156861*^9, 3.532874552443672*^9}, {
   3.5328756408887835`*^9, 3.5328756426671867`*^9}, {3.5328756840852594`*^9, 
   3.532875690340871*^9}, {3.5328775995694237`*^9, 3.532877607619038*^9}, {
   3.5330804511936502`*^9, 3.5330804511936502`*^9}, {3.5330827213068285`*^9, 
   3.5330827218568287`*^9}, {3.5368634501889877`*^9, 
   3.5368634503449883`*^9}, {3.537654628221073*^9, 3.537654628221073*^9}, {
   3.5376556650325246`*^9, 3.5376556690825305`*^9}, {3.5376558319727583`*^9, 
   3.537655832122759*^9}, {3.5403190180369616`*^9, 3.540319018277975*^9}, {
   3.540320078404611*^9, 3.5403200969116697`*^9}, {3.54497301407633*^9, 
   3.5449730147315316`*^9}, {3.5449730455259857`*^9, 3.54497310485289*^9}, 
   3.5449734027198133`*^9, {3.544973705703545*^9, 3.5449737123023567`*^9}, {
   3.5449743847103376`*^9, 3.5449743847103376`*^9}, {3.5449897190542192`*^9, 
   3.5449897371142445`*^9}, 3.54501330464754*^9, 3.5450140205483975`*^9, {
   3.5450140745556927`*^9, 3.545014074867693*^9}, {3.545014174442668*^9, 
   3.5450141765018716`*^9}, 3.5450143633121996`*^9, {3.5450664467216406`*^9, 
   3.5450664468916407`*^9}, {3.5450708578278165`*^9, 3.545070858737818*^9}, {
   3.5454157832263145`*^9, 3.5454158086387587`*^9}, {3.5454159825478644`*^9, 
   3.54541600849071*^9}, {3.5454302617354054`*^9, 3.5454302618654056`*^9}, 
   3.5454968316872363`*^9, {3.545499279040663*^9, 3.545499279340663*^9}, {
   3.5454995447410345`*^9, 3.5454995452610354`*^9}, 3.54550074137271*^9, {
   3.54550107702318*^9, 3.5455010772431803`*^9}, 3.5455014895037575`*^9, 
   3.5455017531441264`*^9, {3.5455036281367517`*^9, 3.545503628336752*^9}, 
   3.545520129082264*^9, {3.5455208855433235`*^9, 3.545520885973324*^9}, {
   3.545520964293433*^9, 3.5455210208435125`*^9}, {3.545529041549029*^9, 
   3.545529065459062*^9}, {3.5455293083994026`*^9, 3.5455293521794634`*^9}, {
   3.5455294046695375`*^9, 3.5455294056895385`*^9}, {3.5455295327397165`*^9, 
   3.5455295327397165`*^9}, {3.545577377053981*^9, 3.5455774501340833`*^9}, 
   3.545577510944168*^9, 3.546115756337967*^9, 3.5468835263890266`*^9, {
   3.5468896017375326`*^9, 3.546889622177561*^9}, {3.547236869117025*^9, 
   3.5472368693270254`*^9}, {3.548429413604306*^9, 3.5484294301543293`*^9}, {
   3.5647912866931725`*^9, 3.564791286713173*^9}, {3.5647994580909114`*^9, 
   3.5647994617101173`*^9}, {3.564799492566972*^9, 3.564799492738572*^9}, {
   3.565133898837299*^9, 3.5651339045273066`*^9}, {3.5651347245684547`*^9, 
   3.565134724798455*^9}, {3.601148547819869*^9, 3.6011485724099035`*^9}, {
   3.6012484220791683`*^9, 3.601248426109174*^9}, {3.601257654807307*^9, 
   3.601257654807307*^9}, 3.61203699602431*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
     RowBox[{"(", 
      RowBox[{"A", "+", 
       RowBox[{"B", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
       RowBox[{"C", " ", 
        SuperscriptBox[
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"A", " ", 
          SuperscriptBox["b", "2"]}], "-", 
         RowBox[{"a", " ", "b", " ", "B"}], "+", 
         RowBox[{
          SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
       "\[And]", " ", 
       RowBox[{"m", "\[LessEqual]", 
        RowBox[{"-", "1"}]}], " ", "\[And]", " ", 
       RowBox[{
        RowBox[{
         SuperscriptBox["a", "2"], "+", 
         SuperscriptBox["b", "2"]}], "\[Equal]", "0"}]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.4965117206885777`*^9, 3.5192471175756674`*^9, {3.5192587973599553`*^9, 
   3.5192588081399703`*^9}, {3.519321495105319*^9, 3.5193214991925263`*^9}, 
   3.5193247857072587`*^9, {3.5194051897010813`*^9, 3.5194051902791147`*^9}, {
   3.5194055551249824`*^9, 3.5194055708408813`*^9}, {3.519405711316916*^9, 
   3.51940572904193*^9}, {3.519410772192028*^9, 3.5194107776364374`*^9}, {
   3.519413782484516*^9, 3.519413782827717*^9}, {3.519767588664174*^9, 
   3.519767611128214*^9}, {3.5197937941314034`*^9, 3.519793794568204*^9}, {
   3.5197947626591043`*^9, 3.519794764749508*^9}, {3.519794865354085*^9, 
   3.519794874355301*^9}, {3.523218063086026*^9, 3.523218070356036*^9}, {
   3.523316514684347*^9, 3.523316514684347*^9}, {3.5233165475343933`*^9, 
   3.5233165475343933`*^9}, {3.5233184331871357`*^9, 
   3.5233184334271364`*^9}, {3.5273599103435545`*^9, 3.527359910608755*^9}, {
   3.5326424241029196`*^9, 3.53264242435292*^9}, {3.5326584126048727`*^9, 
   3.532658412794873*^9}, {3.532726454021428*^9, 3.5327264542814283`*^9}, {
   3.5327554047263184`*^9, 3.532755404993334*^9}, {3.532798666112293*^9, 
   3.532798676322307*^9}, {3.5328778343810363`*^9, 3.5328778343810363`*^9}, 
   3.5330758294771795`*^9, {3.5330827570868783`*^9, 3.533082757336879*^9}, {
   3.534975097668671*^9, 3.534975097888672*^9}, {3.536863459221404*^9, 
   3.536863459439804*^9}, {3.540264711953947*^9, 3.5402647253239655`*^9}, {
   3.540264783504047*^9, 3.540264783504047*^9}, {3.540502117527218*^9, 
   3.540502117997218*^9}, {3.540502551737826*^9, 3.540502551937826*^9}, {
   3.5449015667548094`*^9, 3.5449015920048447`*^9}, {3.544901714065016*^9, 
   3.544901727955035*^9}, 3.544911958524931*^9, {3.5449122781717815`*^9, 
   3.5449122781717815`*^9}, {3.5455187047402697`*^9, 
   3.5455187052402706`*^9}, {3.545518898070541*^9, 3.545518911130559*^9}, {
   3.5455202868524847`*^9, 3.5455202868524847`*^9}, {3.545520317832528*^9, 
   3.545520339272558*^9}, {3.545520578812894*^9, 3.545520601212925*^9}, {
   3.545520817443228*^9, 3.545520820373232*^9}, {3.545578383065389*^9, 
   3.545578398615411*^9}, {3.5461099987254543`*^9, 3.54611000814787*^9}, {
   3.546110085383606*^9, 3.546110115616459*^9}, {3.546113143676578*^9, 
   3.5461131599162064`*^9}, {3.546113861589839*^9, 3.546113902539911*^9}, {
   3.546114166008774*^9, 3.546114166008774*^9}, {3.5461157697071905`*^9, 
   3.546115770081591*^9}, {3.546881285463888*^9, 3.546881338233962*^9}, 
   3.5468817975046053`*^9, {3.5468820373549414`*^9, 3.5468820373549414`*^9}, {
   3.546889918607976*^9, 3.5468899209679794`*^9}, 3.5470663160684643`*^9, {
   3.547066678178971*^9, 3.547066688428985*^9}, {3.5647716171818285`*^9, 
   3.564771617201829*^9}, {3.5647716949019375`*^9, 3.5647716949019375`*^9}, {
   3.5651309594131837`*^9, 3.5651309594131837`*^9}, {3.565132016144663*^9, 
   3.565132026774678*^9}, {3.5651326566155596`*^9, 3.565132666125573*^9}, {
   3.565133028816081*^9, 3.565133028816081*^9}, {3.565133095376174*^9, 
   3.565133098726179*^9}, {3.5651336799369926`*^9, 3.565133681676995*^9}, {
   3.601148547849869*^9, 3.6011485724099035`*^9}, {3.60125765706231*^9, 
   3.601257668577327*^9}, 3.6120369982243133`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[TextData[{
 "Derivation: Algebraic expansion, symmetric tangent recurrence 2b with ",
 Cell[BoxData[
  RowBox[{"m", "\[Rule]", "0"}]]],
 " and symmetric tangent recurrence 2a with ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"A", "\[Rule]", "0"}], ",", 
   RowBox[{"B", "\[Rule]", "1"}], ",", 
   RowBox[{"m", "\[Rule]", "1"}]}]]]
}], "Subsubsection",
 CellChangeTimes->{
  3.4964418130913677`*^9, {3.496442428087432*^9, 3.4964424330170403`*^9}, 
   3.496528894323882*^9, 3.4975761400971813`*^9, {3.4975778216221414`*^9, 
   3.497577829892153*^9}, {3.4975827089309845`*^9, 3.4975827138509912`*^9}, 
   3.497639710530856*^9, 3.4976397495153246`*^9, 3.4976637968549566`*^9, {
   3.497664121273126*^9, 3.4976641274819374`*^9}, {3.4976641823940334`*^9, 
   3.497664184390837*^9}, 3.497664250254153*^9, 3.497664775694276*^9, 
   3.4976649459685745`*^9, {3.49876308829488*^9, 3.4987630935048876`*^9}, {
   3.4987648355173264`*^9, 3.498764841027334*^9}, {3.4991797271436334`*^9, 
   3.4991797277208347`*^9}, {3.4992121758819084`*^9, 3.499212176211909*^9}, {
   3.4993573070196304`*^9, 3.4993573078152323`*^9}, 3.5013524902049417`*^9, 
   3.501621399618896*^9, 3.5016217326739454`*^9, 3.5016376139935837`*^9, 
   3.5016377276810865`*^9, {3.5017223219869995`*^9, 3.501722324888605*^9}, {
   3.5017223746994925`*^9, 3.5017223749958925`*^9}, {3.5017230174518213`*^9, 
   3.5017230272330384`*^9}, {3.501807621678917*^9, 3.5018076224433184`*^9}, {
   3.501808030072035*^9, 3.5018080306804357`*^9}, 3.501878569854258*^9, 
   3.5033381093974123`*^9, {3.503342624699747*^9, 3.5033426251833477`*^9}, {
   3.503448463759152*^9, 3.503448463759152*^9}, {3.504647766288793*^9, 
   3.5046477668787937`*^9}, {3.504661003048687*^9, 3.5046610092586956`*^9}, {
   3.5051602282224827`*^9, 3.5051602386124973`*^9}, 3.505160363032671*^9, 
   3.5051604219327536`*^9, {3.5133572813048143`*^9, 3.513357282474816*^9}, {
   3.5133574128149986`*^9, 3.513357413114999*^9}, {3.513362650459633*^9, 
   3.5133626562096415`*^9}, {3.513363938231436*^9, 3.513363938621436*^9}, 
   3.5468770068978987`*^9, {3.549235721007359*^9, 3.549235728978815*^9}, {
   3.5492357730393353`*^9, 3.5492357860190773`*^9}, {3.563050908895475*^9, 
   3.563050913115481*^9}, {3.564771655301882*^9, 3.564771655391882*^9}, {
   3.5647719285422645`*^9, 3.5647719299922667`*^9}, {3.565129719279092*^9, 
   3.565129736159116*^9}, {3.5651326870956025`*^9, 3.5651326876956034`*^9}, {
   3.5651327314056644`*^9, 3.5651327326656666`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"A", " ", 
      SuperscriptBox["b", "2"]}], "-", 
     RowBox[{"a", " ", "b", " ", "B"}], "+", 
     RowBox[{
      SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
   "\[And]", " ", 
   RowBox[{"m", "\[LessEqual]", 
    RowBox[{"-", "1"}]}], " ", "\[And]", " ", 
   RowBox[{
    RowBox[{
     SuperscriptBox["a", "2"], "+", 
     SuperscriptBox["b", "2"]}], "\[Equal]", "0"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.4964417379912624`*^9, 3.4964417471112747`*^9}, {
   3.496441910111503*^9, 3.4964419275315275`*^9}, 3.496451213764926*^9, 
   3.496528833243797*^9, {3.4975777080419827`*^9, 3.4975777244120054`*^9}, {
   3.4975778737222147`*^9, 3.4975778945722437`*^9}, {3.4976396822792068`*^9, 
   3.49763968986082*^9}, {3.497639771433363*^9, 3.49763978104298*^9}, {
   3.4976638984111347`*^9, 3.4976639153215647`*^9}, {3.4976645939383564`*^9, 
   3.497664598805565*^9}, {3.4976646480080514`*^9, 3.4976646992229414`*^9}, {
   3.4976647966763124`*^9, 3.497664832415975*^9}, 3.497664903037299*^9, {
   3.4986779242986293`*^9, 3.498677947778662*^9}, {3.499179738750054*^9, 
   3.499179740902858*^9}, {3.499213316063504*^9, 3.4992133345435305`*^9}, {
   3.499358051967739*^9, 3.4993580717017736`*^9}, {3.4993638312082148`*^9, 
   3.499363839028226*^9}, {3.499368533262825*^9, 3.499368534762827*^9}, {
   3.500319714354707*^9, 3.500319719674714*^9}, {3.500321207536797*^9, 
   3.5003212188668127`*^9}, {3.5006579695436983`*^9, 3.5006579872737236`*^9}, 
   3.500658071043841*^9, {3.500658346374226*^9, 3.5006583478942285`*^9}, 
   3.5032824113109846`*^9, {3.5032824648346786`*^9, 3.503282468063884*^9}, {
   3.5032843334763603`*^9, 3.503284338296769*^9}, {3.503284722176643*^9, 
   3.503284722176643*^9}, {3.5032847700531273`*^9, 3.5032847726115313`*^9}, {
   3.5032859516928024`*^9, 3.5032859536896057`*^9}, {3.5032882255395966`*^9, 
   3.5032882353520136`*^9}, {3.503342656945004*^9, 3.503342658083806*^9}, {
   3.5033430130280294`*^9, 3.5033430176300373`*^9}, {3.5045501751978064`*^9, 
   3.5045501763378077`*^9}, {3.5046477778688097`*^9, 
   3.5046477849788194`*^9}, {3.504647915179002*^9, 3.5046479188090067`*^9}, {
   3.5046482353994503`*^9, 3.5046482441394625`*^9}, 3.504648571929921*^9, {
   3.5046611414588804`*^9, 3.504661141748881*^9}, 3.5046616085895343`*^9, {
   3.504665890797531*^9, 3.5046659026075473`*^9}, 3.504668082440852*^9, {
   3.5046682773029976`*^9, 3.5046682783160553`*^9}, 3.513362665319654*^9, {
   3.5137110306197805`*^9, 3.513711032522984*^9}, 3.5468771166080523`*^9, {
   3.5468801229822607`*^9, 3.5468801229822607`*^9}, {3.547066893349272*^9, 
   3.547066893349272*^9}, {3.5647716949219375`*^9, 3.5647716949219375`*^9}, {
   3.5648685616170993`*^9, 3.5648685618671*^9}, {3.565130955943179*^9, 
   3.565130955943179*^9}, {3.565132030144683*^9, 3.565132030144683*^9}, {
   3.5651331097761946`*^9, 3.5651331097761946`*^9}, {3.565133688807005*^9, 
   3.565133688807005*^9}, {3.6012616897131004`*^9, 3.6012616897131004`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
    RowBox[{"(", 
     RowBox[{"A", "+", 
      RowBox[{"B", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
      RowBox[{"C", " ", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", "\[IndentingNewLine]", 
      RowBox[{"\[Integral]", 
       RowBox[{
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", 
            RowBox[{"Tan", "[", 
             RowBox[{"e", "+", 
              RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
        RowBox[{"(", 
         RowBox[{"A", "+", 
          RowBox[{"B", " ", 
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], 
        RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}], "+", 
  RowBox[{"C", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
     SuperscriptBox[
      RowBox[{"Tan", "[", 
       RowBox[{"e", "+", 
        RowBox[{"f", " ", "x"}]}], "]"}], "2"], 
     RowBox[{"\[DifferentialD]", 
      RowBox[{"x", "   ", "\[LongRightArrow]", "  ", "\[IndentingNewLine]", 
       RowBox[{"-", 
        FractionBox[
         RowBox[{
          RowBox[{"(", 
           RowBox[{
            RowBox[{"a", " ", "A"}], "+", 
            RowBox[{"b", " ", "B"}], "-", 
            RowBox[{"a", " ", "C"}]}], ")"}], 
          RowBox[{"Tan", "[", 
           RowBox[{"e", "+", 
            RowBox[{"f", " ", "x"}]}], "]"}], 
          SuperscriptBox[
           RowBox[{"(", 
            RowBox[{"a", "+", 
             RowBox[{"b", " ", 
              RowBox[{"Tan", "[", 
               RowBox[{"e", "+", 
                RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"]}], 
         RowBox[{"2", " ", "a", " ", "f", " ", "m"}]]}]}]}]}]}]}], "+", 
  RowBox[{
   FractionBox["1", 
    RowBox[{"2", 
     SuperscriptBox["a", "2"], " ", "m"}]], 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], 
      RowBox[{"m", "+", "1"}]], 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"b", " ", "B"}], " ", "-", 
         RowBox[{"a", " ", "C"}]}], ")"}], "+", 
       RowBox[{"a", " ", "A", " ", 
        RowBox[{"(", 
         RowBox[{
          RowBox[{"2", " ", "m"}], "+", "1"}], ")"}]}], "-", 
       RowBox[{
        RowBox[{"(", 
         RowBox[{
          RowBox[{"b", " ", "C", " ", 
           RowBox[{"(", 
            RowBox[{"m", "-", "1"}], ")"}]}], "+", 
          RowBox[{
           RowBox[{"(", 
            RowBox[{
             RowBox[{"A", " ", "b"}], "-", 
             RowBox[{"a", " ", "B"}]}], ")"}], " ", 
           RowBox[{"(", 
            RowBox[{"m", "+", "1"}], ")"}]}]}], ")"}], " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.496441851501421*^9, 
   3.496441872881451*^9}, {3.496528916553913*^9, 3.4965289272239285`*^9}, {
   3.497576165747217*^9, 3.4975761801372375`*^9}, {3.497577761382057*^9, 
   3.4975777938921027`*^9}, 3.49763959300025*^9, 3.497639634480723*^9, 
   3.4976398417426867`*^9, {3.49766381031778*^9, 3.4976638350282235`*^9}, {
   3.49766435104593*^9, 3.497664390373599*^9}, {3.4976647278489914`*^9, 
   3.4976647419046164`*^9}, {3.497664869980841*^9, 3.497664887437272*^9}, {
   3.498761171082196*^9, 3.498761171082196*^9}, {3.4987645684369526`*^9, 
   3.4987646442070584`*^9}, {3.49883740188522*^9, 3.4988375207310295`*^9}, {
   3.498837939529365*^9, 3.498837958467798*^9}, 3.498875518717125*^9, {
   3.4991794250335026`*^9, 3.499179452879552*^9}, {3.4991796787211485`*^9, 
   3.499179681529153*^9}, {3.4991961982482667`*^9, 3.4991962392763386`*^9}, {
   3.499196653100665*^9, 3.4991966955639396`*^9}, {3.499197135282312*^9, 
   3.499197176918785*^9}, {3.499197690444487*^9, 3.499197694048094*^9}, {
   3.499212356332161*^9, 3.4992123865622034`*^9}, {3.499212421412252*^9, 
   3.499212423032254*^9}, {3.4992126233625345`*^9, 3.499212696332637*^9}, {
   3.499357348905704*^9, 3.4993574380910606`*^9}, {3.4993588274861555`*^9, 
   3.4993589039462624`*^9}, {3.499359110096551*^9, 3.4993591137965565`*^9}, {
   3.4993752824791145`*^9, 3.4993752993591385`*^9}, {3.4994041904561324`*^9, 
   3.4994041927361355`*^9}, {3.499404246686211*^9, 3.499404275156251*^9}, {
   3.499404742476905*^9, 3.4994047525269194`*^9}, {3.500317978262911*^9, 
   3.500317979417313*^9}, 3.500318137887991*^9, {3.5013559487293763`*^9, 
   3.501355958849391*^9}, {3.502675715869453*^9, 3.502675816299594*^9}, {
   3.5026759394697657`*^9, 3.50267594261977*^9}, {3.502676353440346*^9, 
   3.50267644207047*^9}, {3.5040325316737633`*^9, 3.504032555885006*^9}, 
   3.5040712495285273`*^9, 3.5056768305258074`*^9, {3.5091340250266056`*^9, 
   3.5091340255166063`*^9}, {3.509134110686725*^9, 3.509134210656865*^9}, 
   3.5091358140991096`*^9, {3.5091363851499095`*^9, 3.5091363926999197`*^9}, {
   3.5091366480802774`*^9, 3.509136673460313*^9}, {3.509136756420429*^9, 
   3.509136781060464*^9}, {3.509136898910629*^9, 3.509136952490704*^9}, 
   3.509137023530803*^9, {3.5091371495009794`*^9, 3.509137183211027*^9}, {
   3.5091372666811438`*^9, 3.5091373023411937`*^9}, {3.509137374621295*^9, 
   3.5091373799413023`*^9}, {3.5091374289113708`*^9, 
   3.5091374307213736`*^9}, {3.509154869729108*^9, 3.5091548849591293`*^9}, {
   3.509154923779184*^9, 3.509154943939212*^9}, 3.5091550691693873`*^9, {
   3.5091550996494303`*^9, 3.5091551134994497`*^9}, {3.509155210989586*^9, 
   3.5091552779496794`*^9}, {3.509155484369969*^9, 3.5091554935699816`*^9}, {
   3.509156447241317*^9, 3.5091565172614145`*^9}, {3.509157219352398*^9, 
   3.5091572214624004`*^9}, {3.509200648372362*^9, 3.509200660899184*^9}, {
   3.5092402115853205`*^9, 3.5092402329885583`*^9}, {3.509240301316678*^9, 
   3.509240304764284*^9}, {3.509240391219636*^9, 3.5092403955564437`*^9}, 
   3.509479727927079*^9, {3.509479774026716*^9, 3.5094797761458373`*^9}, {
   3.509479865673958*^9, 3.5094799080873833`*^9}, 3.5094801863542995`*^9, {
   3.509560902857679*^9, 3.509560949397744*^9}, {3.5095609846277933`*^9, 
   3.509561101197957*^9}, {3.510172397162881*^9, 3.5101724322785425`*^9}, {
   3.513277045993078*^9, 3.5132771366732054`*^9}, {3.5132772238133273`*^9, 
   3.5132772240033274`*^9}, {3.513277256793373*^9, 3.513277286693415*^9}, 
   3.513277413733593*^9, {3.5133608764671497`*^9, 3.5133611341975107`*^9}, 
   3.513361221247632*^9, 3.513361389647868*^9, 3.5133614242179165`*^9, {
   3.513361473277985*^9, 3.5133615092980356`*^9}, {3.513361645148226*^9, 
   3.5133616486282306`*^9}, {3.513361686898284*^9, 3.5133617743684063`*^9}, {
   3.5133618315084867`*^9, 3.513361836948494*^9}, {3.5133618886085663`*^9, 
   3.513361892068571*^9}, {3.513362042418782*^9, 3.5133620451087856`*^9}, {
   3.5133621033488674`*^9, 3.513362110098877*^9}, {3.5133621564489417`*^9, 
   3.513362233499049*^9}, {3.5133623711192417`*^9, 3.5133624100492964`*^9}, {
   3.5151891941849537`*^9, 3.5151891968349576`*^9}, 3.51597703524818*^9, {
   3.5468767935375996`*^9, 3.5468767935575995`*^9}, {3.546876826437646*^9, 
   3.5468768412976665`*^9}, 3.5468770068978987`*^9, {3.5468770732779913`*^9, 
   3.5468770732879915`*^9}, {3.5470669657893734`*^9, 
   3.5470670164794445`*^9}, {3.5492350307358775`*^9, 
   3.5492350316009274`*^9}, {3.549235551923688*^9, 3.5492355686956472`*^9}, {
   3.5492356163913755`*^9, 3.5492356254278917`*^9}, {3.549235676736827*^9, 
   3.549235680219026*^9}, {3.5492360296880145`*^9, 3.5492360303850546`*^9}, {
   3.5647716172118287`*^9, 3.564771630951848*^9}, {3.565130002049488*^9, 
   3.565130023599518*^9}, {3.5651301230696573`*^9, 3.5651302854398847`*^9}, {
   3.5651327539856963`*^9, 3.5651327862257414`*^9}, {3.601148547899869*^9, 
   3.6011485724499035`*^9}, {3.6012577669324675`*^9, 3.601257804692522*^9}, {
   3.601259939218087*^9, 3.601259946798098*^9}, {3.6012611299722967`*^9, 
   3.6012611553673334`*^9}, 3.6012611896423826`*^9, 3.601261443572747*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, {3.5003184730429688`*^9, 3.50031848112298*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"(", 
      RowBox[{"A_.", "+", 
       RowBox[{"B_.", "*", 
        RowBox[{"tan", "[", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], "]"}]}], "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}]}], ",", 
    "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"-", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "*", "A"}], "+", 
        RowBox[{"b", "*", "B"}], "-", 
        RowBox[{"a", "*", "C"}]}], ")"}]}], "*", 
     RowBox[{"Tan", "[", 
      RowBox[{"e", "+", 
       RowBox[{"f", "*", "x"}]}], "]"}], "*", 
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", 
          RowBox[{"Tan", "[", 
           RowBox[{"e", "+", 
            RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", "m"}], "/", 
      RowBox[{"(", 
       RowBox[{"2", "*", "a", "*", "f", "*", "m"}], ")"}]}]}], " ", "+", " ", 
    "\n", "  ", 
    RowBox[{
     RowBox[{"1", "/", 
      RowBox[{"(", 
       RowBox[{"2", "*", 
        RowBox[{"a", "^", "2"}], "*", "m"}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", 
            RowBox[{"Tan", "[", 
             RowBox[{"e", "+", 
              RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"m", "+", "1"}], ")"}]}], "*", 
        RowBox[{"Simp", "[", 
         RowBox[{
          RowBox[{
           RowBox[{"(", 
            RowBox[{
             RowBox[{"b", "*", "B"}], "-", 
             RowBox[{"a", "*", "C"}]}], ")"}], "+", 
           RowBox[{"a", "*", "A", "*", 
            RowBox[{"(", 
             RowBox[{
              RowBox[{"2", "*", "m"}], "+", "1"}], ")"}]}], "-", 
           RowBox[{
            RowBox[{"(", 
             RowBox[{
              RowBox[{"b", "*", "C", "*", 
               RowBox[{"(", 
                RowBox[{"m", "-", "1"}], ")"}]}], "+", 
              RowBox[{
               RowBox[{"(", 
                RowBox[{
                 RowBox[{"A", "*", "b"}], "-", 
                 RowBox[{"a", "*", "B"}]}], ")"}], "*", 
               RowBox[{"(", 
                RowBox[{"m", "+", "1"}], ")"}]}]}], ")"}], "*", 
            RowBox[{"Tan", "[", 
             RowBox[{"e", "+", 
              RowBox[{"f", "*", "x"}]}], "]"}]}]}], ",", "x"}], "]"}]}], ",", 
       "x"}], "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "B", ",", "C"}], 
       "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}], "-", 
       RowBox[{"a", "*", "b", "*", "B"}], "+", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}]}], ",", "0"}], "]"}], " ", "&&", 
    " ", 
    RowBox[{"LeQ", "[", 
     RowBox[{"m", ",", 
      RowBox[{"-", "1"}]}], "]"}], " ", "&&", " ", 
    RowBox[{"EqQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "^", "2"}], "+", 
       RowBox[{"b", "^", "2"}]}], ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, {3.4993575889901257`*^9, 
   3.4993576670838633`*^9}, {3.499365544950614*^9, 3.4993655661406436`*^9}, {
   3.4993657073008413`*^9, 3.499365707870842*^9}, 3.4993699181547637`*^9, {
   3.5003179803689146`*^9, 3.500317981117716*^9}, {3.5003185042030125`*^9, 
   3.5003185122330236`*^9}, {3.500319341644185*^9, 3.500319361824213*^9}, {
   3.5003214645171566`*^9, 3.500321470987166*^9}, {3.500346709430724*^9, 
   3.5003467096207247`*^9}, {3.5003469262310276`*^9, 3.500346943681052*^9}, {
   3.500347283661528*^9, 3.5003472916715393`*^9}, 3.5003511720069714`*^9, {
   3.5004321452599263`*^9, 3.500432146579002*^9}, {3.5006580987338796`*^9, 
   3.500658104833888*^9}, 3.5006581787039914`*^9, 3.5006582507440925`*^9, {
   3.500658340704218*^9, 3.500658342374221*^9}, {3.5032823981913614`*^9, 
   3.5032824239782066`*^9}, {3.503284295973894*^9, 3.5032842999675016`*^9}, 
   3.50328472618585*^9, {3.5032847612235117`*^9, 3.5032847653107185`*^9}, {
   3.5032859419427853`*^9, 3.5032859987736855`*^9}, 3.503288024252443*^9, {
   3.503288171017501*^9, 3.503288172421503*^9}, {3.503341611009965*^9, 
   3.503341645408025*^9}, {3.5033426638402157`*^9, 3.5033426688790245`*^9}, {
   3.5033429883019857`*^9, 3.503342995072398*^9}, {3.5034424372238855`*^9, 
   3.5034424486275053`*^9}, 3.504647847568907*^9, {3.5046479050189877`*^9, 
   3.5046479069789906`*^9}, {3.50464799969912*^9, 3.5046480037791257`*^9}, {
   3.504648218389426*^9, 3.50464822861944*^9}, {3.504650376712448*^9, 
   3.504650400642481*^9}, {3.504661447439309*^9, 3.504661450419313*^9}, {
   3.50466152664942*^9, 3.504661532049427*^9}, 3.5046616176795473`*^9, {
   3.504667601861365*^9, 3.504667610658868*^9}, 3.5133625475194893`*^9, 
   3.513362635969613*^9, {3.5133626767196703`*^9, 3.513362707549713*^9}, 
   3.51337577199282*^9, {3.5137110231317673`*^9, 3.5137110270317745`*^9}, 
   3.513800185841586*^9, {3.515977051909009*^9, 3.5159770649974327`*^9}, 
   3.516820871556763*^9, 3.51682396359402*^9, {3.541089407067484*^9, 
   3.5410894336343307`*^9}, 3.5410896907383823`*^9, {3.5455833646323633`*^9, 
   3.5455833646423635`*^9}, {3.546876866687702*^9, 3.5468769765878563`*^9}, {
   3.5468771798681407`*^9, 3.5468771798781404`*^9}, 3.546879289361094*^9, {
   3.5468793525911827`*^9, 3.546879353621184*^9}, {3.5468799097619624`*^9, 
   3.5468799184319744`*^9}, {3.5468806430229893`*^9, 3.5468806435229893`*^9}, 
   3.5468810755735946`*^9, {3.546881117243653*^9, 3.5468811178336535`*^9}, {
   3.547067154899638*^9, 3.5470671858296814`*^9}, {3.547067224629736*^9, 
   3.547067228049741*^9}, {3.5647716173118286`*^9, 3.564771655441882*^9}, 
   3.56477171099196*^9, {3.56513064641039*^9, 3.5651306497203946`*^9}, 
   3.565130970433199*^9, 3.5651320335346875`*^9, {3.565132814405781*^9, 
   3.565132831385804*^9}, {3.5651328626258483`*^9, 3.5651328984958982`*^9}, 
   3.565133131086224*^9, {3.565133164706271*^9, 3.565133183146297*^9}, 
   3.5651336945870132`*^9, {3.601152719156457*^9, 3.601152719166457*^9}, {
   3.601257219116681*^9, 3.6012572198166823`*^9}, 3.601261423122718*^9, {
   3.601261534917878*^9, 3.601261558572912*^9}, {3.706139449985252*^9, 
   3.7061394652552733`*^9}, {3.715455355623282*^9, 3.7154553732212887`*^9}},
 Background->GrayLevel[0.85]],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"(", 
      RowBox[{"A_.", "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}]}], ",", 
    "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"-", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "*", "A"}], "-", 
        RowBox[{"a", "*", "C"}]}], ")"}]}], "*", 
     RowBox[{"Tan", "[", 
      RowBox[{"e", "+", 
       RowBox[{"f", "*", "x"}]}], "]"}], "*", 
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", 
          RowBox[{"Tan", "[", 
           RowBox[{"e", "+", 
            RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", "m"}], "/", 
      RowBox[{"(", 
       RowBox[{"2", "*", "a", "*", "f", "*", "m"}], ")"}]}]}], " ", "+", " ", 
    "\n", "  ", 
    RowBox[{
     RowBox[{"1", "/", 
      RowBox[{"(", 
       RowBox[{"2", "*", 
        RowBox[{"a", "^", "2"}], "*", "m"}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", 
            RowBox[{"Tan", "[", 
             RowBox[{"e", "+", 
              RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"m", "+", "1"}], ")"}]}], "*", 
        RowBox[{"Simp", "[", 
         RowBox[{
          RowBox[{
           RowBox[{
            RowBox[{"-", "a"}], "*", "C"}], "+", 
           RowBox[{"a", "*", "A", "*", 
            RowBox[{"(", 
             RowBox[{
              RowBox[{"2", "*", "m"}], "+", "1"}], ")"}]}], "-", 
           RowBox[{
            RowBox[{"(", 
             RowBox[{
              RowBox[{"b", "*", "C", "*", 
               RowBox[{"(", 
                RowBox[{"m", "-", "1"}], ")"}]}], "+", 
              RowBox[{"A", "*", "b", "*", 
               RowBox[{"(", 
                RowBox[{"m", "+", "1"}], ")"}]}]}], ")"}], "*", 
            RowBox[{"Tan", "[", 
             RowBox[{"e", "+", 
              RowBox[{"f", "*", "x"}]}], "]"}]}]}], ",", "x"}], "]"}]}], ",", 
       "x"}], "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "C"}], "}"}],
       ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}], "+", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}]}], ",", "0"}], "]"}], " ", "&&", 
    " ", 
    RowBox[{"LeQ", "[", 
     RowBox[{"m", ",", 
      RowBox[{"-", "1"}]}], "]"}], " ", "&&", " ", 
    RowBox[{"EqQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "^", "2"}], "+", 
       RowBox[{"b", "^", "2"}]}], ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, {3.4993575889901257`*^9, 
   3.4993576670838633`*^9}, {3.499365544950614*^9, 3.4993655661406436`*^9}, {
   3.4993657073008413`*^9, 3.499365707870842*^9}, 3.4993699181547637`*^9, {
   3.5003179803689146`*^9, 3.500317981117716*^9}, {3.5003185042030125`*^9, 
   3.5003185122330236`*^9}, {3.500319341644185*^9, 3.500319361824213*^9}, {
   3.5003214645171566`*^9, 3.500321470987166*^9}, {3.500346709430724*^9, 
   3.5003467096207247`*^9}, {3.5003469262310276`*^9, 3.500346943681052*^9}, {
   3.500347283661528*^9, 3.5003472916715393`*^9}, 3.5003511720069714`*^9, {
   3.5004321452599263`*^9, 3.500432146579002*^9}, {3.5006580987338796`*^9, 
   3.500658104833888*^9}, 3.5006581787039914`*^9, 3.5006582507440925`*^9, {
   3.500658340704218*^9, 3.500658342374221*^9}, {3.5032823981913614`*^9, 
   3.5032824239782066`*^9}, {3.503284295973894*^9, 3.5032842999675016`*^9}, 
   3.50328472618585*^9, {3.5032847612235117`*^9, 3.5032847653107185`*^9}, {
   3.5032859419427853`*^9, 3.5032859987736855`*^9}, 3.503288024252443*^9, {
   3.503288171017501*^9, 3.503288172421503*^9}, {3.503341611009965*^9, 
   3.503341645408025*^9}, {3.5033426638402157`*^9, 3.5033426688790245`*^9}, {
   3.5033429883019857`*^9, 3.503342995072398*^9}, {3.5034424372238855`*^9, 
   3.5034424486275053`*^9}, 3.504647847568907*^9, {3.5046479050189877`*^9, 
   3.5046479069789906`*^9}, {3.50464799969912*^9, 3.5046480037791257`*^9}, {
   3.504648218389426*^9, 3.50464822861944*^9}, {3.504650376712448*^9, 
   3.504650400642481*^9}, {3.504661447439309*^9, 3.504661450419313*^9}, {
   3.50466152664942*^9, 3.504661532049427*^9}, 3.5046616176795473`*^9, {
   3.504667601861365*^9, 3.504667610658868*^9}, 3.5133625475194893`*^9, 
   3.513362635969613*^9, {3.5133626767196703`*^9, 3.513362707549713*^9}, 
   3.51337577199282*^9, {3.5137110231317673`*^9, 3.5137110270317745`*^9}, 
   3.513800185841586*^9, {3.515977051909009*^9, 3.5159770649974327`*^9}, 
   3.516820871556763*^9, 3.51682396359402*^9, {3.541089407067484*^9, 
   3.5410894336343307`*^9}, 3.5410896907383823`*^9, {3.5455833646323633`*^9, 
   3.5455833646423635`*^9}, {3.546876866687702*^9, 3.5468769765878563`*^9}, {
   3.5468771798681407`*^9, 3.5468771798781404`*^9}, 3.546879289361094*^9, {
   3.5468793525911827`*^9, 3.546879353621184*^9}, {3.5468799097619624`*^9, 
   3.5468799184319744`*^9}, {3.5468806430229893`*^9, 3.5468806435229893`*^9}, 
   3.5468810755735946`*^9, {3.546881117243653*^9, 3.5468811178336535`*^9}, {
   3.547067154899638*^9, 3.5470671858296814`*^9}, {3.547067224629736*^9, 
   3.547067228049741*^9}, {3.5647716173118286`*^9, 3.564771655441882*^9}, 
   3.56477171099196*^9, {3.56513064641039*^9, 3.5651306497203946`*^9}, 
   3.565130970433199*^9, 3.5651320335346875`*^9, {3.565132814405781*^9, 
   3.565132831385804*^9}, {3.5651328626258483`*^9, 3.5651328984958982`*^9}, 
   3.565133131086224*^9, {3.565133164706271*^9, 3.565133183146297*^9}, 
   3.5651336945870132`*^9, {3.601152719156457*^9, 3.601152719166457*^9}, {
   3.601257219116681*^9, 3.6012572198166823`*^9}, 3.601261423122718*^9, {
   3.601261534917878*^9, 3.601261558572912*^9}, {3.6012616449630365`*^9, 
   3.6012616758880806`*^9}, {3.706139450005252*^9, 3.706139465265273*^9}, {
   3.7154553587244596`*^9, 3.715455381742776*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["2.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
     RowBox[{"(", 
      RowBox[{"A", "+", 
       RowBox[{"B", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
       RowBox[{"C", " ", 
        SuperscriptBox[
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"A", " ", 
          SuperscriptBox["b", "2"]}], "-", 
         RowBox[{"a", " ", "b", " ", "B"}], "+", 
         RowBox[{
          SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
       "\[And]", " ", 
       RowBox[{"m", "\[LessEqual]", 
        RowBox[{"-", "1"}]}], " ", "\[And]", " ", 
       RowBox[{
        RowBox[{
         SuperscriptBox["a", "2"], "+", 
         SuperscriptBox["b", "2"]}], "\[NotEqual]", "0"}]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.5192470594155855`*^9, {3.519247288125906*^9, 3.5192473207459517`*^9}, {
   3.519250976620308*^9, 3.5192509795375133`*^9}, 3.5192520188581386`*^9, {
   3.5193215829334736`*^9, 3.519321595725496*^9}, {3.5194112531544743`*^9, 
   3.5194112537784753`*^9}, {3.519411537636574*^9, 3.519411568711828*^9}, {
   3.519793882583559*^9, 3.519793883878361*^9}, {3.5214731238817034`*^9, 
   3.521473124371704*^9}, {3.523316505044334*^9, 3.523316505044334*^9}, {
   3.5233184226511183`*^9, 3.523318422861119*^9}, 3.5328367822219863`*^9, {
   3.5328742732655816`*^9, 3.5328742968684235`*^9}, {3.532874332701686*^9, 
   3.532874341843302*^9}, {3.532874546156861*^9, 3.532874552443672*^9}, {
   3.5328756408887835`*^9, 3.5328756426671867`*^9}, {3.5328756840852594`*^9, 
   3.532875690340871*^9}, {3.5328775995694237`*^9, 3.532877607619038*^9}, {
   3.5330804511936502`*^9, 3.5330804511936502`*^9}, {3.5330827213068285`*^9, 
   3.5330827218568287`*^9}, {3.5368634501889877`*^9, 
   3.5368634503449883`*^9}, {3.537654628221073*^9, 3.537654628221073*^9}, {
   3.5376556650325246`*^9, 3.5376556690825305`*^9}, {3.5376558319727583`*^9, 
   3.537655832122759*^9}, {3.5403190180369616`*^9, 3.540319018277975*^9}, {
   3.540320078404611*^9, 3.5403200969116697`*^9}, {3.54497301407633*^9, 
   3.5449730147315316`*^9}, {3.5449730455259857`*^9, 3.54497310485289*^9}, 
   3.5449734027198133`*^9, {3.544973705703545*^9, 3.5449737123023567`*^9}, {
   3.5449743847103376`*^9, 3.5449743847103376`*^9}, {3.5449897190542192`*^9, 
   3.5449897371142445`*^9}, 3.54501330464754*^9, 3.5450140205483975`*^9, {
   3.5450140745556927`*^9, 3.545014074867693*^9}, {3.545014174442668*^9, 
   3.5450141765018716`*^9}, 3.5450143633121996`*^9, {3.5450664467216406`*^9, 
   3.5450664468916407`*^9}, {3.5450708578278165`*^9, 3.545070858737818*^9}, {
   3.5454157832263145`*^9, 3.5454158086387587`*^9}, {3.5454159825478644`*^9, 
   3.54541600849071*^9}, {3.5454302617354054`*^9, 3.5454302618654056`*^9}, 
   3.5454968316872363`*^9, {3.545499279040663*^9, 3.545499279340663*^9}, {
   3.5454995447410345`*^9, 3.5454995452610354`*^9}, 3.54550074137271*^9, {
   3.54550107702318*^9, 3.5455010772431803`*^9}, 3.5455014895037575`*^9, 
   3.5455017531441264`*^9, {3.5455036281367517`*^9, 3.545503628336752*^9}, 
   3.545520129082264*^9, {3.5455208855433235`*^9, 3.545520885973324*^9}, {
   3.545520964293433*^9, 3.5455210208435125`*^9}, {3.545529041549029*^9, 
   3.545529065459062*^9}, {3.5455293083994026`*^9, 3.5455293521794634`*^9}, {
   3.5455294046695375`*^9, 3.5455294056895385`*^9}, {3.5455295327397165`*^9, 
   3.5455295327397165`*^9}, {3.545577377053981*^9, 3.5455774501340833`*^9}, 
   3.545577510944168*^9, 3.546115756337967*^9, 3.5468835263890266`*^9, {
   3.5468896017375326`*^9, 3.546889622177561*^9}, {3.547236869117025*^9, 
   3.5472368693270254`*^9}, {3.548429413604306*^9, 3.5484294301543293`*^9}, {
   3.5647912866931725`*^9, 3.564791286713173*^9}, {3.5647994580909114`*^9, 
   3.5647994617101173`*^9}, {3.564799492566972*^9, 3.564799492738572*^9}, {
   3.565133898837299*^9, 3.5651339700073986`*^9}, {3.601148548029869*^9, 
   3.6011485724599037`*^9}, {3.6012576581223116`*^9, 3.601257675837337*^9}, 
   3.612036998954314*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["1.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"A", "+", 
       RowBox[{"B", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
       RowBox[{"C", " ", 
        SuperscriptBox[
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}]], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"A", " ", 
          SuperscriptBox["b", "2"]}], "-", 
         RowBox[{"a", " ", "b", " ", "B"}], "+", 
         RowBox[{
          SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
       "\[And]", " ", 
       RowBox[{
        RowBox[{
         SuperscriptBox["a", "2"], "+", 
         SuperscriptBox["b", "2"]}], "\[NotEqual]", "0"}]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.5192470594155855`*^9, {3.519247288125906*^9, 3.5192473207459517`*^9}, {
   3.519250976620308*^9, 3.5192509795375133`*^9}, 3.5192520188581386`*^9, {
   3.5193215829334736`*^9, 3.519321595725496*^9}, {3.5194112531544743`*^9, 
   3.5194112537784753`*^9}, {3.519411537636574*^9, 3.519411568711828*^9}, {
   3.519793882583559*^9, 3.519793883878361*^9}, {3.5214731238817034`*^9, 
   3.521473124371704*^9}, {3.523316505044334*^9, 3.523316505044334*^9}, {
   3.5233184226511183`*^9, 3.523318422861119*^9}, 3.5328367822219863`*^9, {
   3.5328742732655816`*^9, 3.5328742968684235`*^9}, {3.532874332701686*^9, 
   3.532874341843302*^9}, {3.532874546156861*^9, 3.532874552443672*^9}, {
   3.5328756408887835`*^9, 3.5328756426671867`*^9}, {3.5328756840852594`*^9, 
   3.532875690340871*^9}, {3.5328775995694237`*^9, 3.532877607619038*^9}, {
   3.5330804511936502`*^9, 3.5330804511936502`*^9}, {3.5330827213068285`*^9, 
   3.5330827218568287`*^9}, {3.5368634501889877`*^9, 
   3.5368634503449883`*^9}, {3.537654628221073*^9, 3.537654628221073*^9}, {
   3.5376556650325246`*^9, 3.5376556690825305`*^9}, {3.5376558319727583`*^9, 
   3.537655832122759*^9}, {3.5403190180369616`*^9, 3.540319018277975*^9}, {
   3.540320078404611*^9, 3.5403200969116697`*^9}, {3.54497301407633*^9, 
   3.5449730147315316`*^9}, {3.5449730455259857`*^9, 3.54497310485289*^9}, 
   3.5449734027198133`*^9, {3.544973705703545*^9, 3.5449737123023567`*^9}, {
   3.5449743847103376`*^9, 3.5449743847103376`*^9}, {3.5449897190542192`*^9, 
   3.5449897371142445`*^9}, 3.54501330464754*^9, 3.5450140205483975`*^9, {
   3.5450140745556927`*^9, 3.545014074867693*^9}, {3.545014174442668*^9, 
   3.5450141765018716`*^9}, 3.5450143633121996`*^9, {3.5450664467216406`*^9, 
   3.5450664468916407`*^9}, {3.5450708578278165`*^9, 3.545070858737818*^9}, {
   3.5454157832263145`*^9, 3.5454158086387587`*^9}, {3.5454159825478644`*^9, 
   3.54541600849071*^9}, {3.5454302617354054`*^9, 3.5454302618654056`*^9}, 
   3.5454968316872363`*^9, {3.545499279040663*^9, 3.545499279340663*^9}, {
   3.5454995447410345`*^9, 3.5454995452610354`*^9}, 3.54550074137271*^9, {
   3.54550107702318*^9, 3.5455010772431803`*^9}, 3.5455014895037575`*^9, 
   3.5455017531441264`*^9, {3.5455036281367517`*^9, 3.545503628336752*^9}, 
   3.545520129082264*^9, {3.5455208855433235`*^9, 3.545520885973324*^9}, {
   3.545520964293433*^9, 3.5455210208435125`*^9}, {3.545529041549029*^9, 
   3.545529065459062*^9}, {3.5455293083994026`*^9, 3.5455293521794634`*^9}, {
   3.5455294046695375`*^9, 3.5455294056895385`*^9}, {3.5455295327397165`*^9, 
   3.5455295327397165`*^9}, {3.545577377053981*^9, 3.5455774501340833`*^9}, 
   3.545577510944168*^9, 3.546115756337967*^9, 3.5468835263890266`*^9, {
   3.5468896017375326`*^9, 3.546889622177561*^9}, {3.547236869117025*^9, 
   3.5472368693270254`*^9}, {3.564791286583173*^9, 3.564791286603173*^9}, {
   3.5647987393040485`*^9, 3.564798778725318*^9}, {3.5647988986271286`*^9, 
   3.5647989025739355`*^9}, {3.564852323624595*^9, 3.5648523308546057`*^9}, {
   3.564852889375387*^9, 3.564852901385404*^9}, {3.5648531209557114`*^9, 
   3.564853135155731*^9}, {3.565134770378519*^9, 3.5651347726285224`*^9}, {
   3.601148548069869*^9, 3.6011485480898695`*^9}, 3.612036999554315*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"A", "+", 
       RowBox[{"B", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
       RowBox[{"C", " ", 
        SuperscriptBox[
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}]], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{
        RowBox[{
         SuperscriptBox["a", "2"], "+", 
         SuperscriptBox["b", "2"]}], "\[NotEqual]", "0"}], " ", "\[And]", " ", 
       RowBox[{
        RowBox[{
         RowBox[{"A", " ", "b"}], "-", 
         RowBox[{"a", " ", "B"}], "-", 
         RowBox[{"b", " ", "C"}]}], "\[Equal]", "0"}]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.4965117206885777`*^9, 3.5192471175756674`*^9, {3.5192587973599553`*^9, 
   3.5192588081399703`*^9}, {3.519321495105319*^9, 3.5193214991925263`*^9}, 
   3.5193247857072587`*^9, {3.5194051897010813`*^9, 3.5194051902791147`*^9}, {
   3.5194055551249824`*^9, 3.5194055708408813`*^9}, {3.519405711316916*^9, 
   3.51940572904193*^9}, {3.519410772192028*^9, 3.5194107776364374`*^9}, {
   3.519413782484516*^9, 3.519413782827717*^9}, {3.519767588664174*^9, 
   3.519767611128214*^9}, {3.5197937941314034`*^9, 3.519793794568204*^9}, {
   3.5197947626591043`*^9, 3.519794764749508*^9}, {3.519794865354085*^9, 
   3.519794874355301*^9}, {3.523218063086026*^9, 3.523218070356036*^9}, {
   3.523316514684347*^9, 3.523316514684347*^9}, {3.5233165475343933`*^9, 
   3.5233165475343933`*^9}, {3.5233184331871357`*^9, 
   3.5233184334271364`*^9}, {3.5273599103435545`*^9, 3.527359910608755*^9}, {
   3.5326424241029196`*^9, 3.53264242435292*^9}, {3.5326584126048727`*^9, 
   3.532658412794873*^9}, {3.532726454021428*^9, 3.5327264542814283`*^9}, {
   3.5327554047263184`*^9, 3.532755404993334*^9}, {3.532798666112293*^9, 
   3.532798676322307*^9}, {3.5328778343810363`*^9, 3.5328778343810363`*^9}, 
   3.5330758294771795`*^9, {3.5330827570868783`*^9, 3.533082757336879*^9}, {
   3.534975097668671*^9, 3.534975097888672*^9}, {3.536863459221404*^9, 
   3.536863459439804*^9}, {3.540264711953947*^9, 3.5402647253239655`*^9}, {
   3.540264783504047*^9, 3.540264783504047*^9}, {3.540502117527218*^9, 
   3.540502117997218*^9}, {3.540502551737826*^9, 3.540502551937826*^9}, {
   3.5449015667548094`*^9, 3.5449015920048447`*^9}, {3.544901714065016*^9, 
   3.544901727955035*^9}, 3.544911958524931*^9, {3.5449122781717815`*^9, 
   3.5449122781717815`*^9}, {3.5455187047402697`*^9, 
   3.5455187052402706`*^9}, {3.545518898070541*^9, 3.545518911130559*^9}, {
   3.5455202868524847`*^9, 3.5455202868524847`*^9}, {3.545520317832528*^9, 
   3.545520339272558*^9}, {3.545520578812894*^9, 3.545520601212925*^9}, {
   3.545520817443228*^9, 3.545520820373232*^9}, {3.545578383065389*^9, 
   3.545578398615411*^9}, {3.5461099987254543`*^9, 3.54611000814787*^9}, {
   3.546110085383606*^9, 3.546110115616459*^9}, {3.546113143676578*^9, 
   3.5461131599162064`*^9}, {3.546113861589839*^9, 3.546113902539911*^9}, {
   3.546114166008774*^9, 3.546114166008774*^9}, {3.5461157697071905`*^9, 
   3.546115770081591*^9}, {3.546881285463888*^9, 3.546881338233962*^9}, 
   3.5468817194244957`*^9, {3.546881968724845*^9, 3.546881968724845*^9}, {
   3.5468888794365215`*^9, 3.5468888796365213`*^9}, {3.546889516597413*^9, 
   3.546889516927414*^9}, {3.546889590097516*^9, 3.546889590507517*^9}, {
   3.547236595916642*^9, 3.547236597206644*^9}, {3.5472366757867546`*^9, 
   3.5472366760767546`*^9}, {3.547236867657023*^9, 3.547236867847023*^9}, {
   3.5647912863631725`*^9, 3.5647912863831725`*^9}, {3.564797402568901*^9, 
   3.564797402568901*^9}, {3.5647980041215577`*^9, 3.564798019253584*^9}, 
   3.564798816648985*^9, {3.5647990627082167`*^9, 3.564799073690636*^9}, 
   3.564799438356877*^9, {3.5648522390844765`*^9, 3.5648522572945023`*^9}, {
   3.564852335724612*^9, 3.564852371854663*^9}, {3.564852907135412*^9, 
   3.564852909345415*^9}, {3.564853177935791*^9, 3.5648531867258034`*^9}, {
   3.564853649116451*^9, 3.5648536503164525`*^9}, {3.564854210437237*^9, 
   3.564854212077239*^9}, 3.565134773868524*^9, {3.6011485481098695`*^9, 
   3.6011485481298695`*^9}, 3.612037000124316*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Algebraic expansion", "Subsubsection",
 CellChangeTimes->{
  3.4964418130913677`*^9, {3.496442428087432*^9, 3.4964424330170403`*^9}, 
   3.496528894323882*^9, 3.4975761400971813`*^9, {3.4975778216221414`*^9, 
   3.497577829892153*^9}, {3.4975827089309845`*^9, 3.4975827138509912`*^9}, 
   3.497639710530856*^9, 3.4976397495153246`*^9, 3.4976637968549566`*^9, {
   3.497664121273126*^9, 3.4976641274819374`*^9}, {3.4976641823940334`*^9, 
   3.497664184390837*^9}, 3.497664250254153*^9, 3.497664775694276*^9, 
   3.4976649459685745`*^9, {3.49876308829488*^9, 3.4987630935048876`*^9}, {
   3.4987648355173264`*^9, 3.498764841027334*^9}, {3.4991797271436334`*^9, 
   3.4991797277208347`*^9}, {3.4992121758819084`*^9, 3.499212176211909*^9}, {
   3.4993573070196304`*^9, 3.4993573078152323`*^9}, 3.5013524902049417`*^9, 
   3.501621399618896*^9, 3.5016217326739454`*^9, 3.5016376139935837`*^9, 
   3.5016377276810865`*^9, {3.5017223219869995`*^9, 3.501722324888605*^9}, {
   3.501722402935542*^9, 3.5017224033411427`*^9}, 3.5017229817901583`*^9, 
   3.5017231707844906`*^9, 3.5017234546797895`*^9, {3.5017235643791823`*^9, 
   3.501723566750386*^9}, {3.5017250640438175`*^9, 3.5017250769294395`*^9}, {
   3.501725550171871*^9, 3.501725550671072*^9}, {3.5017278201126595`*^9, 
   3.501727826898671*^9}, {3.5017278619207325`*^9, 3.5017279153196263`*^9}, {
   3.501727996595769*^9, 3.50172799715737*^9}, {3.501728047857459*^9, 
   3.501728055751073*^9}, {3.5017281686076717`*^9, 3.501728169060073*^9}, {
   3.5017287195538397`*^9, 3.50172872538825*^9}, {3.50172921540071*^9, 
   3.5017292168983135`*^9}, {3.501729246943966*^9, 3.5017292537299776`*^9}, {
   3.5017295437928877`*^9, 3.5017295745093417`*^9}, {3.5017296960647554`*^9, 
   3.501729696314356*^9}, {3.5017298761514716`*^9, 3.5017298771342735`*^9}, {
   3.501729942248788*^9, 3.501729967068431*^9}, {3.5017301202763004`*^9, 
   3.501730120603901*^9}, {3.501730162162374*^9, 3.5017301694943867`*^9}, {
   3.5017304653801064`*^9, 3.501730494037357*^9}, {3.501730604179551*^9, 
   3.5017306171587734`*^9}, {3.5034185653536963`*^9, 
   3.5034185683937006`*^9}, {3.5034226237593784`*^9, 3.5034226290093856`*^9}, 
   3.5053451044944324`*^9, {3.509482100959809*^9, 3.509482110306343*^9}, 
   3.5120745501540403`*^9, 3.5141617710379753`*^9, 3.5146489464527893`*^9, 
   3.5165648188978043`*^9, {3.516564852032263*^9, 3.516564854543867*^9}, {
   3.5468781337194757`*^9, 3.5468781337194757`*^9}, {3.5492380822994175`*^9, 
   3.5492380913959374`*^9}, {3.5630499188890886`*^9, 3.563049921919093*^9}, 
   3.564791286463173*^9, 3.564791361663278*^9, {3.5647981516822166`*^9, 
   3.5647981560502243`*^9}}],

Cell[TextData[{
 "Basis:",
 " If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{"A", " ", "b"}], "-", 
    RowBox[{"a", " ", "B"}], "-", 
    RowBox[{"b", " ", "C"}]}], "\[Equal]", "0"}]]],
 ", then",
 " ",
 Cell[BoxData[
  RowBox[{
   FractionBox[
    RowBox[{"A", "+", 
     RowBox[{"B", " ", "z"}], "+", 
     RowBox[{"C", " ", 
      SuperscriptBox["z", "2"]}]}], 
    RowBox[{"a", "+", 
     RowBox[{"b", " ", "z"}]}]], "\[Equal]", 
   RowBox[{
    FractionBox[
     RowBox[{
      RowBox[{"a", " ", "A"}], "+", 
      RowBox[{"b", " ", "B"}], "-", 
      RowBox[{"a", " ", "C"}]}], 
     RowBox[{
      SuperscriptBox["a", "2"], "+", 
      SuperscriptBox["b", "2"]}]], "+", 
    FractionBox[
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{"A", " ", 
         SuperscriptBox["b", "2"]}], "-", 
        RowBox[{"a", " ", "b", " ", "B"}], "+", 
        RowBox[{
         SuperscriptBox["a", "2"], " ", "C"}]}], ")"}], " ", 
      RowBox[{"(", 
       RowBox[{"1", "+", 
        SuperscriptBox["z", "2"]}], ")"}]}], 
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        SuperscriptBox["a", "2"], "+", 
        SuperscriptBox["b", "2"]}], ")"}], " ", 
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "z"}]}], ")"}]}]]}]}]],
  CellChangeTimes->{{3.5647982023355055`*^9, 3.5647982028347063`*^9}, {
   3.564798236281165*^9, 3.5647982594628057`*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4964418130913677`*^9, {3.496442428087432*^9, 3.4964424330170403`*^9}, 
   3.496528894323882*^9, 3.4975761400971813`*^9, {3.4975778216221414`*^9, 
   3.497577829892153*^9}, {3.4975827089309845`*^9, 3.4975827138509912`*^9}, 
   3.497639710530856*^9, 3.4976397495153246`*^9, 3.4976637968549566`*^9, {
   3.497664121273126*^9, 3.4976641274819374`*^9}, {3.4976641823940334`*^9, 
   3.497664184390837*^9}, 3.497664250254153*^9, 3.497664775694276*^9, 
   3.4976649459685745`*^9, {3.49876308829488*^9, 3.4987630935048876`*^9}, {
   3.4987648355173264`*^9, 3.498764841027334*^9}, {3.4991797271436334`*^9, 
   3.4991797277208347`*^9}, {3.4992121758819084`*^9, 3.499212176211909*^9}, {
   3.4993573070196304`*^9, 3.4993573078152323`*^9}, 3.5013524902049417`*^9, 
   3.501621399618896*^9, 3.5016217326739454`*^9, 3.5016376139935837`*^9, 
   3.5016377276810865`*^9, {3.5017223219869995`*^9, 3.501722324888605*^9}, {
   3.501722402935542*^9, 3.5017224033411427`*^9}, 3.5017229817901583`*^9, 
   3.5017231707844906`*^9, 3.5017234546797895`*^9, {3.5017235643791823`*^9, 
   3.501723566750386*^9}, {3.5017250640438175`*^9, 3.5017250769294395`*^9}, {
   3.501725550171871*^9, 3.501725550671072*^9}, {3.5017278201126595`*^9, 
   3.501727826898671*^9}, {3.5017278619207325`*^9, 3.5017279153196263`*^9}, {
   3.501727996595769*^9, 3.50172799715737*^9}, {3.501728047857459*^9, 
   3.501728055751073*^9}, {3.5017281686076717`*^9, 3.501728169060073*^9}, {
   3.5017287195538397`*^9, 3.50172872538825*^9}, {3.50172921540071*^9, 
   3.5017292168983135`*^9}, {3.501729246943966*^9, 3.5017292537299776`*^9}, {
   3.5017295437928877`*^9, 3.5017295745093417`*^9}, {3.5017296960647554`*^9, 
   3.501729696314356*^9}, {3.5017298761514716`*^9, 3.5017298771342735`*^9}, {
   3.501729942248788*^9, 3.501729967068431*^9}, {3.5017301202763004`*^9, 
   3.501730120603901*^9}, {3.501730162162374*^9, 3.5017301694943867`*^9}, {
   3.5017304653801064`*^9, 3.501730494037357*^9}, {3.501730604179551*^9, 
   3.5017306171587734`*^9}, {3.5034185653536963`*^9, 
   3.5034185683937006`*^9}, {3.5034226237593784`*^9, 3.5034226290093856`*^9}, 
   3.5053451044944324`*^9, {3.509482100959809*^9, 3.509482110306343*^9}, 
   3.5120745501540403`*^9, 3.5141617710379753`*^9, 3.5146489464527893`*^9, 
   3.5165648188978043`*^9, {3.516564852032263*^9, 3.516564854543867*^9}, {
   3.5468781337194757`*^9, 3.5468781337194757`*^9}, {3.5492380822994175`*^9, 
   3.5492380913959374`*^9}, {3.5630499188890886`*^9, 3.563049921919093*^9}, 
   3.564791286463173*^9, 3.564791361663278*^9, {3.5647981516822166`*^9, 
   3.564798163850238*^9}, 3.564798264501615*^9, {3.5648517524537954`*^9, 
   3.5648517524537954`*^9}, {3.5648523910046897`*^9, 3.564852411174718*^9}}],

Cell[TextData[{
 "Note: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{
     SuperscriptBox["a", "2"], "+", 
     SuperscriptBox["b", "2"]}], "\[NotEqual]", "0"}], " ", "\[And]", " ", 
   RowBox[{
    RowBox[{
     RowBox[{"A", " ", "b"}], "-", 
     RowBox[{"a", " ", "B"}], "-", 
     RowBox[{"b", " ", "C"}]}], "\[Equal]", "0"}]}]]],
 ", then ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{"A", " ", 
     SuperscriptBox["b", "2"]}], "-", 
    RowBox[{"a", " ", "b", " ", "B"}], "+", 
    RowBox[{
     SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}]]],
 "."
}], "Subsubsection",
 CellChangeTimes->{
  3.4964418130913677`*^9, {3.496442428087432*^9, 3.4964424330170403`*^9}, 
   3.496528894323882*^9, 3.4975761400971813`*^9, {3.4975778216221414`*^9, 
   3.497577829892153*^9}, {3.4975827089309845`*^9, 3.4975827138509912`*^9}, 
   3.497639710530856*^9, 3.4976397495153246`*^9, 3.4976637968549566`*^9, {
   3.497664121273126*^9, 3.4976641274819374`*^9}, {3.4976641823940334`*^9, 
   3.497664184390837*^9}, 3.497664250254153*^9, 3.497664775694276*^9, 
   3.4976649459685745`*^9, {3.49876308829488*^9, 3.4987630935048876`*^9}, {
   3.4987648355173264`*^9, 3.498764841027334*^9}, {3.4991797271436334`*^9, 
   3.4991797277208347`*^9}, {3.4992121758819084`*^9, 3.499212176211909*^9}, {
   3.4993573070196304`*^9, 3.4993573078152323`*^9}, 3.5013524902049417`*^9, 
   3.501621399618896*^9, 3.5016217326739454`*^9, 3.5016376139935837`*^9, 
   3.5016377276810865`*^9, {3.5017223219869995`*^9, 3.501722324888605*^9}, {
   3.501722402935542*^9, 3.5017224033411427`*^9}, 3.5017229817901583`*^9, 
   3.5017231707844906`*^9, 3.5017234546797895`*^9, {3.5017235643791823`*^9, 
   3.501723566750386*^9}, {3.5017250640438175`*^9, 3.5017250769294395`*^9}, {
   3.501725550171871*^9, 3.501725550671072*^9}, {3.5017278201126595`*^9, 
   3.501727826898671*^9}, {3.5017278619207325`*^9, 3.5017279153196263`*^9}, {
   3.501727996595769*^9, 3.50172799715737*^9}, {3.501728047857459*^9, 
   3.501728055751073*^9}, {3.5017281686076717`*^9, 3.501728169060073*^9}, {
   3.5017287195538397`*^9, 3.50172872538825*^9}, {3.50172921540071*^9, 
   3.5017292168983135`*^9}, {3.501729246943966*^9, 3.5017292537299776`*^9}, {
   3.5017295437928877`*^9, 3.5017295745093417`*^9}, {3.5017296960647554`*^9, 
   3.501729696314356*^9}, {3.5017298761514716`*^9, 3.5017298771342735`*^9}, {
   3.501729942248788*^9, 3.501729967068431*^9}, {3.5017301202763004`*^9, 
   3.501730120603901*^9}, {3.501730162162374*^9, 3.5017301694943867`*^9}, {
   3.5017304653801064`*^9, 3.501730494037357*^9}, {3.501730604179551*^9, 
   3.5017306171587734`*^9}, {3.5034185653536963`*^9, 
   3.5034185683937006`*^9}, {3.5034226237593784`*^9, 3.5034226290093856`*^9}, 
   3.5053451044944324`*^9, {3.509482100959809*^9, 3.509482110306343*^9}, 
   3.5120745501540403`*^9, 3.5141617710379753`*^9, 3.5146489464527893`*^9, 
   3.5165648188978043`*^9, {3.516564852032263*^9, 3.516564854543867*^9}, {
   3.5468781337194757`*^9, 3.5468781337194757`*^9}, {3.5492380822994175`*^9, 
   3.5492380913959374`*^9}, {3.5630499188890886`*^9, 3.563049921919093*^9}, 
   3.564791286463173*^9, 3.564791361663278*^9, {3.5647981516822166`*^9, 
   3.5647981560502243`*^9}, {3.5648537210065517`*^9, 3.5648537545865984`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{
     SuperscriptBox["a", "2"], "+", 
     SuperscriptBox["b", "2"]}], "\[NotEqual]", "0"}], " ", "\[And]", " ", 
   RowBox[{
    RowBox[{
     RowBox[{"A", " ", "b"}], "-", 
     RowBox[{"a", " ", "B"}], "-", 
     RowBox[{"b", " ", "C"}]}], "\[Equal]", "0"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.4964417379912624`*^9, 3.4964417471112747`*^9}, {
   3.496441910111503*^9, 3.4964419275315275`*^9}, 3.496451213764926*^9, 
   3.496528833243797*^9, {3.4975777080419827`*^9, 3.4975777244120054`*^9}, {
   3.4975778737222147`*^9, 3.4975778945722437`*^9}, {3.4976396822792068`*^9, 
   3.49763968986082*^9}, {3.497639771433363*^9, 3.49763978104298*^9}, {
   3.4976638984111347`*^9, 3.4976639153215647`*^9}, {3.4976645939383564`*^9, 
   3.497664598805565*^9}, {3.4976646480080514`*^9, 3.4976646992229414`*^9}, {
   3.4976647966763124`*^9, 3.497664832415975*^9}, 3.497664903037299*^9, {
   3.4986779242986293`*^9, 3.498677947778662*^9}, {3.499179738750054*^9, 
   3.499179740902858*^9}, {3.499213316063504*^9, 3.4992133345435305`*^9}, {
   3.499358051967739*^9, 3.4993580717017736`*^9}, {3.4993592718267775`*^9, 
   3.4993592794067883`*^9}, {3.499363873578274*^9, 3.4993638796182823`*^9}, {
   3.50031965968463*^9, 3.5003196639146357`*^9}, {3.500345120088499*^9, 
   3.500345127428509*^9}, {3.500410576940289*^9, 3.5004105779663477`*^9}, {
   3.5004106327664824`*^9, 3.500410643898119*^9}, {3.500429569940626*^9, 
   3.500429571675726*^9}, {3.503421192847375*^9, 3.503421192847375*^9}, {
   3.5034383255889688`*^9, 3.5034383282289724`*^9}, {3.504331822596939*^9, 
   3.5043318234969406`*^9}, {3.504988941601285*^9, 3.504988942181286*^9}, 
   3.5087861934230595`*^9, 3.509482964630208*^9, {3.509936156008872*^9, 
   3.5099361567288733`*^9}, {3.510505182442281*^9, 3.510505183128682*^9}, {
   3.512074844314452*^9, 3.5120748451244535`*^9}, 3.514614982497452*^9, {
   3.514641288021989*^9, 3.514641289753592*^9}, {3.5146414380210533`*^9, 
   3.514641438286254*^9}, {3.514652945017925*^9, 3.514652945327925*^9}, {
   3.5165642587977285`*^9, 3.5165642592877293`*^9}, 3.516564792622958*^9, {
   3.5165899054683557`*^9, 3.5165899061283565`*^9}, 3.5468776988488674`*^9, {
   3.5472366029866524`*^9, 3.5472366029866524`*^9}, {3.5630499536991377`*^9, 
   3.5630499543791385`*^9}, {3.56479740768571*^9, 3.56479740768571*^9}, {
   3.5647990973090773`*^9, 3.5647990973090773`*^9}, {3.5648523486946306`*^9, 
   3.5648523486946306`*^9}, {3.564852404874709*^9, 3.564852404874709*^9}, {
   3.564853733596569*^9, 3.564853733596569*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     RowBox[{"A", "+", 
      RowBox[{"B", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
      RowBox[{"C", " ", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}]}]], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{
        RowBox[{"(", 
         RowBox[{
          RowBox[{"a", " ", "A"}], "+", 
          RowBox[{"b", " ", "B"}], "-", 
          RowBox[{"a", " ", "C"}]}], ")"}], "x"}], 
       RowBox[{
        SuperscriptBox["a", "2"], "+", 
        SuperscriptBox["b", "2"]}]]}]}]}]}], "+", 
  RowBox[{
   FractionBox[
    RowBox[{
     RowBox[{"A", " ", 
      SuperscriptBox["b", "2"]}], "-", 
     RowBox[{"a", " ", "b", " ", "B"}], "+", 
     RowBox[{
      SuperscriptBox["a", "2"], " ", "C"}]}], 
    RowBox[{
     SuperscriptBox["a", "2"], "+", 
     SuperscriptBox["b", "2"]}]], 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"1", "+", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}], 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.496441851501421*^9, 
   3.496441872881451*^9}, {3.496528916553913*^9, 3.4965289272239285`*^9}, {
   3.497576165747217*^9, 3.4975761801372375`*^9}, {3.497577761382057*^9, 
   3.4975777938921027`*^9}, 3.49763959300025*^9, 3.497639634480723*^9, 
   3.4976398417426867`*^9, {3.49766381031778*^9, 3.4976638350282235`*^9}, {
   3.49766435104593*^9, 3.497664390373599*^9}, {3.4976647278489914`*^9, 
   3.4976647419046164`*^9}, {3.497664869980841*^9, 3.497664887437272*^9}, {
   3.498761171082196*^9, 3.498761171082196*^9}, {3.4987645684369526`*^9, 
   3.4987646442070584`*^9}, {3.49883740188522*^9, 3.4988375207310295`*^9}, {
   3.498837939529365*^9, 3.498837958467798*^9}, 3.498875518717125*^9, {
   3.4991794250335026`*^9, 3.499179452879552*^9}, {3.4991796787211485`*^9, 
   3.499179681529153*^9}, {3.4991961982482667`*^9, 3.4991962392763386`*^9}, {
   3.499196653100665*^9, 3.4991966955639396`*^9}, {3.499197135282312*^9, 
   3.499197176918785*^9}, {3.499197690444487*^9, 3.499197694048094*^9}, {
   3.499212356332161*^9, 3.4992123865622034`*^9}, {3.499212421412252*^9, 
   3.499212423032254*^9}, {3.4992126233625345`*^9, 3.499212696332637*^9}, {
   3.499357348905704*^9, 3.4993574380910606`*^9}, {3.4993588274861555`*^9, 
   3.4993589039462624`*^9}, {3.499359110096551*^9, 3.4993591137965565`*^9}, {
   3.4993752824791145`*^9, 3.4993752993591385`*^9}, {3.4994041904561324`*^9, 
   3.4994041927361355`*^9}, {3.499404246686211*^9, 3.499404275156251*^9}, {
   3.499404742476905*^9, 3.4994047525269194`*^9}, {3.500317978262911*^9, 
   3.500317979417313*^9}, 3.500318137887991*^9, {3.5030149938276834`*^9, 
   3.5030149992252927`*^9}, 3.505344641363784*^9, {3.505344693023856*^9, 
   3.5053447149738874`*^9}, 3.5053535459551945`*^9, {3.5094696268243294`*^9, 
   3.509469698805446*^9}, {3.509469791674758*^9, 3.509469802752392*^9}, {
   3.509469914234768*^9, 3.5094699164558954`*^9}, {3.509470445456152*^9, 
   3.509470483604334*^9}, {3.5094706454305897`*^9, 3.50947066641179*^9}, {
   3.5094707559529114`*^9, 3.5094707698497066`*^9}, 3.5094708527324467`*^9, {
   3.5094710367959747`*^9, 3.5094710383620644`*^9}, 3.5094711873785877`*^9, 
   3.5094801966418877`*^9, {3.5101642321454153`*^9, 3.5101642363106227`*^9}, {
   3.5101642797722993`*^9, 3.5101643150283613`*^9}, {3.5101646049236703`*^9, 
   3.5101646129576845`*^9}, {3.5151898918659306`*^9, 
   3.5151898944059343`*^9}, {3.5165648897831287`*^9, 3.516564962825657*^9}, {
   3.516565006535733*^9, 3.516565084796671*^9}, {3.5165651268107443`*^9, 
   3.5165651388851657`*^9}, 3.5165651693052187`*^9, {3.516565336786712*^9, 
   3.516565347659931*^9}, {3.546876826047645*^9, 3.5468768406276655`*^9}, {
   3.5468781393194838`*^9, 3.546878153329503*^9}, {3.547236589106633*^9, 
   3.5472365892966332`*^9}, {3.5647912864731727`*^9, 
   3.5647912997331915`*^9}, {3.564795825689332*^9, 3.5647958933778505`*^9}, {
   3.564796794107833*^9, 3.5647968031558485`*^9}, {3.5647980300644035`*^9, 
   3.5647981331337843`*^9}, {3.5648517832938385`*^9, 
   3.5648518269238997`*^9}, {3.5648523795146737`*^9, 3.564852379754674*^9}, {
   3.601148548159869*^9, 3.60114854818987*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{"A_", "+", 
       RowBox[{"B_.", "*", 
        RowBox[{"tan", "[", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], "]"}]}], "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}], "/", 
     RowBox[{"(", 
      RowBox[{"a_.", "+", 
       RowBox[{"b_.", "*", 
        RowBox[{"tan", "[", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}]}], ",", "x_Symbol"}],
    "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"a", "*", "A"}], "+", 
       RowBox[{"b", "*", "B"}], "-", 
       RowBox[{"a", "*", "C"}]}], ")"}], "*", 
     RowBox[{"x", "/", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "+", 
        RowBox[{"b", "^", "2"}]}], ")"}]}]}], " ", "+", " ", "\n", "  ", 
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{"A", "*", 
         RowBox[{"b", "^", "2"}]}], "-", 
        RowBox[{"a", "*", "b", "*", "B"}], "+", 
        RowBox[{
         RowBox[{"a", "^", "2"}], "*", "C"}]}], ")"}], "/", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "+", 
        RowBox[{"b", "^", "2"}]}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"(", 
         RowBox[{"1", "+", 
          RowBox[{
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}], "^", "2"}]}], ")"}], "/", 
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", "*", 
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}]}], ",", "x"}], 
      "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "B", ",", "C"}], 
       "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "^", "2"}], "+", 
       RowBox[{"b", "^", "2"}]}], ",", "0"}], "]"}], " ", "&&", " ", 
    RowBox[{"EqQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"A", "*", "b"}], "-", 
       RowBox[{"a", "*", "B"}], "-", 
       RowBox[{"b", "*", "C"}]}], ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.499358991716386*^9, {3.4993590256364326`*^9, 3.499359034196445*^9}, 
   3.499359158156618*^9, {3.499359230066719*^9, 3.4993592406667337`*^9}, {
   3.499363855658249*^9, 3.499363864218261*^9}, {3.4993753165091624`*^9, 
   3.499375327739178*^9}, {3.500317979448513*^9, 3.500317979729314*^9}, {
   3.500318356413175*^9, 3.5003183706092*^9}, {3.500319044263768*^9, 
   3.5003190515337787`*^9}, {3.500321448217134*^9, 3.5003214542871428`*^9}, {
   3.500432003250804*^9, 3.5004320043278656`*^9}, {3.5010431659740324`*^9, 
   3.5010431675511227`*^9}, 3.50267709922139*^9, {3.502677262251618*^9, 
   3.502677280331643*^9}, 3.5027670673403025`*^9, {3.5030150059333043`*^9, 
   3.5030150148565197`*^9}, {3.503071953798221*^9, 3.5030719623082333`*^9}, {
   3.5035497609313307`*^9, 3.5035497734613485`*^9}, 3.503876194868781*^9, {
   3.504156461196662*^9, 3.5041564626786647`*^9}, 3.505344641373784*^9, {
   3.5053446930538564`*^9, 3.505344714983887*^9}, 3.5053535482551975`*^9, {
   3.5094719943757453`*^9, 3.509471999082014*^9}, {3.5094721711418557`*^9, 
   3.5094721778162374`*^9}, {3.50947240515324*^9, 3.509472413765733*^9}, 
   3.5094733026345735`*^9, {3.509475150960292*^9, 3.5094751544394903`*^9}, 
   3.509481784211692*^9, {3.5094818179926243`*^9, 3.5094818327964706`*^9}, 
   3.5094819101238937`*^9, {3.5094828616743193`*^9, 3.5094828762901554`*^9}, 
   3.5098964710997996`*^9, {3.510504660652564*^9, 3.5105046611829653`*^9}, {
   3.5105160467861633`*^9, 3.510516049001367*^9}, 3.5118083030934734`*^9, 
   3.5144744283131104`*^9, {3.5146412598015394`*^9, 3.514641261969943*^9}, {
   3.5146414410006585`*^9, 3.5146414416090593`*^9}, 3.541093940855841*^9, 
   3.541094696875569*^9, 3.541094735735237*^9, 3.5413120519045644`*^9, 
   3.541403645528081*^9, {3.5454921201091385`*^9, 3.5454921201191387`*^9}, 
   3.5468791175908537`*^9, {3.546880397332645*^9, 3.5468803977526455`*^9}, 
   3.546880958953431*^9, 3.547236605526656*^9, 3.5472478596726274`*^9, {
   3.5472503225360756`*^9, 3.5472503235260773`*^9}, 3.547344998753607*^9, {
   3.564791286513173*^9, 3.5647913151132126`*^9}, 3.5647968853679934`*^9, 
   3.564797409729314*^9, {3.5647991227527227`*^9, 3.564799153749977*^9}, 
   3.5647992540113525`*^9, 3.564851915024023*^9, {3.5648522623645096`*^9, 
   3.5648522771545305`*^9}, {3.56485242686474*^9, 3.5648524488747706`*^9}, 
   3.564853777516631*^9, {3.601152719236457*^9, 3.601152719246457*^9}, {
   3.6012572243666887`*^9, 3.60125722496169*^9}, {3.6014927563268585`*^9, 
   3.601492758398977*^9}, 3.6014928383435497`*^9, {3.706139450035252*^9, 
   3.706139465295274*^9}, {3.7154553955955687`*^9, 3.715455399593797*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["2.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"A", "+", 
       RowBox[{"B", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
       RowBox[{"C", " ", 
        SuperscriptBox[
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}]], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"A", " ", 
          SuperscriptBox["b", "2"]}], "-", 
         RowBox[{"a", " ", "b", " ", "B"}], "+", 
         RowBox[{
          SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
       "\[And]", " ", 
       RowBox[{
        RowBox[{
         SuperscriptBox["a", "2"], "+", 
         SuperscriptBox["b", "2"]}], "\[NotEqual]", "0"}], " ", "\[And]", " ", 
       RowBox[{
        RowBox[{
         RowBox[{"A", " ", "b"}], "-", 
         RowBox[{"a", " ", "B"}], "-", 
         RowBox[{"b", " ", "C"}]}], "\[NotEqual]", "0"}]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.5192470594155855`*^9, {3.519247288125906*^9, 3.5192473207459517`*^9}, {
   3.519250976620308*^9, 3.5192509795375133`*^9}, 3.5192520188581386`*^9, {
   3.5193215829334736`*^9, 3.519321595725496*^9}, {3.5194112531544743`*^9, 
   3.5194112537784753`*^9}, {3.519411537636574*^9, 3.519411568711828*^9}, {
   3.519793882583559*^9, 3.519793883878361*^9}, {3.5214731238817034`*^9, 
   3.521473124371704*^9}, {3.523316505044334*^9, 3.523316505044334*^9}, {
   3.5233184226511183`*^9, 3.523318422861119*^9}, 3.5328367822219863`*^9, {
   3.5328742732655816`*^9, 3.5328742968684235`*^9}, {3.532874332701686*^9, 
   3.532874341843302*^9}, {3.532874546156861*^9, 3.532874552443672*^9}, {
   3.5328756408887835`*^9, 3.5328756426671867`*^9}, {3.5328756840852594`*^9, 
   3.532875690340871*^9}, {3.5328775995694237`*^9, 3.532877607619038*^9}, {
   3.5330804511936502`*^9, 3.5330804511936502`*^9}, {3.5330827213068285`*^9, 
   3.5330827218568287`*^9}, {3.5368634501889877`*^9, 
   3.5368634503449883`*^9}, {3.537654628221073*^9, 3.537654628221073*^9}, {
   3.5376556650325246`*^9, 3.5376556690825305`*^9}, {3.5376558319727583`*^9, 
   3.537655832122759*^9}, {3.5403190180369616`*^9, 3.540319018277975*^9}, {
   3.540320078404611*^9, 3.5403200969116697`*^9}, {3.54497301407633*^9, 
   3.5449730147315316`*^9}, {3.5449730455259857`*^9, 3.54497310485289*^9}, 
   3.5449734027198133`*^9, {3.544973705703545*^9, 3.5449737123023567`*^9}, {
   3.5449743847103376`*^9, 3.5449743847103376`*^9}, {3.5449897190542192`*^9, 
   3.5449897371142445`*^9}, 3.54501330464754*^9, 3.5450140205483975`*^9, {
   3.5450140745556927`*^9, 3.545014074867693*^9}, {3.545014174442668*^9, 
   3.5450141765018716`*^9}, 3.5450143633121996`*^9, {3.5450664467216406`*^9, 
   3.5450664468916407`*^9}, {3.5450708578278165`*^9, 3.545070858737818*^9}, {
   3.5454157832263145`*^9, 3.5454158086387587`*^9}, {3.5454159825478644`*^9, 
   3.54541600849071*^9}, {3.5454302617354054`*^9, 3.5454302618654056`*^9}, 
   3.5454968316872363`*^9, {3.545499279040663*^9, 3.545499279340663*^9}, {
   3.5454995447410345`*^9, 3.5454995452610354`*^9}, 3.54550074137271*^9, {
   3.54550107702318*^9, 3.5455010772431803`*^9}, 3.5455014895037575`*^9, 
   3.5455017531441264`*^9, {3.5455036281367517`*^9, 3.545503628336752*^9}, 
   3.545520129082264*^9, {3.5455208855433235`*^9, 3.545520885973324*^9}, {
   3.545520964293433*^9, 3.5455210208435125`*^9}, {3.545529041549029*^9, 
   3.545529065459062*^9}, {3.5455293083994026`*^9, 3.5455293521794634`*^9}, {
   3.5455294046695375`*^9, 3.5455294056895385`*^9}, {3.5455295327397165`*^9, 
   3.5455295327397165`*^9}, {3.545577377053981*^9, 3.5455774501340833`*^9}, 
   3.545577510944168*^9, 3.546115756337967*^9, 3.5468835263890266`*^9, {
   3.5468896017375326`*^9, 3.546889622177561*^9}, {3.547236869117025*^9, 
   3.5472368693270254`*^9}, {3.564791286583173*^9, 3.564791286603173*^9}, {
   3.5647987393040485`*^9, 3.564798778725318*^9}, {3.5647988986271286`*^9, 
   3.5647989025739355`*^9}, {3.564852323624595*^9, 3.5648523308546057`*^9}, {
   3.564852889375387*^9, 3.564852901385404*^9}, {3.5648531209557114`*^9, 
   3.564853135155731*^9}, {3.565134770378519*^9, 3.5651347726285224`*^9}, {
   3.601148548069869*^9, 3.6011485480898695`*^9}, {3.6014950720373096`*^9, 
   3.6014950863271265`*^9}, 3.6120370006843166`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"A", "+", 
       RowBox[{"B", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
       RowBox[{"C", " ", 
        SuperscriptBox[
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], 
      RowBox[{"Tan", "[", 
       RowBox[{"e", "+", 
        RowBox[{"f", " ", "x"}]}], "]"}]], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{"A", "-", "C"}], "\[NotEqual]", "0"}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.4965117206885777`*^9, 3.5192471175756674`*^9, {3.5192587973599553`*^9, 
   3.5192588081399703`*^9}, {3.519321495105319*^9, 3.5193214991925263`*^9}, 
   3.5193247857072587`*^9, {3.5194051897010813`*^9, 3.5194051902791147`*^9}, {
   3.5194055551249824`*^9, 3.5194055708408813`*^9}, {3.519405711316916*^9, 
   3.51940572904193*^9}, {3.519410772192028*^9, 3.5194107776364374`*^9}, {
   3.519413782484516*^9, 3.519413782827717*^9}, {3.519767588664174*^9, 
   3.519767611128214*^9}, {3.5197937941314034`*^9, 3.519793794568204*^9}, {
   3.5197947626591043`*^9, 3.519794764749508*^9}, {3.519794865354085*^9, 
   3.519794874355301*^9}, {3.523218063086026*^9, 3.523218070356036*^9}, {
   3.523316514684347*^9, 3.523316514684347*^9}, {3.5233165475343933`*^9, 
   3.5233165475343933`*^9}, {3.5233184331871357`*^9, 
   3.5233184334271364`*^9}, {3.5273599103435545`*^9, 3.527359910608755*^9}, {
   3.5326424241029196`*^9, 3.53264242435292*^9}, {3.5326584126048727`*^9, 
   3.532658412794873*^9}, {3.532726454021428*^9, 3.5327264542814283`*^9}, {
   3.5327554047263184`*^9, 3.532755404993334*^9}, {3.532798666112293*^9, 
   3.532798676322307*^9}, {3.5328778343810363`*^9, 3.5328778343810363`*^9}, 
   3.5330758294771795`*^9, {3.5330827570868783`*^9, 3.533082757336879*^9}, {
   3.534975097668671*^9, 3.534975097888672*^9}, {3.536863459221404*^9, 
   3.536863459439804*^9}, {3.540264711953947*^9, 3.5402647253239655`*^9}, {
   3.540264783504047*^9, 3.540264783504047*^9}, {3.540502117527218*^9, 
   3.540502117997218*^9}, {3.540502551737826*^9, 3.540502551937826*^9}, {
   3.5449015667548094`*^9, 3.5449015920048447`*^9}, {3.544901714065016*^9, 
   3.544901727955035*^9}, 3.544911958524931*^9, {3.5449122781717815`*^9, 
   3.5449122781717815`*^9}, {3.5455187047402697`*^9, 
   3.5455187052402706`*^9}, {3.545518898070541*^9, 3.545518911130559*^9}, {
   3.5455202868524847`*^9, 3.5455202868524847`*^9}, {3.545520317832528*^9, 
   3.545520339272558*^9}, {3.545520578812894*^9, 3.545520601212925*^9}, {
   3.545520817443228*^9, 3.545520820373232*^9}, {3.545578383065389*^9, 
   3.545578398615411*^9}, {3.5461099987254543`*^9, 3.54611000814787*^9}, {
   3.546110085383606*^9, 3.546110115616459*^9}, {3.546113143676578*^9, 
   3.5461131599162064`*^9}, {3.546113861589839*^9, 3.546113902539911*^9}, {
   3.546114166008774*^9, 3.546114166008774*^9}, {3.5461157697071905`*^9, 
   3.546115770081591*^9}, {3.546881285463888*^9, 3.546881338233962*^9}, 
   3.5468817194244957`*^9, {3.546881968724845*^9, 3.546881968724845*^9}, {
   3.5468888794365215`*^9, 3.5468888796365213`*^9}, {3.546889516597413*^9, 
   3.546889516927414*^9}, {3.546889590097516*^9, 3.546889590507517*^9}, {
   3.547236595916642*^9, 3.547236597206644*^9}, {3.5472366757867546`*^9, 
   3.5472366760767546`*^9}, {3.547236867657023*^9, 3.547236867847023*^9}, {
   3.5647912863631725`*^9, 3.5647912863831725`*^9}, {3.564797402568901*^9, 
   3.564797402568901*^9}, {3.5647980041215577`*^9, 3.564798019253584*^9}, 
   3.564798816648985*^9, {3.5647990627082167`*^9, 3.564799073690636*^9}, 
   3.564799438356877*^9, {3.5648522390844765`*^9, 3.5648522572945023`*^9}, {
   3.564852335724612*^9, 3.564852337674615*^9}, 3.564853272725924*^9, {
   3.5648534680461974`*^9, 3.5648534680461974`*^9}, 3.565134774708525*^9, {
   3.60114854822987*^9, 3.6011485482498693`*^9}, {3.6014950907303786`*^9, 
   3.601495128555542*^9}, {3.601495274197872*^9, 3.601495274197872*^9}, {
   3.6014965349759846`*^9, 3.601496550335863*^9}, {3.6014966518146677`*^9, 
   3.6014966518146677`*^9}, 3.6120370012543173`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Algebraic expansion", "Subsubsection",
 CellChangeTimes->{
  3.4964418130913677`*^9, {3.496442428087432*^9, 3.4964424330170403`*^9}, 
   3.496528894323882*^9, 3.4975761400971813`*^9, {3.4975778216221414`*^9, 
   3.497577829892153*^9}, {3.4975827089309845`*^9, 3.4975827138509912`*^9}, 
   3.497639710530856*^9, 3.4976397495153246`*^9, 3.4976637968549566`*^9, {
   3.497664121273126*^9, 3.4976641274819374`*^9}, {3.4976641823940334`*^9, 
   3.497664184390837*^9}, 3.497664250254153*^9, 3.497664775694276*^9, 
   3.4976649459685745`*^9, {3.49876308829488*^9, 3.4987630935048876`*^9}, {
   3.4987648355173264`*^9, 3.498764841027334*^9}, {3.4991797271436334`*^9, 
   3.4991797277208347`*^9}, {3.4992121758819084`*^9, 3.499212176211909*^9}, {
   3.4993573070196304`*^9, 3.4993573078152323`*^9}, 3.5013524902049417`*^9, 
   3.501621399618896*^9, 3.5016217326739454`*^9, 3.5016376139935837`*^9, 
   3.5016377276810865`*^9, {3.5017223219869995`*^9, 3.501722324888605*^9}, {
   3.501722402935542*^9, 3.5017224033411427`*^9}, 3.5017229817901583`*^9, 
   3.5017231707844906`*^9, 3.5017234546797895`*^9, {3.5017235643791823`*^9, 
   3.501723566750386*^9}, {3.5017250640438175`*^9, 3.5017250769294395`*^9}, {
   3.501725550171871*^9, 3.501725550671072*^9}, {3.5017278201126595`*^9, 
   3.501727826898671*^9}, {3.5017278619207325`*^9, 3.5017279153196263`*^9}, {
   3.501727996595769*^9, 3.50172799715737*^9}, {3.501728047857459*^9, 
   3.501728055751073*^9}, {3.5017281686076717`*^9, 3.501728169060073*^9}, {
   3.5017287195538397`*^9, 3.50172872538825*^9}, {3.50172921540071*^9, 
   3.5017292168983135`*^9}, {3.501729246943966*^9, 3.5017292537299776`*^9}, {
   3.5017295437928877`*^9, 3.5017295745093417`*^9}, {3.5017296960647554`*^9, 
   3.501729696314356*^9}, {3.5017298761514716`*^9, 3.5017298771342735`*^9}, {
   3.501729942248788*^9, 3.501729967068431*^9}, {3.5017301202763004`*^9, 
   3.501730120603901*^9}, {3.501730162162374*^9, 3.5017301694943867`*^9}, {
   3.5017304653801064`*^9, 3.501730494037357*^9}, {3.501730604179551*^9, 
   3.5017306171587734`*^9}, {3.5034185653536963`*^9, 
   3.5034185683937006`*^9}, {3.5034226237593784`*^9, 3.5034226290093856`*^9}, 
   3.5053451044944324`*^9, {3.509482100959809*^9, 3.509482110306343*^9}, 
   3.5120745501540403`*^9, 3.5141617710379753`*^9, 3.5146489464527893`*^9, 
   3.5165648188978043`*^9, {3.516564852032263*^9, 3.516564854543867*^9}, {
   3.5468781337194757`*^9, 3.5468781337194757`*^9}, {3.5492380822994175`*^9, 
   3.5492380913959374`*^9}, {3.5630499188890886`*^9, 3.563049921919093*^9}, 
   3.564791286463173*^9, 3.564791361663278*^9, {3.5647981516822166`*^9, 
   3.5647981560502243`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"A", "-", "C"}], "\[NotEqual]", "0"}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.4964417379912624`*^9, 3.4964417471112747`*^9}, {
   3.496441910111503*^9, 3.4964419275315275`*^9}, 3.496451213764926*^9, 
   3.496528833243797*^9, {3.4975777080419827`*^9, 3.4975777244120054`*^9}, {
   3.4975778737222147`*^9, 3.4975778945722437`*^9}, {3.4976396822792068`*^9, 
   3.49763968986082*^9}, {3.497639771433363*^9, 3.49763978104298*^9}, {
   3.4976638984111347`*^9, 3.4976639153215647`*^9}, {3.4976645939383564`*^9, 
   3.497664598805565*^9}, {3.4976646480080514`*^9, 3.4976646992229414`*^9}, {
   3.4976647966763124`*^9, 3.497664832415975*^9}, 3.497664903037299*^9, {
   3.4986779242986293`*^9, 3.498677947778662*^9}, {3.499179738750054*^9, 
   3.499179740902858*^9}, {3.499213316063504*^9, 3.4992133345435305`*^9}, {
   3.499358051967739*^9, 3.4993580717017736`*^9}, {3.4993592718267775`*^9, 
   3.4993592794067883`*^9}, {3.499363873578274*^9, 3.4993638796182823`*^9}, {
   3.50031965968463*^9, 3.5003196639146357`*^9}, {3.500345120088499*^9, 
   3.500345127428509*^9}, {3.500410576940289*^9, 3.5004105779663477`*^9}, {
   3.5004106327664824`*^9, 3.500410643898119*^9}, {3.500429569940626*^9, 
   3.500429571675726*^9}, {3.503421192847375*^9, 3.503421192847375*^9}, {
   3.5034383255889688`*^9, 3.5034383282289724`*^9}, {3.504331822596939*^9, 
   3.5043318234969406`*^9}, {3.504988941601285*^9, 3.504988942181286*^9}, 
   3.5087861934230595`*^9, 3.509482964630208*^9, {3.509936156008872*^9, 
   3.5099361567288733`*^9}, {3.510505182442281*^9, 3.510505183128682*^9}, {
   3.512074844314452*^9, 3.5120748451244535`*^9}, 3.514614982497452*^9, {
   3.514641288021989*^9, 3.514641289753592*^9}, {3.5146414380210533`*^9, 
   3.514641438286254*^9}, {3.514652945017925*^9, 3.514652945327925*^9}, {
   3.5165642587977285`*^9, 3.5165642592877293`*^9}, 3.516564792622958*^9, {
   3.5165899054683557`*^9, 3.5165899061283565`*^9}, 3.5468776988488674`*^9, {
   3.5472366029866524`*^9, 3.5472366029866524`*^9}, {3.5630499536991377`*^9, 
   3.5630499543791385`*^9}, {3.56479740768571*^9, 3.56479740768571*^9}, {
   3.5647990973090773`*^9, 3.5647990973090773`*^9}, {3.5648523486946306`*^9, 
   3.5648523486946306`*^9}, {3.5648534960262365`*^9, 
   3.5648534960262365`*^9}, {3.601495219121722*^9, 3.6014952467983055`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     RowBox[{"A", "+", 
      RowBox[{"B", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
      RowBox[{"C", " ", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], 
     RowBox[{"Tan", "[", 
      RowBox[{"e", "+", 
       RowBox[{"f", " ", "x"}]}], "]"}]], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", "B"}]}], " ", "x"}]}], "+", 
  RowBox[{"A", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox["1", 
      RowBox[{"Tan", "[", 
       RowBox[{"e", "+", 
        RowBox[{"f", " ", "x"}]}], "]"}]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}], "+", 
  RowBox[{"C", 
   RowBox[{"\[Integral]", 
    RowBox[{
     RowBox[{"Tan", "[", 
      RowBox[{"e", "+", 
       RowBox[{"f", " ", "x"}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.496441851501421*^9, 
   3.496441872881451*^9}, {3.496528916553913*^9, 3.4965289272239285`*^9}, {
   3.497576165747217*^9, 3.4975761801372375`*^9}, {3.497577761382057*^9, 
   3.4975777938921027`*^9}, 3.49763959300025*^9, 3.497639634480723*^9, 
   3.4976398417426867`*^9, {3.49766381031778*^9, 3.4976638350282235`*^9}, {
   3.49766435104593*^9, 3.497664390373599*^9}, {3.4976647278489914`*^9, 
   3.4976647419046164`*^9}, {3.497664869980841*^9, 3.497664887437272*^9}, {
   3.498761171082196*^9, 3.498761171082196*^9}, {3.4987645684369526`*^9, 
   3.4987646442070584`*^9}, {3.49883740188522*^9, 3.4988375207310295`*^9}, {
   3.498837939529365*^9, 3.498837958467798*^9}, 3.498875518717125*^9, {
   3.4991794250335026`*^9, 3.499179452879552*^9}, {3.4991796787211485`*^9, 
   3.499179681529153*^9}, {3.4991961982482667`*^9, 3.4991962392763386`*^9}, {
   3.499196653100665*^9, 3.4991966955639396`*^9}, {3.499197135282312*^9, 
   3.499197176918785*^9}, {3.499197690444487*^9, 3.499197694048094*^9}, {
   3.499212356332161*^9, 3.4992123865622034`*^9}, {3.499212421412252*^9, 
   3.499212423032254*^9}, {3.4992126233625345`*^9, 3.499212696332637*^9}, {
   3.499357348905704*^9, 3.4993574380910606`*^9}, {3.4993588274861555`*^9, 
   3.4993589039462624`*^9}, {3.499359110096551*^9, 3.4993591137965565`*^9}, {
   3.4993752824791145`*^9, 3.4993752993591385`*^9}, {3.4994041904561324`*^9, 
   3.4994041927361355`*^9}, {3.499404246686211*^9, 3.499404275156251*^9}, {
   3.499404742476905*^9, 3.4994047525269194`*^9}, {3.500317978262911*^9, 
   3.500317979417313*^9}, 3.500318137887991*^9, {3.5030149938276834`*^9, 
   3.5030149992252927`*^9}, 3.505344641363784*^9, {3.505344693023856*^9, 
   3.5053447149738874`*^9}, 3.5053535459551945`*^9, {3.5094696268243294`*^9, 
   3.509469698805446*^9}, {3.509469791674758*^9, 3.509469802752392*^9}, {
   3.509469914234768*^9, 3.5094699164558954`*^9}, {3.509470445456152*^9, 
   3.509470483604334*^9}, {3.5094706454305897`*^9, 3.50947066641179*^9}, {
   3.5094707559529114`*^9, 3.5094707698497066`*^9}, 3.5094708527324467`*^9, {
   3.5094710367959747`*^9, 3.5094710383620644`*^9}, 3.5094711873785877`*^9, 
   3.5094801966418877`*^9, {3.5101642321454153`*^9, 3.5101642363106227`*^9}, {
   3.5101642797722993`*^9, 3.5101643150283613`*^9}, {3.5101646049236703`*^9, 
   3.5101646129576845`*^9}, {3.5151898918659306`*^9, 
   3.5151898944059343`*^9}, {3.5165648897831287`*^9, 3.516564962825657*^9}, {
   3.516565006535733*^9, 3.516565084796671*^9}, {3.5165651268107443`*^9, 
   3.5165651388851657`*^9}, 3.5165651693052187`*^9, {3.516565336786712*^9, 
   3.516565347659931*^9}, {3.546876826047645*^9, 3.5468768406276655`*^9}, {
   3.5468781393194838`*^9, 3.546878153329503*^9}, {3.547236589106633*^9, 
   3.5472365892966332`*^9}, {3.5647912864731727`*^9, 
   3.5647912997331915`*^9}, {3.564795825689332*^9, 3.5647958933778505`*^9}, {
   3.564796794107833*^9, 3.5647968031558485`*^9}, {3.5647980300644035`*^9, 
   3.5647981331337843`*^9}, {3.5648517832938385`*^9, 
   3.5648518269238997`*^9}, {3.6011485482798696`*^9, 3.6011485483198695`*^9}, 
   3.6014951569191647`*^9, {3.6014953535584116`*^9, 3.6014953883564014`*^9}, {
   3.6014966684316177`*^9, 3.601496677687147*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{"A_", "+", 
       RowBox[{"B_.", "*", 
        RowBox[{"tan", "[", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], "]"}]}], "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}], "/", 
     RowBox[{"tan", "[", 
      RowBox[{"e_.", "+", 
       RowBox[{"f_.", "*", "x_"}]}], "]"}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"B", "*", "x"}], "+", 
    RowBox[{"A", "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{"1", "/", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", "*", "x"}]}], "]"}]}], ",", "x"}], "]"}]}], " ", "+", 
    " ", 
    RowBox[{"C", "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", "*", "x"}]}], "]"}], ",", "x"}], "]"}]}]}], " ", "/;", 
   "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"e", ",", "f", ",", "A", ",", "B", ",", "C"}], "}"}], ",", 
      "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{"A", ",", "C"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.499358991716386*^9, {3.4993590256364326`*^9, 3.499359034196445*^9}, 
   3.499359158156618*^9, {3.499359230066719*^9, 3.4993592406667337`*^9}, {
   3.499363855658249*^9, 3.499363864218261*^9}, {3.4993753165091624`*^9, 
   3.499375327739178*^9}, {3.500317979448513*^9, 3.500317979729314*^9}, {
   3.500318356413175*^9, 3.5003183706092*^9}, {3.500319044263768*^9, 
   3.5003190515337787`*^9}, {3.500321448217134*^9, 3.5003214542871428`*^9}, {
   3.500432003250804*^9, 3.5004320043278656`*^9}, {3.5010431659740324`*^9, 
   3.5010431675511227`*^9}, 3.50267709922139*^9, {3.502677262251618*^9, 
   3.502677280331643*^9}, 3.5027670673403025`*^9, {3.5030150059333043`*^9, 
   3.5030150148565197`*^9}, {3.503071953798221*^9, 3.5030719623082333`*^9}, {
   3.5035497609313307`*^9, 3.5035497734613485`*^9}, 3.503876194868781*^9, {
   3.504156461196662*^9, 3.5041564626786647`*^9}, 3.505344641373784*^9, {
   3.5053446930538564`*^9, 3.505344714983887*^9}, 3.5053535482551975`*^9, {
   3.5094719943757453`*^9, 3.509471999082014*^9}, {3.5094721711418557`*^9, 
   3.5094721778162374`*^9}, {3.50947240515324*^9, 3.509472413765733*^9}, 
   3.5094733026345735`*^9, {3.509475150960292*^9, 3.5094751544394903`*^9}, 
   3.509481784211692*^9, {3.5094818179926243`*^9, 3.5094818327964706`*^9}, 
   3.5094819101238937`*^9, {3.5094828616743193`*^9, 3.5094828762901554`*^9}, 
   3.5098964710997996`*^9, {3.510504660652564*^9, 3.5105046611829653`*^9}, {
   3.5105160467861633`*^9, 3.510516049001367*^9}, 3.5118083030934734`*^9, 
   3.5144744283131104`*^9, {3.5146412598015394`*^9, 3.514641261969943*^9}, {
   3.5146414410006585`*^9, 3.5146414416090593`*^9}, 3.541093940855841*^9, 
   3.541094696875569*^9, 3.541094735735237*^9, 3.5413120519045644`*^9, 
   3.541403645528081*^9, {3.5454921201091385`*^9, 3.5454921201191387`*^9}, 
   3.5468791175908537`*^9, {3.546880397332645*^9, 3.5468803977526455`*^9}, 
   3.546880958953431*^9, 3.547236605526656*^9, 3.5472478596726274`*^9, {
   3.5472503225360756`*^9, 3.5472503235260773`*^9}, 3.547344998753607*^9, {
   3.564791286513173*^9, 3.5647913151132126`*^9}, 3.5647968853679934`*^9, 
   3.564797409729314*^9, {3.5647991227527227`*^9, 3.564799153749977*^9}, 
   3.5647992540113525`*^9, 3.564851915024023*^9, {3.5648520782342515`*^9, 
   3.5648520984042797`*^9}, {3.564852146954348*^9, 3.56485216268437*^9}, {
   3.5648522889845467`*^9, 3.56485229822456*^9}, 3.564853534606291*^9, {
   3.564854502867646*^9, 3.564854507377652*^9}, 3.5648545655477343`*^9, {
   3.601152719326457*^9, 3.6011527193364573`*^9}, {3.6012572305016975`*^9, 
   3.6012572311766987`*^9}, {3.6014925811658397`*^9, 3.6014925941635833`*^9}, 
   3.601495193228241*^9, {3.601495522789091*^9, 3.60149554812254*^9}, {
   3.6014965997336884`*^9, 3.6014966069171*^9}, {3.6014967261499195`*^9, 
   3.6014967398947053`*^9}, 3.7061394653152733`*^9, 3.7154554035180216`*^9, 
   3.7154555590519176`*^9, 3.7155212839536066`*^9},
 Background->GrayLevel[0.85]],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{"A_", "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}], "/", 
     RowBox[{"tan", "[", 
      RowBox[{"e_.", "+", 
       RowBox[{"f_.", "*", "x_"}]}], "]"}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"A", "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{"1", "/", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", "*", "x"}]}], "]"}]}], ",", "x"}], "]"}]}], " ", "+", 
    " ", 
    RowBox[{"C", "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", "*", "x"}]}], "]"}], ",", "x"}], "]"}]}]}], " ", "/;", 
   "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"e", ",", "f", ",", "A", ",", "C"}], "}"}], ",", "x"}], "]"}], 
    " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{"A", ",", "C"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.499358991716386*^9, {3.4993590256364326`*^9, 3.499359034196445*^9}, 
   3.499359158156618*^9, {3.499359230066719*^9, 3.4993592406667337`*^9}, {
   3.499363855658249*^9, 3.499363864218261*^9}, {3.4993753165091624`*^9, 
   3.499375327739178*^9}, {3.500317979448513*^9, 3.500317979729314*^9}, {
   3.500318356413175*^9, 3.5003183706092*^9}, {3.500319044263768*^9, 
   3.5003190515337787`*^9}, {3.500321448217134*^9, 3.5003214542871428`*^9}, {
   3.500432003250804*^9, 3.5004320043278656`*^9}, {3.5010431659740324`*^9, 
   3.5010431675511227`*^9}, 3.50267709922139*^9, {3.502677262251618*^9, 
   3.502677280331643*^9}, 3.5027670673403025`*^9, {3.5030150059333043`*^9, 
   3.5030150148565197`*^9}, {3.503071953798221*^9, 3.5030719623082333`*^9}, {
   3.5035497609313307`*^9, 3.5035497734613485`*^9}, 3.503876194868781*^9, {
   3.504156461196662*^9, 3.5041564626786647`*^9}, 3.505344641373784*^9, {
   3.5053446930538564`*^9, 3.505344714983887*^9}, 3.5053535482551975`*^9, {
   3.5094719943757453`*^9, 3.509471999082014*^9}, {3.5094721711418557`*^9, 
   3.5094721778162374`*^9}, {3.50947240515324*^9, 3.509472413765733*^9}, 
   3.5094733026345735`*^9, {3.509475150960292*^9, 3.5094751544394903`*^9}, 
   3.509481784211692*^9, {3.5094818179926243`*^9, 3.5094818327964706`*^9}, 
   3.5094819101238937`*^9, {3.5094828616743193`*^9, 3.5094828762901554`*^9}, 
   3.5098964710997996`*^9, {3.510504660652564*^9, 3.5105046611829653`*^9}, {
   3.5105160467861633`*^9, 3.510516049001367*^9}, 3.5118083030934734`*^9, 
   3.5144744283131104`*^9, {3.5146412598015394`*^9, 3.514641261969943*^9}, {
   3.5146414410006585`*^9, 3.5146414416090593`*^9}, 3.541093940855841*^9, 
   3.541094696875569*^9, 3.541094735735237*^9, 3.5413120519045644`*^9, 
   3.541403645528081*^9, {3.5454921201091385`*^9, 3.5454921201191387`*^9}, 
   3.5468791175908537`*^9, {3.546880397332645*^9, 3.5468803977526455`*^9}, 
   3.546880958953431*^9, 3.547236605526656*^9, 3.5472478596726274`*^9, {
   3.5472503225360756`*^9, 3.5472503235260773`*^9}, 3.547344998753607*^9, {
   3.564791286513173*^9, 3.5647913151132126`*^9}, 3.5647968853679934`*^9, 
   3.564797409729314*^9, {3.5647991227527227`*^9, 3.564799153749977*^9}, 
   3.5647992540113525`*^9, 3.564851915024023*^9, {3.5648520782342515`*^9, 
   3.5648520984042797`*^9}, {3.564852146954348*^9, 3.56485216268437*^9}, {
   3.5648522889845467`*^9, 3.56485229822456*^9}, 3.564853534606291*^9, {
   3.564854502867646*^9, 3.564854507377652*^9}, 3.5648545655477343`*^9, {
   3.601152719326457*^9, 3.6011527193364573`*^9}, {3.6012572305016975`*^9, 
   3.6012572311766987`*^9}, {3.6014925811658397`*^9, 3.6014925941635833`*^9}, 
   3.601495193228241*^9, {3.601495522789091*^9, 3.60149554812254*^9}, 
   3.7061394653252735`*^9, 3.7154554050471087`*^9, 3.715455561067033*^9, 
   3.71552128559161*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"A", "+", 
       RowBox[{"B", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
       RowBox[{"C", " ", 
        SuperscriptBox[
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}]], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"A", " ", 
          SuperscriptBox["b", "2"]}], "-", 
         RowBox[{"a", " ", "b", " ", "B"}], "+", 
         RowBox[{
          SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
       "\[And]", " ", 
       RowBox[{
        RowBox[{
         SuperscriptBox["a", "2"], "+", 
         SuperscriptBox["b", "2"]}], "\[NotEqual]", "0"}], " ", "\[And]", " ", 
       RowBox[{
        RowBox[{
         RowBox[{"A", " ", "b"}], "-", 
         RowBox[{"a", " ", "B"}], "-", 
         RowBox[{"b", " ", "C"}]}], "\[NotEqual]", "0"}]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.4965117206885777`*^9, 3.5192471175756674`*^9, {3.5192587973599553`*^9, 
   3.5192588081399703`*^9}, {3.519321495105319*^9, 3.5193214991925263`*^9}, 
   3.5193247857072587`*^9, {3.5194051897010813`*^9, 3.5194051902791147`*^9}, {
   3.5194055551249824`*^9, 3.5194055708408813`*^9}, {3.519405711316916*^9, 
   3.51940572904193*^9}, {3.519410772192028*^9, 3.5194107776364374`*^9}, {
   3.519413782484516*^9, 3.519413782827717*^9}, {3.519767588664174*^9, 
   3.519767611128214*^9}, {3.5197937941314034`*^9, 3.519793794568204*^9}, {
   3.5197947626591043`*^9, 3.519794764749508*^9}, {3.519794865354085*^9, 
   3.519794874355301*^9}, {3.523218063086026*^9, 3.523218070356036*^9}, {
   3.523316514684347*^9, 3.523316514684347*^9}, {3.5233165475343933`*^9, 
   3.5233165475343933`*^9}, {3.5233184331871357`*^9, 
   3.5233184334271364`*^9}, {3.5273599103435545`*^9, 3.527359910608755*^9}, {
   3.5326424241029196`*^9, 3.53264242435292*^9}, {3.5326584126048727`*^9, 
   3.532658412794873*^9}, {3.532726454021428*^9, 3.5327264542814283`*^9}, {
   3.5327554047263184`*^9, 3.532755404993334*^9}, {3.532798666112293*^9, 
   3.532798676322307*^9}, {3.5328778343810363`*^9, 3.5328778343810363`*^9}, 
   3.5330758294771795`*^9, {3.5330827570868783`*^9, 3.533082757336879*^9}, {
   3.534975097668671*^9, 3.534975097888672*^9}, {3.536863459221404*^9, 
   3.536863459439804*^9}, {3.540264711953947*^9, 3.5402647253239655`*^9}, {
   3.540264783504047*^9, 3.540264783504047*^9}, {3.540502117527218*^9, 
   3.540502117997218*^9}, {3.540502551737826*^9, 3.540502551937826*^9}, {
   3.5449015667548094`*^9, 3.5449015920048447`*^9}, {3.544901714065016*^9, 
   3.544901727955035*^9}, 3.544911958524931*^9, {3.5449122781717815`*^9, 
   3.5449122781717815`*^9}, {3.5455187047402697`*^9, 
   3.5455187052402706`*^9}, {3.545518898070541*^9, 3.545518911130559*^9}, {
   3.5455202868524847`*^9, 3.5455202868524847`*^9}, {3.545520317832528*^9, 
   3.545520339272558*^9}, {3.545520578812894*^9, 3.545520601212925*^9}, {
   3.545520817443228*^9, 3.545520820373232*^9}, {3.545578383065389*^9, 
   3.545578398615411*^9}, {3.5461099987254543`*^9, 3.54611000814787*^9}, {
   3.546110085383606*^9, 3.546110115616459*^9}, {3.546113143676578*^9, 
   3.5461131599162064`*^9}, {3.546113861589839*^9, 3.546113902539911*^9}, {
   3.546114166008774*^9, 3.546114166008774*^9}, {3.5461157697071905`*^9, 
   3.546115770081591*^9}, {3.546881285463888*^9, 3.546881338233962*^9}, 
   3.5468817194244957`*^9, {3.546881968724845*^9, 3.546881968724845*^9}, {
   3.5468888794365215`*^9, 3.5468888796365213`*^9}, {3.546889516597413*^9, 
   3.546889516927414*^9}, {3.546889590097516*^9, 3.546889590507517*^9}, {
   3.547236595916642*^9, 3.547236597206644*^9}, {3.5472366757867546`*^9, 
   3.5472366760767546`*^9}, {3.547236867657023*^9, 3.547236867847023*^9}, {
   3.5647912863631725`*^9, 3.5647912863831725`*^9}, {3.564797402568901*^9, 
   3.564797402568901*^9}, {3.5647980041215577`*^9, 3.564798019253584*^9}, 
   3.564798816648985*^9, {3.5647990627082167`*^9, 3.564799073690636*^9}, 
   3.564799438356877*^9, {3.5648522390844765`*^9, 3.5648522572945023`*^9}, {
   3.564852335724612*^9, 3.564852337674615*^9}, 3.564853272725924*^9, {
   3.5648534680461974`*^9, 3.5648534680461974`*^9}, 3.565134774708525*^9, {
   3.60114854822987*^9, 3.6011485482498693`*^9}, 3.6014950907303786`*^9, 
   3.612037001794318*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Algebraic expansion", "Subsubsection",
 CellChangeTimes->{
  3.4964418130913677`*^9, {3.496442428087432*^9, 3.4964424330170403`*^9}, 
   3.496528894323882*^9, 3.4975761400971813`*^9, {3.4975778216221414`*^9, 
   3.497577829892153*^9}, {3.4975827089309845`*^9, 3.4975827138509912`*^9}, 
   3.497639710530856*^9, 3.4976397495153246`*^9, 3.4976637968549566`*^9, {
   3.497664121273126*^9, 3.4976641274819374`*^9}, {3.4976641823940334`*^9, 
   3.497664184390837*^9}, 3.497664250254153*^9, 3.497664775694276*^9, 
   3.4976649459685745`*^9, {3.49876308829488*^9, 3.4987630935048876`*^9}, {
   3.4987648355173264`*^9, 3.498764841027334*^9}, {3.4991797271436334`*^9, 
   3.4991797277208347`*^9}, {3.4992121758819084`*^9, 3.499212176211909*^9}, {
   3.4993573070196304`*^9, 3.4993573078152323`*^9}, 3.5013524902049417`*^9, 
   3.501621399618896*^9, 3.5016217326739454`*^9, 3.5016376139935837`*^9, 
   3.5016377276810865`*^9, {3.5017223219869995`*^9, 3.501722324888605*^9}, {
   3.501722402935542*^9, 3.5017224033411427`*^9}, 3.5017229817901583`*^9, 
   3.5017231707844906`*^9, 3.5017234546797895`*^9, {3.5017235643791823`*^9, 
   3.501723566750386*^9}, {3.5017250640438175`*^9, 3.5017250769294395`*^9}, {
   3.501725550171871*^9, 3.501725550671072*^9}, {3.5017278201126595`*^9, 
   3.501727826898671*^9}, {3.5017278619207325`*^9, 3.5017279153196263`*^9}, {
   3.501727996595769*^9, 3.50172799715737*^9}, {3.501728047857459*^9, 
   3.501728055751073*^9}, {3.5017281686076717`*^9, 3.501728169060073*^9}, {
   3.5017287195538397`*^9, 3.50172872538825*^9}, {3.50172921540071*^9, 
   3.5017292168983135`*^9}, {3.501729246943966*^9, 3.5017292537299776`*^9}, {
   3.5017295437928877`*^9, 3.5017295745093417`*^9}, {3.5017296960647554`*^9, 
   3.501729696314356*^9}, {3.5017298761514716`*^9, 3.5017298771342735`*^9}, {
   3.501729942248788*^9, 3.501729967068431*^9}, {3.5017301202763004`*^9, 
   3.501730120603901*^9}, {3.501730162162374*^9, 3.5017301694943867`*^9}, {
   3.5017304653801064`*^9, 3.501730494037357*^9}, {3.501730604179551*^9, 
   3.5017306171587734`*^9}, {3.5034185653536963`*^9, 
   3.5034185683937006`*^9}, {3.5034226237593784`*^9, 3.5034226290093856`*^9}, 
   3.5053451044944324`*^9, {3.509482100959809*^9, 3.509482110306343*^9}, 
   3.5120745501540403`*^9, 3.5141617710379753`*^9, 3.5146489464527893`*^9, 
   3.5165648188978043`*^9, {3.516564852032263*^9, 3.516564854543867*^9}, {
   3.5468781337194757`*^9, 3.5468781337194757`*^9}, {3.5492380822994175`*^9, 
   3.5492380913959374`*^9}, {3.5630499188890886`*^9, 3.563049921919093*^9}, 
   3.564791286463173*^9, 3.564791361663278*^9, {3.5647981516822166`*^9, 
   3.5647981560502243`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   FractionBox[
    RowBox[{"A", "+", 
     RowBox[{"B", " ", "z"}], "+", 
     RowBox[{"C", " ", 
      SuperscriptBox["z", "2"]}]}], 
    RowBox[{"a", "+", 
     RowBox[{"b", " ", "z"}]}]], "\[Equal]", 
   RowBox[{
    FractionBox[
     RowBox[{
      RowBox[{"a", " ", "A"}], "+", 
      RowBox[{"b", " ", "B"}], "-", 
      RowBox[{"a", " ", "C"}]}], 
     RowBox[{
      SuperscriptBox["a", "2"], "+", 
      SuperscriptBox["b", "2"]}]], "-", 
    FractionBox[
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{"A", " ", "b"}], "-", 
        RowBox[{"a", " ", "B"}], "-", 
        RowBox[{"b", " ", "C"}]}], ")"}], " ", "z"}], 
     RowBox[{
      SuperscriptBox["a", "2"], "+", 
      SuperscriptBox["b", "2"]}]], "+", 
    FractionBox[
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{"A", " ", 
         SuperscriptBox["b", "2"]}], "-", 
        RowBox[{"a", " ", "b", " ", "B"}], "+", 
        RowBox[{
         SuperscriptBox["a", "2"], " ", "C"}]}], ")"}], " ", 
      RowBox[{"(", 
       RowBox[{"1", "+", 
        SuperscriptBox["z", "2"]}], ")"}]}], 
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        SuperscriptBox["a", "2"], "+", 
        SuperscriptBox["b", "2"]}], ")"}], " ", 
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "z"}]}], ")"}]}]]}]}]],
  CellChangeTimes->{{3.5647982023355055`*^9, 3.5647982028347063`*^9}, {
   3.564798236281165*^9, 3.5647982594628057`*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4964418130913677`*^9, {3.496442428087432*^9, 3.4964424330170403`*^9}, 
   3.496528894323882*^9, 3.4975761400971813`*^9, {3.4975778216221414`*^9, 
   3.497577829892153*^9}, {3.4975827089309845`*^9, 3.4975827138509912`*^9}, 
   3.497639710530856*^9, 3.4976397495153246`*^9, 3.4976637968549566`*^9, {
   3.497664121273126*^9, 3.4976641274819374`*^9}, {3.4976641823940334`*^9, 
   3.497664184390837*^9}, 3.497664250254153*^9, 3.497664775694276*^9, 
   3.4976649459685745`*^9, {3.49876308829488*^9, 3.4987630935048876`*^9}, {
   3.4987648355173264`*^9, 3.498764841027334*^9}, {3.4991797271436334`*^9, 
   3.4991797277208347`*^9}, {3.4992121758819084`*^9, 3.499212176211909*^9}, {
   3.4993573070196304`*^9, 3.4993573078152323`*^9}, 3.5013524902049417`*^9, 
   3.501621399618896*^9, 3.5016217326739454`*^9, 3.5016376139935837`*^9, 
   3.5016377276810865`*^9, {3.5017223219869995`*^9, 3.501722324888605*^9}, {
   3.501722402935542*^9, 3.5017224033411427`*^9}, 3.5017229817901583`*^9, 
   3.5017231707844906`*^9, 3.5017234546797895`*^9, {3.5017235643791823`*^9, 
   3.501723566750386*^9}, {3.5017250640438175`*^9, 3.5017250769294395`*^9}, {
   3.501725550171871*^9, 3.501725550671072*^9}, {3.5017278201126595`*^9, 
   3.501727826898671*^9}, {3.5017278619207325`*^9, 3.5017279153196263`*^9}, {
   3.501727996595769*^9, 3.50172799715737*^9}, {3.501728047857459*^9, 
   3.501728055751073*^9}, {3.5017281686076717`*^9, 3.501728169060073*^9}, {
   3.5017287195538397`*^9, 3.50172872538825*^9}, {3.50172921540071*^9, 
   3.5017292168983135`*^9}, {3.501729246943966*^9, 3.5017292537299776`*^9}, {
   3.5017295437928877`*^9, 3.5017295745093417`*^9}, {3.5017296960647554`*^9, 
   3.501729696314356*^9}, {3.5017298761514716`*^9, 3.5017298771342735`*^9}, {
   3.501729942248788*^9, 3.501729967068431*^9}, {3.5017301202763004`*^9, 
   3.501730120603901*^9}, {3.501730162162374*^9, 3.5017301694943867`*^9}, {
   3.5017304653801064`*^9, 3.501730494037357*^9}, {3.501730604179551*^9, 
   3.5017306171587734`*^9}, {3.5034185653536963`*^9, 
   3.5034185683937006`*^9}, {3.5034226237593784`*^9, 3.5034226290093856`*^9}, 
   3.5053451044944324`*^9, {3.509482100959809*^9, 3.509482110306343*^9}, 
   3.5120745501540403`*^9, 3.5141617710379753`*^9, 3.5146489464527893`*^9, 
   3.5165648188978043`*^9, {3.516564852032263*^9, 3.516564854543867*^9}, {
   3.5468781337194757`*^9, 3.5468781337194757`*^9}, {3.5492380822994175`*^9, 
   3.5492380913959374`*^9}, {3.5630499188890886`*^9, 3.563049921919093*^9}, 
   3.564791286463173*^9, 3.564791361663278*^9, {3.5647981516822166`*^9, 
   3.564798163850238*^9}, 3.564798264501615*^9, {3.5648517524537954`*^9, 
   3.5648517524537954`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"A", " ", 
      SuperscriptBox["b", "2"]}], "-", 
     RowBox[{"a", " ", "b", " ", "B"}], "+", 
     RowBox[{
      SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
   "\[And]", " ", 
   RowBox[{
    RowBox[{
     SuperscriptBox["a", "2"], "+", 
     SuperscriptBox["b", "2"]}], "\[NotEqual]", "0"}], " ", "\[And]", " ", 
   RowBox[{
    RowBox[{
     RowBox[{"A", " ", "b"}], "-", 
     RowBox[{"a", " ", "B"}], "-", 
     RowBox[{"b", " ", "C"}]}], "\[NotEqual]", "0"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.4964417379912624`*^9, 3.4964417471112747`*^9}, {
   3.496441910111503*^9, 3.4964419275315275`*^9}, 3.496451213764926*^9, 
   3.496528833243797*^9, {3.4975777080419827`*^9, 3.4975777244120054`*^9}, {
   3.4975778737222147`*^9, 3.4975778945722437`*^9}, {3.4976396822792068`*^9, 
   3.49763968986082*^9}, {3.497639771433363*^9, 3.49763978104298*^9}, {
   3.4976638984111347`*^9, 3.4976639153215647`*^9}, {3.4976645939383564`*^9, 
   3.497664598805565*^9}, {3.4976646480080514`*^9, 3.4976646992229414`*^9}, {
   3.4976647966763124`*^9, 3.497664832415975*^9}, 3.497664903037299*^9, {
   3.4986779242986293`*^9, 3.498677947778662*^9}, {3.499179738750054*^9, 
   3.499179740902858*^9}, {3.499213316063504*^9, 3.4992133345435305`*^9}, {
   3.499358051967739*^9, 3.4993580717017736`*^9}, {3.4993592718267775`*^9, 
   3.4993592794067883`*^9}, {3.499363873578274*^9, 3.4993638796182823`*^9}, {
   3.50031965968463*^9, 3.5003196639146357`*^9}, {3.500345120088499*^9, 
   3.500345127428509*^9}, {3.500410576940289*^9, 3.5004105779663477`*^9}, {
   3.5004106327664824`*^9, 3.500410643898119*^9}, {3.500429569940626*^9, 
   3.500429571675726*^9}, {3.503421192847375*^9, 3.503421192847375*^9}, {
   3.5034383255889688`*^9, 3.5034383282289724`*^9}, {3.504331822596939*^9, 
   3.5043318234969406`*^9}, {3.504988941601285*^9, 3.504988942181286*^9}, 
   3.5087861934230595`*^9, 3.509482964630208*^9, {3.509936156008872*^9, 
   3.5099361567288733`*^9}, {3.510505182442281*^9, 3.510505183128682*^9}, {
   3.512074844314452*^9, 3.5120748451244535`*^9}, 3.514614982497452*^9, {
   3.514641288021989*^9, 3.514641289753592*^9}, {3.5146414380210533`*^9, 
   3.514641438286254*^9}, {3.514652945017925*^9, 3.514652945327925*^9}, {
   3.5165642587977285`*^9, 3.5165642592877293`*^9}, 3.516564792622958*^9, {
   3.5165899054683557`*^9, 3.5165899061283565`*^9}, 3.5468776988488674`*^9, {
   3.5472366029866524`*^9, 3.5472366029866524`*^9}, {3.5630499536991377`*^9, 
   3.5630499543791385`*^9}, {3.56479740768571*^9, 3.56479740768571*^9}, {
   3.5647990973090773`*^9, 3.5647990973090773`*^9}, {3.5648523486946306`*^9, 
   3.5648523486946306`*^9}, {3.5648534960262365`*^9, 3.5648534960262365`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     RowBox[{"A", "+", 
      RowBox[{"B", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
      RowBox[{"C", " ", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}]}]], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{
        RowBox[{"(", 
         RowBox[{
          RowBox[{"a", " ", "A"}], "+", 
          RowBox[{"b", " ", "B"}], "-", 
          RowBox[{"a", " ", "C"}]}], ")"}], "x"}], 
       RowBox[{
        SuperscriptBox["a", "2"], "+", 
        SuperscriptBox["b", "2"]}]]}]}]}]}], "-", 
  RowBox[{
   FractionBox[
    RowBox[{
     RowBox[{"A", " ", "b"}], "-", 
     RowBox[{"a", " ", "B"}], "-", 
     RowBox[{"b", " ", "C"}]}], 
    RowBox[{
     SuperscriptBox["a", "2"], "+", 
     SuperscriptBox["b", "2"]}]], 
   RowBox[{"\[Integral]", 
    RowBox[{
     RowBox[{"Tan", "[", 
      RowBox[{"e", "+", 
       RowBox[{"f", " ", "x"}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}], "+", 
  RowBox[{
   FractionBox[
    RowBox[{
     RowBox[{"A", " ", 
      SuperscriptBox["b", "2"]}], "-", 
     RowBox[{"a", " ", "b", " ", "B"}], "+", 
     RowBox[{
      SuperscriptBox["a", "2"], " ", "C"}]}], 
    RowBox[{
     SuperscriptBox["a", "2"], "+", 
     SuperscriptBox["b", "2"]}]], 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"1", "+", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}], 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.496441851501421*^9, 
   3.496441872881451*^9}, {3.496528916553913*^9, 3.4965289272239285`*^9}, {
   3.497576165747217*^9, 3.4975761801372375`*^9}, {3.497577761382057*^9, 
   3.4975777938921027`*^9}, 3.49763959300025*^9, 3.497639634480723*^9, 
   3.4976398417426867`*^9, {3.49766381031778*^9, 3.4976638350282235`*^9}, {
   3.49766435104593*^9, 3.497664390373599*^9}, {3.4976647278489914`*^9, 
   3.4976647419046164`*^9}, {3.497664869980841*^9, 3.497664887437272*^9}, {
   3.498761171082196*^9, 3.498761171082196*^9}, {3.4987645684369526`*^9, 
   3.4987646442070584`*^9}, {3.49883740188522*^9, 3.4988375207310295`*^9}, {
   3.498837939529365*^9, 3.498837958467798*^9}, 3.498875518717125*^9, {
   3.4991794250335026`*^9, 3.499179452879552*^9}, {3.4991796787211485`*^9, 
   3.499179681529153*^9}, {3.4991961982482667`*^9, 3.4991962392763386`*^9}, {
   3.499196653100665*^9, 3.4991966955639396`*^9}, {3.499197135282312*^9, 
   3.499197176918785*^9}, {3.499197690444487*^9, 3.499197694048094*^9}, {
   3.499212356332161*^9, 3.4992123865622034`*^9}, {3.499212421412252*^9, 
   3.499212423032254*^9}, {3.4992126233625345`*^9, 3.499212696332637*^9}, {
   3.499357348905704*^9, 3.4993574380910606`*^9}, {3.4993588274861555`*^9, 
   3.4993589039462624`*^9}, {3.499359110096551*^9, 3.4993591137965565`*^9}, {
   3.4993752824791145`*^9, 3.4993752993591385`*^9}, {3.4994041904561324`*^9, 
   3.4994041927361355`*^9}, {3.499404246686211*^9, 3.499404275156251*^9}, {
   3.499404742476905*^9, 3.4994047525269194`*^9}, {3.500317978262911*^9, 
   3.500317979417313*^9}, 3.500318137887991*^9, {3.5030149938276834`*^9, 
   3.5030149992252927`*^9}, 3.505344641363784*^9, {3.505344693023856*^9, 
   3.5053447149738874`*^9}, 3.5053535459551945`*^9, {3.5094696268243294`*^9, 
   3.509469698805446*^9}, {3.509469791674758*^9, 3.509469802752392*^9}, {
   3.509469914234768*^9, 3.5094699164558954`*^9}, {3.509470445456152*^9, 
   3.509470483604334*^9}, {3.5094706454305897`*^9, 3.50947066641179*^9}, {
   3.5094707559529114`*^9, 3.5094707698497066`*^9}, 3.5094708527324467`*^9, {
   3.5094710367959747`*^9, 3.5094710383620644`*^9}, 3.5094711873785877`*^9, 
   3.5094801966418877`*^9, {3.5101642321454153`*^9, 3.5101642363106227`*^9}, {
   3.5101642797722993`*^9, 3.5101643150283613`*^9}, {3.5101646049236703`*^9, 
   3.5101646129576845`*^9}, {3.5151898918659306`*^9, 
   3.5151898944059343`*^9}, {3.5165648897831287`*^9, 3.516564962825657*^9}, {
   3.516565006535733*^9, 3.516565084796671*^9}, {3.5165651268107443`*^9, 
   3.5165651388851657`*^9}, 3.5165651693052187`*^9, {3.516565336786712*^9, 
   3.516565347659931*^9}, {3.546876826047645*^9, 3.5468768406276655`*^9}, {
   3.5468781393194838`*^9, 3.546878153329503*^9}, {3.547236589106633*^9, 
   3.5472365892966332`*^9}, {3.5647912864731727`*^9, 
   3.5647912997331915`*^9}, {3.564795825689332*^9, 3.5647958933778505`*^9}, {
   3.564796794107833*^9, 3.5647968031558485`*^9}, {3.5647980300644035`*^9, 
   3.5647981331337843`*^9}, {3.5648517832938385`*^9, 
   3.5648518269238997`*^9}, {3.6011485482798696`*^9, 3.6011485483198695`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{"A_", "+", 
       RowBox[{"B_.", "*", 
        RowBox[{"tan", "[", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], "]"}]}], "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}], "/", 
     RowBox[{"(", 
      RowBox[{"a_.", "+", 
       RowBox[{"b_.", "*", 
        RowBox[{"tan", "[", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}]}], ",", "x_Symbol"}],
    "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"a", "*", "A"}], "+", 
       RowBox[{"b", "*", "B"}], "-", 
       RowBox[{"a", "*", "C"}]}], ")"}], "*", 
     RowBox[{"x", "/", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "+", 
        RowBox[{"b", "^", "2"}]}], ")"}]}]}], " ", "-", " ", 
    "\[IndentingNewLine]", "  ", 
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{"A", "*", "b"}], "-", 
        RowBox[{"a", "*", "B"}], "-", 
        RowBox[{"b", "*", "C"}]}], ")"}], "/", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "+", 
        RowBox[{"b", "^", "2"}]}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", "*", "x"}]}], "]"}], ",", "x"}], "]"}]}], " ", "+", " ",
     "\n", "  ", 
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{"A", "*", 
         RowBox[{"b", "^", "2"}]}], "-", 
        RowBox[{"a", "*", "b", "*", "B"}], "+", 
        RowBox[{
         RowBox[{"a", "^", "2"}], "*", "C"}]}], ")"}], "/", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "+", 
        RowBox[{"b", "^", "2"}]}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"(", 
         RowBox[{"1", "+", 
          RowBox[{
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}], "^", "2"}]}], ")"}], "/", 
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", "*", 
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}]}], ",", "x"}], 
      "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "B", ",", "C"}], 
       "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}], "-", 
       RowBox[{"a", "*", "b", "*", "B"}], "+", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}]}], ",", "0"}], "]"}], " ", "&&", 
    " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "^", "2"}], "+", 
       RowBox[{"b", "^", "2"}]}], ",", "0"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"A", "*", "b"}], "-", 
       RowBox[{"a", "*", "B"}], "-", 
       RowBox[{"b", "*", "C"}]}], ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.499358991716386*^9, {3.4993590256364326`*^9, 3.499359034196445*^9}, 
   3.499359158156618*^9, {3.499359230066719*^9, 3.4993592406667337`*^9}, {
   3.499363855658249*^9, 3.499363864218261*^9}, {3.4993753165091624`*^9, 
   3.499375327739178*^9}, {3.500317979448513*^9, 3.500317979729314*^9}, {
   3.500318356413175*^9, 3.5003183706092*^9}, {3.500319044263768*^9, 
   3.5003190515337787`*^9}, {3.500321448217134*^9, 3.5003214542871428`*^9}, {
   3.500432003250804*^9, 3.5004320043278656`*^9}, {3.5010431659740324`*^9, 
   3.5010431675511227`*^9}, 3.50267709922139*^9, {3.502677262251618*^9, 
   3.502677280331643*^9}, 3.5027670673403025`*^9, {3.5030150059333043`*^9, 
   3.5030150148565197`*^9}, {3.503071953798221*^9, 3.5030719623082333`*^9}, {
   3.5035497609313307`*^9, 3.5035497734613485`*^9}, 3.503876194868781*^9, {
   3.504156461196662*^9, 3.5041564626786647`*^9}, 3.505344641373784*^9, {
   3.5053446930538564`*^9, 3.505344714983887*^9}, 3.5053535482551975`*^9, {
   3.5094719943757453`*^9, 3.509471999082014*^9}, {3.5094721711418557`*^9, 
   3.5094721778162374`*^9}, {3.50947240515324*^9, 3.509472413765733*^9}, 
   3.5094733026345735`*^9, {3.509475150960292*^9, 3.5094751544394903`*^9}, 
   3.509481784211692*^9, {3.5094818179926243`*^9, 3.5094818327964706`*^9}, 
   3.5094819101238937`*^9, {3.5094828616743193`*^9, 3.5094828762901554`*^9}, 
   3.5098964710997996`*^9, {3.510504660652564*^9, 3.5105046611829653`*^9}, {
   3.5105160467861633`*^9, 3.510516049001367*^9}, 3.5118083030934734`*^9, 
   3.5144744283131104`*^9, {3.5146412598015394`*^9, 3.514641261969943*^9}, {
   3.5146414410006585`*^9, 3.5146414416090593`*^9}, 3.541093940855841*^9, 
   3.541094696875569*^9, 3.541094735735237*^9, 3.5413120519045644`*^9, 
   3.541403645528081*^9, {3.5454921201091385`*^9, 3.5454921201191387`*^9}, 
   3.5468791175908537`*^9, {3.546880397332645*^9, 3.5468803977526455`*^9}, 
   3.546880958953431*^9, 3.547236605526656*^9, 3.5472478596726274`*^9, {
   3.5472503225360756`*^9, 3.5472503235260773`*^9}, 3.547344998753607*^9, {
   3.564791286513173*^9, 3.5647913151132126`*^9}, 3.5647968853679934`*^9, 
   3.564797409729314*^9, {3.5647991227527227`*^9, 3.564799153749977*^9}, 
   3.5647992540113525`*^9, 3.564851915024023*^9, {3.5648522623645096`*^9, 
   3.5648522771545305`*^9}, {3.564853500616243*^9, 3.564853507196252*^9}, {
   3.601152719286457*^9, 3.6011527192964573`*^9}, {3.601257227176693*^9, 
   3.6012572284166946`*^9}, {3.6014925570454597`*^9, 3.601492561865736*^9}, 
   3.6014926453625116`*^9, {3.7061394653552732`*^9, 3.7061394653752737`*^9}, {
   3.715455408500306*^9, 3.715455413158573*^9}},
 Background->GrayLevel[0.85]],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{"A_", "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}], "/", 
     RowBox[{"(", 
      RowBox[{"a_", "+", 
       RowBox[{"b_.", "*", 
        RowBox[{"tan", "[", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}]}], ",", "x_Symbol"}],
    "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"a", "*", 
     RowBox[{"(", 
      RowBox[{"A", "-", "C"}], ")"}], "*", 
     RowBox[{"x", "/", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "+", 
        RowBox[{"b", "^", "2"}]}], ")"}]}]}], " ", "-", " ", 
    "\[IndentingNewLine]", "  ", 
    RowBox[{"b", "*", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"A", "-", "C"}], ")"}], "/", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "+", 
        RowBox[{"b", "^", "2"}]}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", "*", "x"}]}], "]"}], ",", "x"}], "]"}]}], " ", "+", " ",
     "\n", "  ", 
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{
         RowBox[{"a", "^", "2"}], "*", "C"}], "+", 
        RowBox[{"A", "*", 
         RowBox[{"b", "^", "2"}]}]}], ")"}], "/", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "+", 
        RowBox[{"b", "^", "2"}]}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"(", 
         RowBox[{"1", "+", 
          RowBox[{
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}], "^", "2"}]}], ")"}], "/", 
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", "*", 
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}]}], ",", "x"}], 
      "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "C"}], "}"}],
       ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}], "+", 
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}]}], ",", "0"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "^", "2"}], "+", 
       RowBox[{"b", "^", "2"}]}], ",", "0"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{"A", ",", "C"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.499358991716386*^9, {3.4993590256364326`*^9, 3.499359034196445*^9}, 
   3.499359158156618*^9, {3.499359230066719*^9, 3.4993592406667337`*^9}, {
   3.499363855658249*^9, 3.499363864218261*^9}, {3.4993753165091624`*^9, 
   3.499375327739178*^9}, {3.500317979448513*^9, 3.500317979729314*^9}, {
   3.500318356413175*^9, 3.5003183706092*^9}, {3.500319044263768*^9, 
   3.5003190515337787`*^9}, {3.500321448217134*^9, 3.5003214542871428`*^9}, {
   3.500432003250804*^9, 3.5004320043278656`*^9}, {3.5010431659740324`*^9, 
   3.5010431675511227`*^9}, 3.50267709922139*^9, {3.502677262251618*^9, 
   3.502677280331643*^9}, 3.5027670673403025`*^9, {3.5030150059333043`*^9, 
   3.5030150148565197`*^9}, {3.503071953798221*^9, 3.5030719623082333`*^9}, {
   3.5035497609313307`*^9, 3.5035497734613485`*^9}, 3.503876194868781*^9, {
   3.504156461196662*^9, 3.5041564626786647`*^9}, 3.505344641373784*^9, {
   3.5053446930538564`*^9, 3.505344714983887*^9}, 3.5053535482551975`*^9, {
   3.5094719943757453`*^9, 3.509471999082014*^9}, {3.5094721711418557`*^9, 
   3.5094721778162374`*^9}, {3.50947240515324*^9, 3.509472413765733*^9}, 
   3.5094733026345735`*^9, {3.509475150960292*^9, 3.5094751544394903`*^9}, 
   3.509481784211692*^9, {3.5094818179926243`*^9, 3.5094818327964706`*^9}, 
   3.5094819101238937`*^9, {3.5094828616743193`*^9, 3.5094828762901554`*^9}, 
   3.5098964710997996`*^9, {3.510504660652564*^9, 3.5105046611829653`*^9}, {
   3.5105160467861633`*^9, 3.510516049001367*^9}, 3.5118083030934734`*^9, 
   3.5144744283131104`*^9, {3.5146412598015394`*^9, 3.514641261969943*^9}, {
   3.5146414410006585`*^9, 3.5146414416090593`*^9}, 3.541093940855841*^9, 
   3.541094696875569*^9, 3.541094735735237*^9, 3.5413120519045644`*^9, 
   3.541403645528081*^9, {3.5454921201091385`*^9, 3.5454921201191387`*^9}, 
   3.5468791175908537`*^9, {3.546880397332645*^9, 3.5468803977526455`*^9}, 
   3.546880958953431*^9, 3.547236605526656*^9, 3.5472478596726274`*^9, {
   3.5472503225360756`*^9, 3.5472503235260773`*^9}, 3.547344998753607*^9, {
   3.564791286513173*^9, 3.5647913151132126`*^9}, 3.5647968853679934`*^9, 
   3.564797409729314*^9, {3.5647991227527227`*^9, 3.564799153749977*^9}, 
   3.5647992540113525`*^9, 3.564851915024023*^9, {3.5648520782342515`*^9, 
   3.5648520984042797`*^9}, {3.564852146954348*^9, 3.56485216268437*^9}, {
   3.5648522889845467`*^9, 3.56485229822456*^9}, 3.564853534606291*^9, {
   3.564854502867646*^9, 3.564854507377652*^9}, 3.5648545655477343`*^9, {
   3.601152719326457*^9, 3.6011527193364573`*^9}, {3.6012572305016975`*^9, 
   3.6012572311766987`*^9}, {3.6014925811658397`*^9, 
   3.6014925941635833`*^9}, {3.706139465385274*^9, 3.7061394654052734`*^9}, {
   3.7154554163117533`*^9, 3.715455424212205*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
     RowBox[{"(", 
      RowBox[{"A", "+", 
       RowBox[{"B", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
       RowBox[{"C", " ", 
        SuperscriptBox[
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"A", " ", 
          SuperscriptBox["b", "2"]}], "-", 
         RowBox[{"a", " ", "b", " ", "B"}], "+", 
         RowBox[{
          SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
       "\[And]", " ", 
       RowBox[{"m", "<", 
        RowBox[{"-", "1"}]}], " ", "\[And]", " ", 
       RowBox[{
        RowBox[{
         SuperscriptBox["a", "2"], "+", 
         SuperscriptBox["b", "2"]}], "\[NotEqual]", "0"}]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.4965117206885777`*^9, 3.5192471175756674`*^9, {3.5192587973599553`*^9, 
   3.5192588081399703`*^9}, {3.519321495105319*^9, 3.5193214991925263`*^9}, 
   3.5193247857072587`*^9, {3.5194051897010813`*^9, 3.5194051902791147`*^9}, {
   3.5194055551249824`*^9, 3.5194055708408813`*^9}, {3.519405711316916*^9, 
   3.51940572904193*^9}, {3.519410772192028*^9, 3.5194107776364374`*^9}, {
   3.519413782484516*^9, 3.519413782827717*^9}, {3.519767588664174*^9, 
   3.519767611128214*^9}, {3.5197937941314034`*^9, 3.519793794568204*^9}, {
   3.5197947626591043`*^9, 3.519794764749508*^9}, {3.519794865354085*^9, 
   3.519794874355301*^9}, {3.523218063086026*^9, 3.523218070356036*^9}, {
   3.523316514684347*^9, 3.523316514684347*^9}, {3.5233165475343933`*^9, 
   3.5233165475343933`*^9}, {3.5233184331871357`*^9, 
   3.5233184334271364`*^9}, {3.5273599103435545`*^9, 3.527359910608755*^9}, {
   3.5326424241029196`*^9, 3.53264242435292*^9}, {3.5326584126048727`*^9, 
   3.532658412794873*^9}, {3.532726454021428*^9, 3.5327264542814283`*^9}, {
   3.5327554047263184`*^9, 3.532755404993334*^9}, {3.532798666112293*^9, 
   3.532798676322307*^9}, {3.5328778343810363`*^9, 3.5328778343810363`*^9}, 
   3.5330758294771795`*^9, {3.5330827570868783`*^9, 3.533082757336879*^9}, {
   3.534975097668671*^9, 3.534975097888672*^9}, {3.536863459221404*^9, 
   3.536863459439804*^9}, {3.540264711953947*^9, 3.5402647253239655`*^9}, {
   3.540264783504047*^9, 3.540264783504047*^9}, {3.540502117527218*^9, 
   3.540502117997218*^9}, {3.540502551737826*^9, 3.540502551937826*^9}, {
   3.5449015667548094`*^9, 3.5449015920048447`*^9}, {3.544901714065016*^9, 
   3.544901727955035*^9}, 3.544911958524931*^9, {3.5449122781717815`*^9, 
   3.5449122781717815`*^9}, {3.5455187047402697`*^9, 
   3.5455187052402706`*^9}, {3.545518898070541*^9, 3.545518911130559*^9}, {
   3.5455202868524847`*^9, 3.5455202868524847`*^9}, {3.545520317832528*^9, 
   3.545520339272558*^9}, {3.545520578812894*^9, 3.545520601212925*^9}, {
   3.545520817443228*^9, 3.545520820373232*^9}, {3.545578383065389*^9, 
   3.545578398615411*^9}, {3.5461099987254543`*^9, 3.54611000814787*^9}, {
   3.546110085383606*^9, 3.546110115616459*^9}, {3.546113143676578*^9, 
   3.5461131599162064`*^9}, {3.546113861589839*^9, 3.546113902539911*^9}, {
   3.546114166008774*^9, 3.546114166008774*^9}, {3.5461157697071905`*^9, 
   3.546115770081591*^9}, {3.546881285463888*^9, 3.546881338233962*^9}, 
   3.5468817120044856`*^9, {3.546881961574835*^9, 3.546881961574835*^9}, {
   3.5468888776065187`*^9, 3.546888877866519*^9}, 3.5468894778073587`*^9, {
   3.5468896374375825`*^9, 3.5468896376275826`*^9}, 3.5472370273872466`*^9, 
   3.5472372810876017`*^9, {3.547237346987694*^9, 3.5472373589577107`*^9}, 
   3.5484290277237654`*^9, {3.5500094804472303`*^9, 3.5500094804472303`*^9}, {
   3.564791287213174*^9, 3.5647912872231736`*^9}, {3.5647913911933193`*^9, 
   3.5647913911933193`*^9}, {3.5647994810385513`*^9, 3.5647994842677574`*^9}, 
   3.5651348878886833`*^9, {3.60114854835987*^9, 3.601148572469904*^9}, {
   3.6012600079581857`*^9, 3.6012600120481915`*^9}, {3.6012606526591115`*^9, 
   3.6012606526591115`*^9}, 3.612037002334319*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[TextData[{
 "Derivation: ",
 "Nondegenerate tangent",
 " recurrence 1a with ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"n", "\[Rule]", "0"}], ",", 
   RowBox[{"p", "\[Rule]", "0"}]}]]]
}], "Subsubsection",
 CellChangeTimes->{
  3.4964418130913677`*^9, {3.496442428087432*^9, 3.4964424330170403`*^9}, 
   3.496528894323882*^9, 3.4975761400971813`*^9, {3.4975778216221414`*^9, 
   3.497577829892153*^9}, {3.4975827089309845`*^9, 3.4975827138509912`*^9}, 
   3.497639710530856*^9, 3.4976397495153246`*^9, 3.4976637968549566`*^9, {
   3.497664121273126*^9, 3.4976641274819374`*^9}, {3.4976641823940334`*^9, 
   3.497664184390837*^9}, 3.497664250254153*^9, 3.497664775694276*^9, 
   3.4976649459685745`*^9, {3.49876308829488*^9, 3.4987630935048876`*^9}, {
   3.4987648355173264`*^9, 3.498764841027334*^9}, {3.4991797271436334`*^9, 
   3.4991797277208347`*^9}, {3.4992121758819084`*^9, 3.499212176211909*^9}, {
   3.4993573070196304`*^9, 3.4993573078152323`*^9}, 3.5013524902049417`*^9, 
   3.501621399618896*^9, 3.5016217326739454`*^9, 3.5016376139935837`*^9, 
   3.5016377276810865`*^9, {3.5017223219869995`*^9, 3.501722324888605*^9}, {
   3.501722402935542*^9, 3.5017224033411427`*^9}, 3.5017229817901583`*^9, 
   3.5017231707844906`*^9, 3.5017234546797895`*^9, {3.5017235643791823`*^9, 
   3.501723566750386*^9}, {3.5017250640438175`*^9, 3.5017250769294395`*^9}, {
   3.504317709628769*^9, 3.5043177230887876`*^9}, {3.505345104484432*^9, 
   3.505345123354459*^9}, {3.5057058835492005`*^9, 3.5057058890792084`*^9}, {
   3.5094820888781176`*^9, 3.5094820947444534`*^9}, 3.5120745389640245`*^9, 
   3.514161770227974*^9, 3.51464892969836*^9, {3.5165647309016504`*^9, 
   3.5165647341464562`*^9}, {3.546878123619462*^9, 3.546878123619462*^9}, {
   3.54923649222547*^9, 3.549236498311818*^9}, {3.563050025619238*^9, 
   3.563050028219242*^9}, 3.5647912873231735`*^9, 3.564791361763278*^9, {
   3.6012600586332583`*^9, 3.6012600586332583`*^9}, {3.601260181543435*^9, 
   3.6012601825234365`*^9}, {3.6043635388340397`*^9, 3.6043635788715973`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"A", " ", 
      SuperscriptBox["b", "2"]}], "-", 
     RowBox[{"a", " ", "b", " ", "B"}], "+", 
     RowBox[{
      SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
   "\[And]", " ", 
   RowBox[{"n", "<", 
    RowBox[{"-", "1"}]}], " ", "\[And]", " ", 
   RowBox[{
    RowBox[{
     SuperscriptBox["a", "2"], "+", 
     SuperscriptBox["b", "2"]}], "\[NotEqual]", "0"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.4964417379912624`*^9, 3.4964417471112747`*^9}, {
   3.496441910111503*^9, 3.4964419275315275`*^9}, 3.496451213764926*^9, 
   3.496528833243797*^9, {3.4975777080419827`*^9, 3.4975777244120054`*^9}, {
   3.4975778737222147`*^9, 3.4975778945722437`*^9}, {3.4976396822792068`*^9, 
   3.49763968986082*^9}, {3.497639771433363*^9, 3.49763978104298*^9}, {
   3.4976638984111347`*^9, 3.4976639153215647`*^9}, {3.4976645939383564`*^9, 
   3.497664598805565*^9}, {3.4976646480080514`*^9, 3.4976646992229414`*^9}, {
   3.4976647966763124`*^9, 3.497664832415975*^9}, 3.497664903037299*^9, {
   3.4986779242986293`*^9, 3.498677947778662*^9}, {3.499179738750054*^9, 
   3.499179740902858*^9}, {3.499213316063504*^9, 3.4992133345435305`*^9}, {
   3.499358051967739*^9, 3.4993580717017736`*^9}, {3.4993592718267775`*^9, 
   3.4993592794067883`*^9}, {3.499363873578274*^9, 3.4993638796182823`*^9}, {
   3.50031965968463*^9, 3.5003196639146357`*^9}, {3.5004320083900976`*^9, 
   3.5004320093321514`*^9}, {3.5013559257793446`*^9, 
   3.5013559257793446`*^9}, {3.5013564069000177`*^9, 3.501356418530034*^9}, {
   3.5026761650200815`*^9, 3.5026762017401333`*^9}, {3.5026762588802133`*^9, 
   3.5026762588802133`*^9}, {3.5026766235607233`*^9, 3.502676653910766*^9}, {
   3.5026767649609213`*^9, 3.502676774560935*^9}, {3.5027672387222033`*^9, 
   3.5027672432618113`*^9}, {3.503549718271271*^9, 3.5035497401413016`*^9}, {
   3.504028628528908*^9, 3.504028628903309*^9}, {3.5040318440957556`*^9, 
   3.5040318489317646`*^9}, 3.5043107424584675`*^9, 3.504750503056654*^9, {
   3.509240781672722*^9, 3.5092407958531466`*^9}, {3.5094792332377844`*^9, 
   3.5094792342168407`*^9}, {3.5094793524296017`*^9, 
   3.5094793545107207`*^9}, {3.509480690341126*^9, 3.509480692496249*^9}, {
   3.5094823892893004`*^9, 3.509482390550372*^9}, {3.5099361428088536`*^9, 
   3.5099361433688545`*^9}, {3.510505173363065*^9, 3.5105051737842655`*^9}, {
   3.5114608103187995`*^9, 3.5114608109988003`*^9}, {3.5120748388444443`*^9, 
   3.512074839664446*^9}, 3.514614908818122*^9, {3.5146496332751956`*^9, 
   3.5146496378616037`*^9}, {3.5146601061679506`*^9, 3.514660106407951*^9}, {
   3.516564255177724*^9, 3.5165642557477245`*^9}, 3.5165647754997287`*^9, 
   3.51658990138835*^9, 3.5468776988488674`*^9, {3.547237363117717*^9, 
   3.547237363117717*^9}, {3.5492369715828876`*^9, 3.549236972017913*^9}, {
   3.5500094836764355`*^9, 3.5500094862972403`*^9}, 3.5647913912033195`*^9, {
   3.5647996225932007`*^9, 3.5647996225932007`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
    RowBox[{"(", 
     RowBox[{"A", "+", 
      RowBox[{"B", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
      RowBox[{"C", " ", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", "\[IndentingNewLine]", 
      FractionBox[
       RowBox[{
        RowBox[{"(", 
         RowBox[{
          RowBox[{"A", " ", 
           SuperscriptBox["b", "2"]}], "-", 
          RowBox[{"a", " ", "b", " ", "B"}], "+", 
          RowBox[{
           SuperscriptBox["a", "2"], " ", "C"}]}], ")"}], 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", 
            RowBox[{"Tan", "[", 
             RowBox[{"e", "+", 
              RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], 
         RowBox[{"m", "+", "1"}]]}], 
       RowBox[{"b", " ", "f", " ", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}], " ", 
        RowBox[{"(", 
         RowBox[{
          SuperscriptBox["a", "2"], "+", 
          SuperscriptBox["b", "2"]}], ")"}]}]]}]}]}]}], "+", 
  RowBox[{
   FractionBox["1", 
    RowBox[{
     SuperscriptBox["a", "2"], "+", 
     SuperscriptBox["b", "2"]}]], 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], 
      RowBox[{"m", "+", "1"}]], 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"b", " ", "B"}], "+", 
       RowBox[{"a", " ", 
        RowBox[{"(", 
         RowBox[{"A", "-", "C"}], ")"}]}], "-", 
       RowBox[{
        RowBox[{"(", 
         RowBox[{
          RowBox[{"A", " ", "b"}], "-", 
          RowBox[{"a", " ", "B"}], "-", 
          RowBox[{"b", " ", "C"}]}], ")"}], " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.496441851501421*^9, 
   3.496441872881451*^9}, {3.496528916553913*^9, 3.4965289272239285`*^9}, {
   3.497576165747217*^9, 3.4975761801372375`*^9}, {3.497577761382057*^9, 
   3.4975777938921027`*^9}, 3.49763959300025*^9, 3.497639634480723*^9, 
   3.4976398417426867`*^9, {3.49766381031778*^9, 3.4976638350282235`*^9}, {
   3.49766435104593*^9, 3.497664390373599*^9}, {3.4976647278489914`*^9, 
   3.4976647419046164`*^9}, {3.497664869980841*^9, 3.497664887437272*^9}, {
   3.498761171082196*^9, 3.498761171082196*^9}, {3.4987645684369526`*^9, 
   3.4987646442070584`*^9}, {3.49883740188522*^9, 3.4988375207310295`*^9}, {
   3.498837939529365*^9, 3.498837958467798*^9}, 3.498875518717125*^9, {
   3.4991794250335026`*^9, 3.499179452879552*^9}, {3.4991796787211485`*^9, 
   3.499179681529153*^9}, {3.4991961982482667`*^9, 3.4991962392763386`*^9}, {
   3.499196653100665*^9, 3.4991966955639396`*^9}, {3.499197135282312*^9, 
   3.499197176918785*^9}, {3.499197690444487*^9, 3.499197694048094*^9}, {
   3.499212356332161*^9, 3.4992123865622034`*^9}, {3.499212421412252*^9, 
   3.499212423032254*^9}, {3.4992126233625345`*^9, 3.499212696332637*^9}, {
   3.499357348905704*^9, 3.4993574380910606`*^9}, {3.4993588274861555`*^9, 
   3.4993589039462624`*^9}, {3.499359110096551*^9, 3.4993591137965565`*^9}, {
   3.4993752824791145`*^9, 3.4993752993591385`*^9}, {3.4994041904561324`*^9, 
   3.4994041927361355`*^9}, {3.499404246686211*^9, 3.499404275156251*^9}, {
   3.499404742476905*^9, 3.4994047525269194`*^9}, {3.500317978262911*^9, 
   3.500317979417313*^9}, 3.500318137887991*^9, {3.5013559487293763`*^9, 
   3.501355958849391*^9}, {3.502675715869453*^9, 3.502675816299594*^9}, {
   3.5026759394697657`*^9, 3.50267594261977*^9}, {3.502676353440346*^9, 
   3.50267644207047*^9}, {3.5040325316737633`*^9, 3.504032555885006*^9}, 
   3.5040712495285273`*^9, 3.5056768305258074`*^9, {3.5091340250266056`*^9, 
   3.5091340255166063`*^9}, {3.509134110686725*^9, 3.509134210656865*^9}, 
   3.5091358140991096`*^9, {3.5091363851499095`*^9, 3.5091363926999197`*^9}, {
   3.5091366480802774`*^9, 3.509136673460313*^9}, {3.509136756420429*^9, 
   3.509136781060464*^9}, {3.509136898910629*^9, 3.509136952490704*^9}, 
   3.509137023530803*^9, {3.5091371495009794`*^9, 3.509137183211027*^9}, {
   3.5091372666811438`*^9, 3.5091373023411937`*^9}, {3.509137374621295*^9, 
   3.5091373799413023`*^9}, {3.5091374289113708`*^9, 
   3.5091374307213736`*^9}, {3.509154869729108*^9, 3.5091548849591293`*^9}, {
   3.509154923779184*^9, 3.509154943939212*^9}, 3.5091550691693873`*^9, {
   3.5091550996494303`*^9, 3.5091551134994497`*^9}, {3.509155210989586*^9, 
   3.5091552779496794`*^9}, {3.509155484369969*^9, 3.5091554935699816`*^9}, {
   3.509156447241317*^9, 3.5091565172614145`*^9}, {3.509157219352398*^9, 
   3.5091572214624004`*^9}, {3.509200648372362*^9, 3.509200660899184*^9}, {
   3.5092402115853205`*^9, 3.5092402329885583`*^9}, {3.509240301316678*^9, 
   3.509240304764284*^9}, {3.509240391219636*^9, 3.5092403955564437`*^9}, 
   3.509479727927079*^9, {3.509479774026716*^9, 3.5094797761458373`*^9}, {
   3.509479865673958*^9, 3.5094799080873833`*^9}, 3.5094801863542995`*^9, {
   3.5101615196270494`*^9, 3.5101615236518564`*^9}, {3.510161654707687*^9, 
   3.5101616568604903`*^9}, {3.5101619292841697`*^9, 3.510161930812972*^9}, {
   3.510161988657874*^9, 3.5101619894222755`*^9}, {3.5151898848059206`*^9, 
   3.515189886945924*^9}, {3.5165643357978363`*^9, 3.5165643884979105`*^9}, {
   3.516564434357974*^9, 3.516564521108096*^9}, {3.516564601038223*^9, 
   3.5165646230342617`*^9}, {3.546876826027645*^9, 3.5468768405676656`*^9}, {
   3.546878099429428*^9, 3.546878112839447*^9}, {3.5472374052677755`*^9, 
   3.547237420127796*^9}, 3.547237631298092*^9, {3.564791287343174*^9, 
   3.5647912999131913`*^9}, {3.5647913912133193`*^9, 
   3.5647913912133193`*^9}, {3.56479622918404*^9, 3.5647962678565083`*^9}, {
   3.564796305530575*^9, 3.5647963091809807`*^9}, {3.5647963600370703`*^9, 
   3.5647963657934804`*^9}, {3.60114854840987*^9, 3.601148572469904*^9}, {
   3.601260014223195*^9, 3.6012600457382402`*^9}, {3.601260315043627*^9, 
   3.6012603279336452`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, {3.509479967012754*^9, 3.5094799793114576`*^9}, 
   3.5647964725288677`*^9, 3.6012604124287663`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}], "^", "m_"}], "*", 
     RowBox[{"(", 
      RowBox[{"A_.", "+", 
       RowBox[{"B_.", "*", 
        RowBox[{"tan", "[", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], "]"}]}], "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}]}], ",", 
    "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}], "-", 
       RowBox[{"a", "*", "b", "*", "B"}], "+", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}]}], ")"}], "*", 
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", 
          RowBox[{"Tan", "[", 
           RowBox[{"e", "+", 
            RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", 
       RowBox[{"(", 
        RowBox[{"m", "+", "1"}], ")"}]}], "/", 
      RowBox[{"(", 
       RowBox[{"b", "*", "f", "*", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}], "*", 
        RowBox[{"(", 
         RowBox[{
          RowBox[{"a", "^", "2"}], "+", 
          RowBox[{"b", "^", "2"}]}], ")"}]}], ")"}]}]}], " ", "+", " ", "\n", 
    "  ", 
    RowBox[{
     RowBox[{"1", "/", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "+", 
        RowBox[{"b", "^", "2"}]}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", 
            RowBox[{"Tan", "[", 
             RowBox[{"e", "+", 
              RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"m", "+", "1"}], ")"}]}], "*", 
        RowBox[{"Simp", "[", 
         RowBox[{
          RowBox[{
           RowBox[{"b", "*", "B"}], "+", 
           RowBox[{"a", "*", 
            RowBox[{"(", 
             RowBox[{"A", "-", "C"}], ")"}]}], "-", 
           RowBox[{
            RowBox[{"(", 
             RowBox[{
              RowBox[{"A", "*", "b"}], "-", 
              RowBox[{"a", "*", "B"}], "-", 
              RowBox[{"b", "*", "C"}]}], ")"}], "*", 
            RowBox[{"Tan", "[", 
             RowBox[{"e", "+", 
              RowBox[{"f", "*", "x"}]}], "]"}]}]}], ",", "x"}], "]"}]}], ",", 
       "x"}], "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "B", ",", "C"}], 
       "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}], "-", 
       RowBox[{"a", "*", "b", "*", "B"}], "+", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}]}], ",", "0"}], "]"}], " ", "&&", 
    " ", 
    RowBox[{"LtQ", "[", 
     RowBox[{"m", ",", 
      RowBox[{"-", "1"}]}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "^", "2"}], "+", 
       RowBox[{"b", "^", "2"}]}], ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.499358991716386*^9, {3.4993590256364326`*^9, 3.499359034196445*^9}, 
   3.499359158156618*^9, {3.499359230066719*^9, 3.4993592406667337`*^9}, {
   3.499363855658249*^9, 3.499363864218261*^9}, {3.4993753165091624`*^9, 
   3.499375327739178*^9}, {3.500317979448513*^9, 3.500317979729314*^9}, {
   3.500318356413175*^9, 3.5003183706092*^9}, {3.500319044263768*^9, 
   3.5003190515337787`*^9}, {3.500321448217134*^9, 3.5003214542871428`*^9}, {
   3.500432003250804*^9, 3.5004320043278656`*^9}, {3.5010431659740324`*^9, 
   3.5010431675511227`*^9}, {3.5013560814295626`*^9, 
   3.5013561004895887`*^9}, {3.50135618678971*^9, 3.5013562181997538`*^9}, {
   3.5013562556298065`*^9, 3.5013562631598167`*^9}, {3.5013563481499357`*^9, 
   3.501356350129938*^9}, {3.5013563916299963`*^9, 3.501356393369999*^9}, 
   3.501356538350202*^9, {3.5013565824102635`*^9, 3.5013565889502726`*^9}, 
   3.502675840959628*^9, {3.5026759066297197`*^9, 3.502675919529738*^9}, {
   3.5026760125798683`*^9, 3.5026760334598975`*^9}, {3.5026762053901386`*^9, 
   3.5026762059201393`*^9}, {3.5026762389901853`*^9, 3.5026762460801954`*^9}, 
   3.5026764690205073`*^9, {3.502676562960639*^9, 3.5026765748706555`*^9}, {
   3.502676709910845*^9, 3.502676745080894*^9}, {3.502677686692212*^9, 
   3.5026776934822216`*^9}, {3.5027670612250915`*^9, 
   3.5027670636898956`*^9}, {3.503071936858197*^9, 3.5030719454782095`*^9}, 
   3.50319810512*^9, {3.503549703821251*^9, 3.503549709891259*^9}, 
   3.50387619429158*^9, {3.5051554747458277`*^9, 3.5051554812558365`*^9}, 
   3.505344641353784*^9, {3.5053446929738564`*^9, 3.505344714963887*^9}, 
   3.505353544425192*^9, {3.50924009739312*^9, 3.509240125441969*^9}, {
   3.509240420719288*^9, 3.509240479234991*^9}, {3.5092405242254696`*^9, 
   3.509240529685479*^9}, 3.509240560292733*^9, 3.5092406309608574`*^9, {
   3.509240739193847*^9, 3.5092407562290773`*^9}, {3.5094733083539004`*^9, 
   3.50947331918552*^9}, {3.509475162067927*^9, 3.509475162249937*^9}, 
   3.509475333159713*^9, {3.5094799670517564`*^9, 3.5094799740681577`*^9}, 
   3.5094800542107415`*^9, {3.509480699247636*^9, 3.5094807001776886`*^9}, {
   3.509481029697536*^9, 3.509481029902548*^9}, {3.5094811407088857`*^9, 
   3.509481160623025*^9}, {3.5094825690095797`*^9, 3.5094826005283823`*^9}, 
   3.5098966309844804`*^9, {3.511460805978793*^9, 3.5114608069587946`*^9}, 
   3.5118083077734804`*^9, 3.5144745072304487`*^9, {3.514649548645047*^9, 
   3.514649558176664*^9}, 3.5146606628987293`*^9, 3.5410939346314306`*^9, {
   3.5410946782179365`*^9, 3.541094681478342*^9}, {3.541094719838809*^9, 
   3.5410947203380103`*^9}, 3.541312045074555*^9, 3.5414036167380404`*^9, {
   3.545492120049138*^9, 3.5454921200591383`*^9}, 3.546879078550799*^9, {
   3.546880369282606*^9, 3.5468803777326174`*^9}, 3.5468809483734164`*^9, {
   3.5472373743077326`*^9, 3.5472373809077415`*^9}, 3.5472377733582907`*^9, {
   3.5472378194183555`*^9, 3.54723782985837*^9}, 3.547345016003631*^9, {
   3.5484292225340385`*^9, 3.548429226594044*^9}, 3.548429535144476*^9, 
   3.550009530570118*^9, {3.564791287403174*^9, 3.5647913153732133`*^9}, {
   3.564791407273342*^9, 3.5647914072933416`*^9}, {3.5647964728252687`*^9, 
   3.5647965047117243`*^9}, {3.5647995947627516`*^9, 
   3.5647995990059586`*^9}, {3.601152719366457*^9, 3.601152719376457*^9}, {
   3.6012572346167035`*^9, 3.6012572352367043`*^9}, {3.601260420143778*^9, 
   3.6012604678588457`*^9}, {3.6102215778959126`*^9, 
   3.6102215828859196`*^9}, {3.7061394654352736`*^9, 
   3.7061394654452734`*^9}, {3.7154554598472433`*^9, 3.715455489265926*^9}},
 Background->GrayLevel[0.85]],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}], "^", "m_"}], "*", 
     RowBox[{"(", 
      RowBox[{"A_.", "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}]}], ",", 
    "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}], "+", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}]}], ")"}], "*", 
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", 
          RowBox[{"Tan", "[", 
           RowBox[{"e", "+", 
            RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", 
       RowBox[{"(", 
        RowBox[{"m", "+", "1"}], ")"}]}], "/", 
      RowBox[{"(", 
       RowBox[{"b", "*", "f", "*", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}], "*", 
        RowBox[{"(", 
         RowBox[{
          RowBox[{"a", "^", "2"}], "+", 
          RowBox[{"b", "^", "2"}]}], ")"}]}], ")"}]}]}], " ", "+", " ", "\n", 
    "  ", 
    RowBox[{
     RowBox[{"1", "/", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "+", 
        RowBox[{"b", "^", "2"}]}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", 
            RowBox[{"Tan", "[", 
             RowBox[{"e", "+", 
              RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"m", "+", "1"}], ")"}]}], "*", 
        RowBox[{"Simp", "[", 
         RowBox[{
          RowBox[{
           RowBox[{"a", "*", 
            RowBox[{"(", 
             RowBox[{"A", "-", "C"}], ")"}]}], "-", 
           RowBox[{
            RowBox[{"(", 
             RowBox[{
              RowBox[{"A", "*", "b"}], "-", 
              RowBox[{"b", "*", "C"}]}], ")"}], "*", 
            RowBox[{"Tan", "[", 
             RowBox[{"e", "+", 
              RowBox[{"f", "*", "x"}]}], "]"}]}]}], ",", "x"}], "]"}]}], ",", 
       "x"}], "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "C"}], "}"}],
       ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}], "+", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}]}], ",", "0"}], "]"}], " ", "&&", 
    " ", 
    RowBox[{"LtQ", "[", 
     RowBox[{"m", ",", 
      RowBox[{"-", "1"}]}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "^", "2"}], "+", 
       RowBox[{"b", "^", "2"}]}], ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.499358991716386*^9, {3.4993590256364326`*^9, 3.499359034196445*^9}, 
   3.499359158156618*^9, {3.499359230066719*^9, 3.4993592406667337`*^9}, {
   3.499363855658249*^9, 3.499363864218261*^9}, {3.4993753165091624`*^9, 
   3.499375327739178*^9}, {3.500317979448513*^9, 3.500317979729314*^9}, {
   3.500318356413175*^9, 3.5003183706092*^9}, {3.500319044263768*^9, 
   3.5003190515337787`*^9}, {3.500321448217134*^9, 3.5003214542871428`*^9}, {
   3.500432003250804*^9, 3.5004320043278656`*^9}, {3.5010431659740324`*^9, 
   3.5010431675511227`*^9}, {3.5013560814295626`*^9, 
   3.5013561004895887`*^9}, {3.50135618678971*^9, 3.5013562181997538`*^9}, {
   3.5013562556298065`*^9, 3.5013562631598167`*^9}, {3.5013563481499357`*^9, 
   3.501356350129938*^9}, {3.5013563916299963`*^9, 3.501356393369999*^9}, 
   3.501356538350202*^9, {3.5013565824102635`*^9, 3.5013565889502726`*^9}, 
   3.502675840959628*^9, {3.5026759066297197`*^9, 3.502675919529738*^9}, {
   3.5026760125798683`*^9, 3.5026760334598975`*^9}, {3.5026762053901386`*^9, 
   3.5026762059201393`*^9}, {3.5026762389901853`*^9, 3.5026762460801954`*^9}, 
   3.5026764690205073`*^9, {3.502676562960639*^9, 3.5026765748706555`*^9}, {
   3.502676709910845*^9, 3.502676745080894*^9}, {3.502677686692212*^9, 
   3.5026776934822216`*^9}, {3.5027670612250915`*^9, 
   3.5027670636898956`*^9}, {3.503071936858197*^9, 3.5030719454782095`*^9}, 
   3.50319810512*^9, {3.503549703821251*^9, 3.503549709891259*^9}, 
   3.50387619429158*^9, {3.5051554747458277`*^9, 3.5051554812558365`*^9}, 
   3.505344641353784*^9, {3.5053446929738564`*^9, 3.505344714963887*^9}, 
   3.505353544425192*^9, {3.50924009739312*^9, 3.509240125441969*^9}, {
   3.509240420719288*^9, 3.509240479234991*^9}, {3.5092405242254696`*^9, 
   3.509240529685479*^9}, 3.509240560292733*^9, 3.5092406309608574`*^9, {
   3.509240739193847*^9, 3.5092407562290773`*^9}, {3.5094733083539004`*^9, 
   3.50947331918552*^9}, {3.509475162067927*^9, 3.509475162249937*^9}, 
   3.509475333159713*^9, {3.5094799670517564`*^9, 3.5094799740681577`*^9}, 
   3.5094800542107415`*^9, {3.509480699247636*^9, 3.5094807001776886`*^9}, {
   3.509481029697536*^9, 3.509481029902548*^9}, {3.5094811407088857`*^9, 
   3.509481160623025*^9}, {3.5094825690095797`*^9, 3.5094826005283823`*^9}, 
   3.5098966309844804`*^9, {3.511460805978793*^9, 3.5114608069587946`*^9}, 
   3.5118083077734804`*^9, 3.5144745072304487`*^9, {3.514649548645047*^9, 
   3.514649558176664*^9}, 3.5146606628987293`*^9, 3.5410939346314306`*^9, {
   3.5410946782179365`*^9, 3.541094681478342*^9}, {3.541094719838809*^9, 
   3.5410947203380103`*^9}, 3.541312045074555*^9, 3.5414036167380404`*^9, {
   3.545492120049138*^9, 3.5454921200591383`*^9}, 3.546879078550799*^9, {
   3.546880369282606*^9, 3.5468803777326174`*^9}, 3.5468809483734164`*^9, {
   3.5472373743077326`*^9, 3.5472373809077415`*^9}, 3.5472377733582907`*^9, {
   3.5472378194183555`*^9, 3.54723782985837*^9}, 3.547345016003631*^9, {
   3.5484292225340385`*^9, 3.548429226594044*^9}, 3.548429535144476*^9, 
   3.550009530570118*^9, {3.564791287403174*^9, 3.5647913153732133`*^9}, {
   3.564791407273342*^9, 3.5647914072933416`*^9}, {3.5647964728252687`*^9, 
   3.5647965047117243`*^9}, {3.5647995947627516`*^9, 
   3.5647995990059586`*^9}, {3.601152719366457*^9, 3.601152719376457*^9}, {
   3.6012572346167035`*^9, 3.6012572352367043`*^9}, {3.601260420143778*^9, 
   3.6012604678588457`*^9}, {3.601260567368989*^9, 3.6012605908790226`*^9}, {
   3.6102215998559437`*^9, 3.6102216026959476`*^9}, {3.706139465465274*^9, 
   3.706139465465274*^9}, {3.715455476086172*^9, 3.715455498210438*^9}, 
   3.7154555296112337`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
     RowBox[{"(", 
      RowBox[{"A", "+", 
       RowBox[{"B", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
       RowBox[{"C", " ", 
        SuperscriptBox[
         RowBox[{"Tan", "[", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"A", " ", 
          SuperscriptBox["b", "2"]}], "-", 
         RowBox[{"a", " ", "b", " ", "B"}], "+", 
         RowBox[{
          SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
       "\[And]", " ", 
       RowBox[{"m", "\[NotLessEqual]", 
        RowBox[{"-", "1"}]}]}]]]], "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.4965117206885777`*^9, 3.5192471175756674`*^9, {3.5192587973599553`*^9, 
   3.5192588081399703`*^9}, {3.519321495105319*^9, 3.5193214991925263`*^9}, 
   3.5193247857072587`*^9, {3.5194051897010813`*^9, 3.5194051902791147`*^9}, {
   3.5194055551249824`*^9, 3.5194055708408813`*^9}, {3.519405711316916*^9, 
   3.51940572904193*^9}, {3.519410772192028*^9, 3.5194107776364374`*^9}, {
   3.519413782484516*^9, 3.519413782827717*^9}, {3.519767588664174*^9, 
   3.519767611128214*^9}, {3.5197937941314034`*^9, 3.519793794568204*^9}, {
   3.5197947626591043`*^9, 3.519794764749508*^9}, {3.519794865354085*^9, 
   3.519794874355301*^9}, {3.523218063086026*^9, 3.523218070356036*^9}, {
   3.523316514684347*^9, 3.523316514684347*^9}, {3.5233165475343933`*^9, 
   3.5233165475343933`*^9}, {3.5233184331871357`*^9, 
   3.5233184334271364`*^9}, {3.5273599103435545`*^9, 3.527359910608755*^9}, {
   3.5326424241029196`*^9, 3.53264242435292*^9}, {3.5326584126048727`*^9, 
   3.532658412794873*^9}, {3.532726454021428*^9, 3.5327264542814283`*^9}, {
   3.5327554047263184`*^9, 3.532755404993334*^9}, {3.532798666112293*^9, 
   3.532798676322307*^9}, {3.5328778343810363`*^9, 3.5328778343810363`*^9}, 
   3.5330758294771795`*^9, {3.5330827570868783`*^9, 3.533082757336879*^9}, {
   3.534975097668671*^9, 3.534975097888672*^9}, {3.536863459221404*^9, 
   3.536863459439804*^9}, {3.540264711953947*^9, 3.5402647253239655`*^9}, {
   3.540264783504047*^9, 3.540264783504047*^9}, {3.540502117527218*^9, 
   3.540502117997218*^9}, {3.540502551737826*^9, 3.540502551937826*^9}, {
   3.5449015667548094`*^9, 3.5449015920048447`*^9}, {3.544901714065016*^9, 
   3.544901727955035*^9}, 3.544911958524931*^9, {3.5449122781717815`*^9, 
   3.5449122781717815`*^9}, {3.5455187047402697`*^9, 
   3.5455187052402706`*^9}, {3.545518898070541*^9, 3.545518911130559*^9}, {
   3.5455202868524847`*^9, 3.5455202868524847`*^9}, {3.545520317832528*^9, 
   3.545520339272558*^9}, {3.545520578812894*^9, 3.545520601212925*^9}, {
   3.545520817443228*^9, 3.545520820373232*^9}, {3.545578383065389*^9, 
   3.545578398615411*^9}, {3.5461099987254543`*^9, 3.54611000814787*^9}, {
   3.546110085383606*^9, 3.546110115616459*^9}, {3.546113143676578*^9, 
   3.5461131599162064`*^9}, {3.546113861589839*^9, 3.546113902539911*^9}, {
   3.546114166008774*^9, 3.546114166008774*^9}, {3.5461157697071905`*^9, 
   3.546115770081591*^9}, {3.546881285463888*^9, 3.546881338233962*^9}, 
   3.5468817194244957`*^9, {3.546881968724845*^9, 3.546881968724845*^9}, {
   3.5468888794365215`*^9, 3.5468888796365213`*^9}, {3.546889516597413*^9, 
   3.546889516927414*^9}, {3.546889590097516*^9, 3.546889590507517*^9}, {
   3.547236595916642*^9, 3.547236597206644*^9}, {3.5472366757867546`*^9, 
   3.5472366760767546`*^9}, {3.547236867657023*^9, 3.547236867847023*^9}, {
   3.5647912863631725`*^9, 3.5647912863831725`*^9}, {3.564797402568901*^9, 
   3.564797402568901*^9}, {3.564798919156765*^9, 3.5647989487188168`*^9}, {
   3.5647994404940805`*^9, 3.564799440837281*^9}, {3.565133873927264*^9, 
   3.565133874177264*^9}, {3.5651342093277335`*^9, 3.5651342198077483`*^9}, {
   3.565134721488451*^9, 3.565134721718451*^9}, {3.6011485484998703`*^9, 
   3.6011485724799037`*^9}, {3.601248429049178*^9, 3.601248433129184*^9}, {
   3.6012576818023453`*^9, 3.6012576818023453`*^9}, {3.6012598101779013`*^9, 
   3.601259821017917*^9}, 3.6120370028543196`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[TextData[{
 "Derivation: Nondegenerate tangent recurrence 1b with ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"m", "\[Rule]", "0"}], ",", 
   RowBox[{"p", "\[Rule]", "0"}]}]]]
}], "Subsubsection",
 CellChangeTimes->{
  3.4964418130913677`*^9, {3.496442428087432*^9, 3.4964424330170403`*^9}, 
   3.496528894323882*^9, 3.4975761400971813`*^9, {3.4975778216221414`*^9, 
   3.497577829892153*^9}, {3.4975827089309845`*^9, 3.4975827138509912`*^9}, 
   3.497639710530856*^9, 3.4976397495153246`*^9, 3.4976637968549566`*^9, {
   3.497664121273126*^9, 3.4976641274819374`*^9}, {3.4976641823940334`*^9, 
   3.497664184390837*^9}, 3.497664250254153*^9, 3.497664775694276*^9, 
   3.4976649459685745`*^9, {3.49876308829488*^9, 3.4987630935048876`*^9}, {
   3.4987648355173264`*^9, 3.498764841027334*^9}, {3.4991797271436334`*^9, 
   3.4991797277208347`*^9}, {3.4992121758819084`*^9, 3.499212176211909*^9}, {
   3.4993573070196304`*^9, 3.4993573078152323`*^9}, 3.5013524902049417`*^9, 
   3.501621399618896*^9, 3.5016217326739454`*^9, 3.5016376139935837`*^9, 
   3.5016377276810865`*^9, {3.5017223219869995`*^9, 3.501722324888605*^9}, {
   3.501722402935542*^9, 3.5017224033411427`*^9}, 3.5017229817901583`*^9, 
   3.5017231707844906`*^9, 3.5017234546797895`*^9, {3.5017235643791823`*^9, 
   3.501723566750386*^9}, {3.5017250640438175`*^9, 3.5017250769294395`*^9}, {
   3.501725550171871*^9, 3.501725550671072*^9}, {3.5017278201126595`*^9, 
   3.501727826898671*^9}, {3.5017278619207325`*^9, 3.5017279153196263`*^9}, {
   3.501727996595769*^9, 3.50172799715737*^9}, {3.501728047857459*^9, 
   3.501728055751073*^9}, {3.5017281686076717`*^9, 3.501728169060073*^9}, {
   3.5017287195538397`*^9, 3.50172872538825*^9}, {3.50172921540071*^9, 
   3.5017292168983135`*^9}, {3.501729246943966*^9, 3.5017292537299776`*^9}, {
   3.5017295437928877`*^9, 3.5017295745093417`*^9}, {3.5017296960647554`*^9, 
   3.501729696314356*^9}, {3.5017298761514716`*^9, 3.5017298771342735`*^9}, {
   3.501729942248788*^9, 3.501729967068431*^9}, {3.5017301202763004`*^9, 
   3.501730120603901*^9}, {3.501730162162374*^9, 3.5017301694943867`*^9}, {
   3.5017304653801064`*^9, 3.501730494037357*^9}, {3.501730604179551*^9, 
   3.5017306171587734`*^9}, {3.5034185653536963`*^9, 
   3.5034185683937006`*^9}, {3.5034226237593784`*^9, 3.5034226290093856`*^9}, 
   3.5053451044944324`*^9, {3.509482100959809*^9, 3.509482110306343*^9}, 
   3.5120745501540403`*^9, 3.5141617710379753`*^9, 3.5146489464527893`*^9, 
   3.5165648188978043`*^9, {3.516564852032263*^9, 3.516564854543867*^9}, {
   3.5468781337194757`*^9, 3.5468781337194757`*^9}, {3.5492380822994175`*^9, 
   3.5492380913959374`*^9}, {3.5630499188890886`*^9, 3.563049921919093*^9}, 
   3.564791286463173*^9, 3.564791361663278*^9, {3.565134336087911*^9, 
   3.5651343534779353`*^9}, {3.565134408888013*^9, 3.565134426908038*^9}, {
   3.5651345205581694`*^9, 3.565134522618172*^9}, {3.6012597242677784`*^9, 
   3.601259747607812*^9}, 3.604363580151599*^9, {3.604363618686654*^9, 
   3.604363625331664*^9}, {3.604363784449394*^9, 3.604363784449394*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"A", " ", 
      SuperscriptBox["b", "2"]}], "-", 
     RowBox[{"a", " ", "b", " ", "B"}], "+", 
     RowBox[{
      SuperscriptBox["a", "2"], " ", "C"}]}], "\[NotEqual]", "0"}], " ", 
   "\[And]", " ", 
   RowBox[{"m", "\[NotLessEqual]", 
    RowBox[{"-", "1"}]}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.4964417379912624`*^9, 3.4964417471112747`*^9}, {
   3.496441910111503*^9, 3.4964419275315275`*^9}, 3.496451213764926*^9, 
   3.496528833243797*^9, {3.4975777080419827`*^9, 3.4975777244120054`*^9}, {
   3.4975778737222147`*^9, 3.4975778945722437`*^9}, {3.4976396822792068`*^9, 
   3.49763968986082*^9}, {3.497639771433363*^9, 3.49763978104298*^9}, {
   3.4976638984111347`*^9, 3.4976639153215647`*^9}, {3.4976645939383564`*^9, 
   3.497664598805565*^9}, {3.4976646480080514`*^9, 3.4976646992229414`*^9}, {
   3.4976647966763124`*^9, 3.497664832415975*^9}, 3.497664903037299*^9, {
   3.4986779242986293`*^9, 3.498677947778662*^9}, {3.499179738750054*^9, 
   3.499179740902858*^9}, {3.499213316063504*^9, 3.4992133345435305`*^9}, {
   3.499358051967739*^9, 3.4993580717017736`*^9}, {3.4993592718267775`*^9, 
   3.4993592794067883`*^9}, {3.499363873578274*^9, 3.4993638796182823`*^9}, {
   3.50031965968463*^9, 3.5003196639146357`*^9}, {3.500345120088499*^9, 
   3.500345127428509*^9}, {3.500410576940289*^9, 3.5004105779663477`*^9}, {
   3.5004106327664824`*^9, 3.500410643898119*^9}, {3.500429569940626*^9, 
   3.500429571675726*^9}, {3.503421192847375*^9, 3.503421192847375*^9}, {
   3.5034383255889688`*^9, 3.5034383282289724`*^9}, {3.504331822596939*^9, 
   3.5043318234969406`*^9}, {3.504988941601285*^9, 3.504988942181286*^9}, 
   3.5087861934230595`*^9, 3.509482964630208*^9, {3.509936156008872*^9, 
   3.5099361567288733`*^9}, {3.510505182442281*^9, 3.510505183128682*^9}, {
   3.512074844314452*^9, 3.5120748451244535`*^9}, 3.514614982497452*^9, {
   3.514641288021989*^9, 3.514641289753592*^9}, {3.5146414380210533`*^9, 
   3.514641438286254*^9}, {3.514652945017925*^9, 3.514652945327925*^9}, {
   3.5165642587977285`*^9, 3.5165642592877293`*^9}, 3.516564792622958*^9, {
   3.5165899054683557`*^9, 3.5165899061283565`*^9}, 3.5468776988488674`*^9, {
   3.5472366029866524`*^9, 3.5472366029866524`*^9}, {3.5630499536991377`*^9, 
   3.5630499543791385`*^9}, {3.56479740768571*^9, 3.56479740768571*^9}, {
   3.564798954303626*^9, 3.564798954303626*^9}, {3.5651345820082555`*^9, 
   3.5651345820082555`*^9}, {3.601259826457925*^9, 3.601259826457925*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
    RowBox[{"(", 
     RowBox[{"A", "+", 
      RowBox[{"B", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}], "+", 
      RowBox[{"C", " ", 
       SuperscriptBox[
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}], "2"]}]}], ")"}], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{"C", 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", 
            RowBox[{"Tan", "[", 
             RowBox[{"e", "+", 
              RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], 
         RowBox[{"m", "+", "1"}]]}], 
       RowBox[{"b", " ", "f", " ", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}]]}]}]}]}], "+", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        RowBox[{"Tan", "[", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], "m"], 
    RowBox[{"(", 
     RowBox[{"A", "-", "C", "+", 
      RowBox[{"B", " ", 
       RowBox[{"Tan", "[", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], "]"}]}]}], ")"}], 
    RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.496441851501421*^9, 
   3.496441872881451*^9}, {3.496528916553913*^9, 3.4965289272239285`*^9}, {
   3.497576165747217*^9, 3.4975761801372375`*^9}, {3.497577761382057*^9, 
   3.4975777938921027`*^9}, 3.49763959300025*^9, 3.497639634480723*^9, 
   3.4976398417426867`*^9, {3.49766381031778*^9, 3.4976638350282235`*^9}, {
   3.49766435104593*^9, 3.497664390373599*^9}, {3.4976647278489914`*^9, 
   3.4976647419046164`*^9}, {3.497664869980841*^9, 3.497664887437272*^9}, {
   3.498761171082196*^9, 3.498761171082196*^9}, {3.4987645684369526`*^9, 
   3.4987646442070584`*^9}, {3.49883740188522*^9, 3.4988375207310295`*^9}, {
   3.498837939529365*^9, 3.498837958467798*^9}, 3.498875518717125*^9, {
   3.4991794250335026`*^9, 3.499179452879552*^9}, {3.4991796787211485`*^9, 
   3.499179681529153*^9}, {3.4991961982482667`*^9, 3.4991962392763386`*^9}, {
   3.499196653100665*^9, 3.4991966955639396`*^9}, {3.499197135282312*^9, 
   3.499197176918785*^9}, {3.499197690444487*^9, 3.499197694048094*^9}, {
   3.499212356332161*^9, 3.4992123865622034`*^9}, {3.499212421412252*^9, 
   3.499212423032254*^9}, {3.4992126233625345`*^9, 3.499212696332637*^9}, {
   3.499357348905704*^9, 3.4993574380910606`*^9}, {3.4993588274861555`*^9, 
   3.4993589039462624`*^9}, {3.499359110096551*^9, 3.4993591137965565`*^9}, {
   3.4993752824791145`*^9, 3.4993752993591385`*^9}, {3.4994041904561324`*^9, 
   3.4994041927361355`*^9}, {3.499404246686211*^9, 3.499404275156251*^9}, {
   3.499404742476905*^9, 3.4994047525269194`*^9}, {3.500317978262911*^9, 
   3.500317979417313*^9}, 3.500318137887991*^9, {3.5030149938276834`*^9, 
   3.5030149992252927`*^9}, 3.505344641363784*^9, {3.505344693023856*^9, 
   3.5053447149738874`*^9}, 3.5053535459551945`*^9, {3.5094696268243294`*^9, 
   3.509469698805446*^9}, {3.509469791674758*^9, 3.509469802752392*^9}, {
   3.509469914234768*^9, 3.5094699164558954`*^9}, {3.509470445456152*^9, 
   3.509470483604334*^9}, {3.5094706454305897`*^9, 3.50947066641179*^9}, {
   3.5094707559529114`*^9, 3.5094707698497066`*^9}, 3.5094708527324467`*^9, {
   3.5094710367959747`*^9, 3.5094710383620644`*^9}, 3.5094711873785877`*^9, 
   3.5094801966418877`*^9, {3.5101642321454153`*^9, 3.5101642363106227`*^9}, {
   3.5101642797722993`*^9, 3.5101643150283613`*^9}, {3.5101646049236703`*^9, 
   3.5101646129576845`*^9}, {3.5151898918659306`*^9, 
   3.5151898944059343`*^9}, {3.5165648897831287`*^9, 3.516564962825657*^9}, {
   3.516565006535733*^9, 3.516565084796671*^9}, {3.5165651268107443`*^9, 
   3.5165651388851657`*^9}, 3.5165651693052187`*^9, {3.516565336786712*^9, 
   3.516565347659931*^9}, {3.546876826047645*^9, 3.5468768406276655`*^9}, {
   3.5468781393194838`*^9, 3.546878153329503*^9}, {3.547236589106633*^9, 
   3.5472365892966332`*^9}, {3.5647912864731727`*^9, 
   3.5647912997331915`*^9}, {3.564795825689332*^9, 3.5647958933778505`*^9}, {
   3.564796794107833*^9, 3.5647968031558485`*^9}, {3.565134269877818*^9, 
   3.5651343242878942`*^9}, {3.6011485485398703`*^9, 3.601148572509904*^9}, {
   3.60125798441278*^9, 3.601257994792795*^9}, {3.6012581958130836`*^9, 
   3.601258197448086*^9}, 3.601258312243251*^9, {3.6012598400679445`*^9, 
   3.601259840812946*^9}, 3.601259884388008*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"(", 
      RowBox[{"A_.", "+", 
       RowBox[{"B_.", "*", 
        RowBox[{"tan", "[", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], "]"}]}], "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}]}], ",", 
    "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"C", "*", 
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", 
          RowBox[{"Tan", "[", 
           RowBox[{"e", "+", 
            RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", 
       RowBox[{"(", 
        RowBox[{"m", "+", "1"}], ")"}]}], "/", 
      RowBox[{"(", 
       RowBox[{"b", "*", "f", "*", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], ")"}]}]}], " ", "+", " ", 
    RowBox[{"Int", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", "*", 
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", "m"}], "*", 
       RowBox[{"Simp", "[", 
        RowBox[{
         RowBox[{"A", "-", "C", "+", 
          RowBox[{"B", "*", 
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}]}]}], ",", "x"}], "]"}]}], ",", 
      "x"}], "]"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "B", ",", "C", ",", 
        "m"}], "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}], "-", 
       RowBox[{"a", "*", "b", "*", "B"}], "+", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}]}], ",", "0"}], "]"}], " ", "&&", 
    " ", 
    RowBox[{"Not", "[", 
     RowBox[{"LeQ", "[", 
      RowBox[{"m", ",", 
       RowBox[{"-", "1"}]}], "]"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.499358991716386*^9, {3.4993590256364326`*^9, 3.499359034196445*^9}, 
   3.499359158156618*^9, {3.499359230066719*^9, 3.4993592406667337`*^9}, {
   3.499363855658249*^9, 3.499363864218261*^9}, {3.4993753165091624`*^9, 
   3.499375327739178*^9}, {3.500317979448513*^9, 3.500317979729314*^9}, {
   3.500318356413175*^9, 3.5003183706092*^9}, {3.500319044263768*^9, 
   3.5003190515337787`*^9}, {3.500321448217134*^9, 3.5003214542871428`*^9}, {
   3.500432003250804*^9, 3.5004320043278656`*^9}, {3.5010431659740324`*^9, 
   3.5010431675511227`*^9}, 3.50267709922139*^9, {3.502677262251618*^9, 
   3.502677280331643*^9}, 3.5027670673403025`*^9, {3.5030150059333043`*^9, 
   3.5030150148565197`*^9}, {3.503071953798221*^9, 3.5030719623082333`*^9}, {
   3.5035497609313307`*^9, 3.5035497734613485`*^9}, 3.503876194868781*^9, {
   3.504156461196662*^9, 3.5041564626786647`*^9}, 3.505344641373784*^9, {
   3.5053446930538564`*^9, 3.505344714983887*^9}, 3.5053535482551975`*^9, {
   3.5094719943757453`*^9, 3.509471999082014*^9}, {3.5094721711418557`*^9, 
   3.5094721778162374`*^9}, {3.50947240515324*^9, 3.509472413765733*^9}, 
   3.5094733026345735`*^9, {3.509475150960292*^9, 3.5094751544394903`*^9}, 
   3.509481784211692*^9, {3.5094818179926243`*^9, 3.5094818327964706`*^9}, 
   3.5094819101238937`*^9, {3.5094828616743193`*^9, 3.5094828762901554`*^9}, 
   3.5098964710997996`*^9, {3.510504660652564*^9, 3.5105046611829653`*^9}, {
   3.5105160467861633`*^9, 3.510516049001367*^9}, 3.5118083030934734`*^9, 
   3.5144744283131104`*^9, {3.5146412598015394`*^9, 3.514641261969943*^9}, {
   3.5146414410006585`*^9, 3.5146414416090593`*^9}, 3.541093940855841*^9, 
   3.541094696875569*^9, 3.541094735735237*^9, 3.5413120519045644`*^9, 
   3.541403645528081*^9, {3.5454921201091385`*^9, 3.5454921201191387`*^9}, 
   3.5468791175908537`*^9, {3.546880397332645*^9, 3.5468803977526455`*^9}, 
   3.546880958953431*^9, 3.547236605526656*^9, 3.5472478596726274`*^9, {
   3.5472503225360756`*^9, 3.5472503235260773`*^9}, 3.547344998753607*^9, {
   3.564791286513173*^9, 3.5647913151132126`*^9}, 3.5647968853679934`*^9, 
   3.564797409729314*^9, 3.564799009122123*^9, {3.5651345970782766`*^9, 
   3.565134637488333*^9}, {3.601152719456457*^9, 3.6011527194664574`*^9}, {
   3.601257240941713*^9, 3.601257241741714*^9}, {3.60125902878928*^9, 
   3.6012590922718706`*^9}, {3.610221477775772*^9, 3.6102214862257843`*^9}, 
   3.7061394654952736`*^9, {3.715455515146406*^9, 3.7154555202166967`*^9}},
 Background->GrayLevel[0.85]],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}]}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"(", 
      RowBox[{"A_.", "+", 
       RowBox[{"C_.", "*", 
        RowBox[{
         RowBox[{"tan", "[", 
          RowBox[{"e_.", "+", 
           RowBox[{"f_.", "*", "x_"}]}], "]"}], "^", "2"}]}]}], ")"}]}], ",", 
    "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"C", "*", 
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", 
          RowBox[{"Tan", "[", 
           RowBox[{"e", "+", 
            RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", 
       RowBox[{"(", 
        RowBox[{"m", "+", "1"}], ")"}]}], "/", 
      RowBox[{"(", 
       RowBox[{"b", "*", "f", "*", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], ")"}]}]}], " ", "+", " ", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{"A", "-", "C"}], ")"}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", "*", 
           RowBox[{"Tan", "[", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], "]"}]}]}], ")"}], "^", "m"}], ",", 
       "x"}], "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "e", ",", "f", ",", "A", ",", "C", ",", "m"}], 
       "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"A", "*", 
        RowBox[{"b", "^", "2"}]}], "+", 
       RowBox[{
        RowBox[{"a", "^", "2"}], "*", "C"}]}], ",", "0"}], "]"}], " ", "&&", 
    " ", 
    RowBox[{"Not", "[", 
     RowBox[{"LeQ", "[", 
      RowBox[{"m", ",", 
       RowBox[{"-", "1"}]}], "]"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976397432129135`*^9, 3.4976633737510133`*^9, 
   3.497663791067346*^9, 3.497664242469739*^9, {3.49766439658241*^9, 
   3.497664398470013*^9}, 3.497664769079864*^9, {3.4986774515679674`*^9, 
   3.49867746046798*^9}, {3.498761171092196*^9, 3.4987611711021957`*^9}, {
   3.4987647642072268`*^9, 3.4987647797772484`*^9}, {3.4987662584793186`*^9, 
   3.498766258909319*^9}, {3.4987729599087005`*^9, 3.4987729607687016`*^9}, 
   3.4988018581019883`*^9, {3.499179579551774*^9, 3.4991796094414263`*^9}, {
   3.499179689157567*^9, 3.499179691637971*^9}, {3.4991797663621025`*^9, 
   3.499179767407304*^9}, {3.499179868308281*^9, 3.499179869805884*^9}, 
   3.499180259000168*^9, {3.499197395157969*^9, 3.499197401054779*^9}, {
   3.499212499112361*^9, 3.49921251269238*^9}, {3.499212715522664*^9, 
   3.4992127382126956`*^9}, {3.4992128038327875`*^9, 
   3.4992128061327906`*^9}, {3.499213982604438*^9, 3.499214025174497*^9}, {
   3.499214066304555*^9, 3.499214072114563*^9}, {3.4992145411652193`*^9, 
   3.4992145475152283`*^9}, 3.4993574899767523`*^9, {3.4993576909831047`*^9, 
   3.4993577157403483`*^9}, 3.4993578086385117`*^9, {3.4993579390391407`*^9, 
   3.4993579651067867`*^9}, {3.4993580774113836`*^9, 3.4993580785501857`*^9}, 
   3.499358991716386*^9, {3.4993590256364326`*^9, 3.499359034196445*^9}, 
   3.499359158156618*^9, {3.499359230066719*^9, 3.4993592406667337`*^9}, {
   3.499363855658249*^9, 3.499363864218261*^9}, {3.4993753165091624`*^9, 
   3.499375327739178*^9}, {3.500317979448513*^9, 3.500317979729314*^9}, {
   3.500318356413175*^9, 3.5003183706092*^9}, {3.500319044263768*^9, 
   3.5003190515337787`*^9}, {3.500321448217134*^9, 3.5003214542871428`*^9}, {
   3.500432003250804*^9, 3.5004320043278656`*^9}, {3.5010431659740324`*^9, 
   3.5010431675511227`*^9}, 3.50267709922139*^9, {3.502677262251618*^9, 
   3.502677280331643*^9}, 3.5027670673403025`*^9, {3.5030150059333043`*^9, 
   3.5030150148565197`*^9}, {3.503071953798221*^9, 3.5030719623082333`*^9}, {
   3.5035497609313307`*^9, 3.5035497734613485`*^9}, 3.503876194868781*^9, {
   3.504156461196662*^9, 3.5041564626786647`*^9}, 3.505344641373784*^9, {
   3.5053446930538564`*^9, 3.505344714983887*^9}, 3.5053535482551975`*^9, {
   3.5094719943757453`*^9, 3.509471999082014*^9}, {3.5094721711418557`*^9, 
   3.5094721778162374`*^9}, {3.50947240515324*^9, 3.509472413765733*^9}, 
   3.5094733026345735`*^9, {3.509475150960292*^9, 3.5094751544394903`*^9}, 
   3.509481784211692*^9, {3.5094818179926243`*^9, 3.5094818327964706`*^9}, 
   3.5094819101238937`*^9, {3.5094828616743193`*^9, 3.5094828762901554`*^9}, 
   3.5098964710997996`*^9, {3.510504660652564*^9, 3.5105046611829653`*^9}, {
   3.5105160467861633`*^9, 3.510516049001367*^9}, 3.5118083030934734`*^9, 
   3.5144744283131104`*^9, {3.5146412598015394`*^9, 3.514641261969943*^9}, {
   3.5146414410006585`*^9, 3.5146414416090593`*^9}, 3.541093940855841*^9, 
   3.541094696875569*^9, 3.541094735735237*^9, 3.5413120519045644`*^9, 
   3.541403645528081*^9, {3.5454921201091385`*^9, 3.5454921201191387`*^9}, 
   3.5468791175908537`*^9, {3.546880397332645*^9, 3.5468803977526455`*^9}, 
   3.546880958953431*^9, 3.547236605526656*^9, 3.5472478596726274`*^9, {
   3.5472503225360756`*^9, 3.5472503235260773`*^9}, 3.547344998753607*^9, {
   3.564791286513173*^9, 3.5647913151132126`*^9}, 3.5647968853679934`*^9, 
   3.564797409729314*^9, 3.564799009122123*^9, {3.5651345970782766`*^9, 
   3.565134637488333*^9}, {3.601152719456457*^9, 3.6011527194664574`*^9}, {
   3.601257240941713*^9, 3.601257241741714*^9}, {3.60125902878928*^9, 
   3.6012590922718706`*^9}, {3.601259394512305*^9, 3.6012594190723405`*^9}, 
   3.706139465505274*^9, {3.715455515165407*^9, 3.7154555230668592`*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]]
}, Closed]]
},
AutoGeneratedPackage->Automatic,
CellGrouping->Manual,
WindowSize->{1908, 946},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
PrintingCopies->1,
PrintingStartingPageNumber->1,
PrintingPageRange->{1, 32000},
PageHeaders->{{
   Cell[
    TextData[{
      StyleBox[
       CounterBox["Page"], "PageNumber"], " ", "|", " ", 
      StyleBox[
       ValueBox["FileName"], "Header"]}], "Header", 
    CellMargins -> {{0, Inherited}, {Inherited, Inherited}}], None, None}, {
   Cell[
    TextData[{
     "Rules", " ", "for", " ", "integrands", " ", "of", " ", "the", " ", 
      "form", " ", "(", "a", "+", "b", " ", "tan", "[", "e", "+", "f", " ", 
      "x", "]", ")", "^", "m", " ", "(", "A", "+", "B", " ", "tan", "(", "e", 
      "+", "f", " ", "x", ")", "+", "C", " ", "tan", "(", "e", "+", "f", " ", 
      "x", ")", "^", "2", ")"}], "Header", 
    CellMargins -> {{0, Inherited}, {Inherited, Inherited}}], None, 
   Cell[
    TextData[
     CounterBox["Page", CounterFunction :> Identity]], "Header", 
    CellMargins -> {{Inherited, 0}, {Inherited, Inherited}}]}},
PageFooters->{{None, None, None}, {None, None, None}},
PageHeaderLines->{False, False},
PageFooterLines->{False, False},
PrintingOptions->{"FacingPages"->False,
"FirstPageFace"->Right,
"FirstPageFooter"->False,
"FirstPageHeader"->False,
"Magnification"->1.,
"PaperOrientation"->"Landscape",
"PaperSize"->{783., 601.8},
"PrintCellBrackets"->False,
"PrintMultipleHorizontalPages"->False,
"PrintRegistrationMarks"->False,
"PrintingMargins"->28.8},
Magnification->1.3999998569488525`,
FrontEndVersion->"8.0 for Microsoft Windows (64-bit) (October 6, 2011)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[545, 20, 7570, 146, 46, "Subsubtitle"],
Cell[CellGroupData[{
Cell[8140, 170, 4889, 81, 53, "Subsubtitle"],
Cell[13032, 253, 3175, 44, 67, "Subsubsection"],
Cell[16210, 299, 4334, 80, 74, "Subsubsection"],
Cell[20547, 381, 2576, 35, 47, "Subsubsection"],
Cell[23126, 418, 4027, 76, 114, "Subsubtitle"],
Cell[27156, 496, 771, 11, 67, "Subsubsection"],
Cell[27930, 509, 5931, 108, 202, "Code"],
Cell[33864, 619, 5986, 110, 222, "Code"],
Cell[39853, 731, 89, 2, 67, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[39979, 738, 5137, 95, 53, "Subsubtitle"],
Cell[45119, 835, 1271, 17, 67, "Subsubsection"],
Cell[46393, 854, 2473, 52, 74, "Subsubsection"],
Cell[48869, 908, 2861, 46, 50, "Subsubsection"],
Cell[51733, 956, 4320, 86, 116, "Subsubtitle"],
Cell[56056, 1044, 771, 11, 67, "Subsubsection"],
Cell[56830, 1057, 6144, 122, 202, "Code"],
Cell[62977, 1181, 5913, 116, 222, "Code"],
Cell[68893, 1299, 89, 2, 67, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[69019, 1306, 5053, 92, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[74097, 1402, 5370, 98, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[79492, 1504, 5435, 103, 53, "Subsubtitle"],
Cell[84930, 1609, 2467, 39, 69, "Subsubsection"],
Cell[87400, 1650, 3640, 61, 67, "Subsubsection"],
Cell[91043, 1713, 9619, 197, 310, "Subsubtitle"],
Cell[100665, 1912, 818, 11, 67, "Subsubsection"],
Cell[101486, 1925, 8216, 177, 251, "Code"],
Cell[109705, 2104, 7808, 165, 271, "Code"],
Cell[117516, 2271, 89, 2, 67, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[117642, 2278, 5436, 102, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[123103, 2384, 5330, 98, 66, "Subsubtitle"],
Cell[CellGroupData[{
Cell[128458, 2486, 5496, 100, 66, "Subsubtitle"],
Cell[133957, 2588, 2657, 36, 67, "Subsubsection"],
Cell[136617, 2626, 4147, 88, 96, "Subsubsection"],
Cell[140767, 2716, 3262, 59, 67, "Subsubsection"],
Cell[144032, 2777, 3173, 52, 67, "Subsubsection"],
Cell[147208, 2831, 5594, 112, 128, "Subsubtitle"],
Cell[152805, 2945, 771, 11, 67, "Subsubsection"],
Cell[153579, 2958, 6722, 139, 251, "Code"],
Cell[160304, 3099, 89, 2, 67, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[160430, 3106, 5580, 104, 66, "Subsubtitle"],
Cell[CellGroupData[{
Cell[166035, 3214, 5246, 91, 66, "Subsubtitle"],
Cell[171284, 3307, 2657, 36, 35, "Subsubsection"],
Cell[173944, 3345, 2941, 44, 26, "Subsubsection"],
Cell[176888, 3391, 5037, 91, 66, "Subsubtitle"],
Cell[181928, 3484, 771, 11, 35, "Subsubsection"],
Cell[182702, 3497, 5831, 106, 104, "Code"],
Cell[188536, 3605, 5537, 100, 114, "Code"],
Cell[194076, 3707, 89, 2, 35, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[194202, 3714, 5676, 107, 66, "Subsubtitle"],
Cell[199881, 3823, 2657, 36, 67, "Subsubsection"],
Cell[202541, 3861, 4206, 89, 96, "Subsubsection"],
Cell[206750, 3952, 3372, 59, 67, "Subsubsection"],
Cell[210125, 4013, 5962, 126, 128, "Subsubtitle"],
Cell[216090, 4141, 771, 11, 67, "Subsubsection"],
Cell[216864, 4154, 7506, 166, 300, "Code"],
Cell[224373, 4322, 7076, 150, 320, "Code"],
Cell[231452, 4474, 89, 2, 67, "Subsubsection"]
}, Closed]]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[231602, 4483, 5439, 103, 53, "Subsubtitle"],
Cell[237044, 4588, 2041, 33, 69, "Subsubsection"],
Cell[239088, 4623, 3598, 61, 67, "Subsubsection"],
Cell[242689, 4686, 7405, 148, 240, "Subsubtitle"],
Cell[250097, 4836, 871, 12, 67, "Subsubsection"],
Cell[250971, 4850, 8608, 182, 251, "Code"],
Cell[259582, 5034, 8334, 174, 271, "Code"],
Cell[267919, 5210, 89, 2, 67, "Subsubsection"]
}, Closed]]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[268069, 5219, 5505, 101, 53, "Subsubtitle"],
Cell[273577, 5322, 3064, 45, 69, "Subsubsection"],
Cell[276644, 5369, 3137, 52, 67, "Subsubsection"],
Cell[279784, 5423, 5620, 109, 128, "Subsubtitle"],
Cell[285407, 5534, 771, 11, 67, "Subsubsection"],
Cell[286181, 5547, 6478, 133, 202, "Code"],
Cell[292662, 5682, 6087, 123, 222, "Code"],
Cell[298752, 5807, 89, 2, 67, "Subsubsection"]
}, Closed]]
}, Closed]]
}
]
*)

(* End of internal cache information *)