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 ArcTan

 http://functions.wolfram.com/01.14.21.0010.01

 Input Form

 Integrate[ArcTan[a z + b]/(c z + d), z] == (1/c) (ArcTan[b + a z] (-Log[1/Sqrt[1 + (b + a z)^2]] + Log[Sin[ArcTan[((-b) c + a d)/c] + ArcTan[b + a z]]]) + (1/2) ((-(1/4)) I (Pi - 2 ArcTan[b + a z])^2 - I (ArcTan[b - (a d)/c] - ArcTan[b + a z])^2 + (Pi - 2 ArcTan[b + a z]) Log[1 + E^(-2 I ArcTan[b + a z])] + 2 (-ArcTan[b - (a d)/c] + ArcTan[b + a z]) Log[1 - E^(2 I (-ArcTan[b - (a d)/c] + ArcTan[b + a z]))] - (Pi - 2 ArcTan[b + a z]) Log[2/Sqrt[1 + (b + a z)^2]] + 2 (ArcTan[b - (a d)/c] - ArcTan[b + a z]) Log[2 Sin[ArcTan[((-b) c + a d)/c] + ArcTan[b + a z]]] - I PolyLog[2, -E^(-2 I ArcTan[b + a z])] - I PolyLog[2, E^(2 I (-ArcTan[b - (a d)/c] + ArcTan[b + a z]))]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], "]"]], RowBox[List[RowBox[List["c", " ", "z"]], "+", "d"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "c"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", FractionBox["1", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], "2"]]]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["Sin", "[", RowBox[List[RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "c"]], "+", RowBox[List["a", " ", "d"]]]], "c"], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], "2"]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["b", "-", FractionBox[RowBox[List["a", " ", "d"]], "c"]]], "]"]], "-", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ArcTan", "[", RowBox[List["b", "-", FractionBox[RowBox[List["a", " ", "d"]], "c"]]], "]"]]]], "+", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ArcTan", "[", RowBox[List["b", "-", FractionBox[RowBox[List["a", " ", "d"]], "c"]]], "]"]]]], "+", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", FractionBox["2", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], "2"]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["b", "-", FractionBox[RowBox[List["a", " ", "d"]], "c"]]], "]"]], "-", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["2", " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "c"]], "+", RowBox[List["a", " ", "d"]]]], "c"], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ArcTan", "[", RowBox[List["b", "-", FractionBox[RowBox[List["a", " ", "d"]], "c"]]], "]"]]]], "+", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 tan - 1 ( b + a z ) d + c z z 1 c ( tan - 1 ( b + a z ) ( log ( sin ( tan - 1 ( a d - b c c ) + tan - 1 ( b + a z ) ) ) - log ( 1 ( b + a z ) 2 + 1 ) ) + 1 2 ( - ( tan - 1 ( b - a d c ) - tan - 1 ( b + a z ) ) 2 + 2 log ( 2 sin ( tan - 1 ( a d - b c c ) + tan - 1 ( b + a z ) ) ) ( tan - 1 ( b - a d c ) - tan - 1 ( b + a z ) ) - 1 4 ( π - 2 tan - 1 ( b + a z ) ) 2 + ( π - 2 tan - 1 ( b + a z ) ) log ( 1 + exp ( - 2 tan - 1 ( b + a z ) ) ) + 2 ( tan - 1 ( b + a z ) - tan - 1 ( b - a d c ) ) log ( 1 - exp ( 2 ( tan - 1 ( b + a z ) - tan - 1 ( b - a d c ) ) ) ) - ( π - 2 tan - 1 ( b + a z ) ) log ( 2 ( b + a z ) 2 + 1 ) - Li PolyLog 2 ( - exp ( - 2 tan - 1 ( b + a z ) ) ) - Li PolyLog 2 ( exp ( 2 ( tan - 1 ( b + a z ) - tan - 1 ( b - a d c ) ) ) ) ) ) z b a z d c z -1 1 c -1 b a z a d -1 b c c -1 b a z -1 1 b a z 2 1 1 2 -1 1 2 -1 b -1 a d c -1 -1 b a z 2 2 2 a d -1 b c c -1 b a z b -1 a d c -1 -1 b a z -1 1 4 -1 2 b a z 2 -1 2 b a z 1 -2 b a z 2 b a z -1 b -1 a d c -1 1 -1 2 b a z -1 b -1 a d c -1 -1 -1 2 b a z 2 b a z 2 1 1 2 -1 -1 PolyLog 2 -1 -2 b a z -1 PolyLog 2 2 b a z -1 b -1 a d c -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], "]"]], RowBox[List[RowBox[List["c_", " ", "z_"]], "+", "d_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", FractionBox["1", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], "2"]]]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["Sin", "[", RowBox[List[RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "c"]], "+", RowBox[List["a", " ", "d"]]]], "c"], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], "2"]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["b", "-", FractionBox[RowBox[List["a", " ", "d"]], "c"]]], "]"]], "-", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ArcTan", "[", RowBox[List["b", "-", FractionBox[RowBox[List["a", " ", "d"]], "c"]]], "]"]]]], "+", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ArcTan", "[", RowBox[List["b", "-", FractionBox[RowBox[List["a", " ", "d"]], "c"]]], "]"]]]], "+", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", FractionBox["2", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], "2"]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["b", "-", FractionBox[RowBox[List["a", " ", "d"]], "c"]]], "]"]], "-", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["2", " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "c"]], "+", RowBox[List["a", " ", "d"]]]], "c"], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ArcTan", "[", RowBox[List["b", "-", FractionBox[RowBox[List["a", " ", "d"]], "c"]]], "]"]]]], "+", RowBox[List["ArcTan", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]]]]]]], "]"]]]]]], ")"]]]]]], "c"]]]]]

 Date Added to functions.wolfram.com (modification date)

 2001-10-29