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 ArcTan

 http://functions.wolfram.com/01.14.21.0008.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2001-10-29