html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 ArcTanh

 http://functions.wolfram.com/01.27.21.0008.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["ArcTanh", "[", RowBox[List["b_", "+", RowBox[List["a_", " ", "z_"]]]], "]"]], "z_"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ArcTanh", "[", "b", "]"]], "-", RowBox[List["ArcTanh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]], "2"]]], "-", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcTanh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTanh", "[", "b", "]"]], "-", RowBox[List["ArcTanh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", RowBox[List["ArcTanh", "[", "b", "]"]]]], "-", RowBox[List["2", " ", RowBox[List["ArcTanh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcTanh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["ArcTanh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcTanh", "[", 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", " ", "\[ImaginaryI]", " ", RowBox[List["ArcTanh", "[", 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[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["ArcTanh", "[", "b", "]"]], "-", RowBox[List["ArcTanh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]]]], "]"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTanh", "[", "b", "]"]], "-", RowBox[List["ArcTanh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["ArcTanh", "[", "b", "]"]], "-", RowBox[List["ArcTanh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", RowBox[List["ArcTanh", "[", "b", "]"]]]], "-", RowBox[List["2", " ", RowBox[List["ArcTanh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["ArcTanh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]

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

 2001-10-29