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 Tanh

 http://functions.wolfram.com/01.21.21.0031.01

 Input Form

 Integrate[Tanh[a ArcCos[z]], z] == (1/(2 (1 + 4 a^2))) (((1 - 2 I a) E^(2 a ArcCos[z]) Hypergeometric2F1[1 - I/(2 a), 1, 2 - I/(2 a), -E^(2 a ArcCos[z])] + (1 + 2 I a) (E^(2 (I + a) ArcCos[z]) Hypergeometric2F1[1 + I/(2 a), 1, 2 + I/(2 a), -E^(2 a ArcCos[z])] + I (I + 2 a) (Hypergeometric2F1[-(I/(2 a)), 1, 1 - I/(2 a), -E^(2 a ArcCos[z])] + E^(2 I ArcCos[z]) Hypergeometric2F1[I/(2 a), 1, 1 + I/(2 a), -E^(2 a ArcCos[z])])))/E^(I ArcCos[z]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Tanh", "[", RowBox[List["a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]], ")"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", "a"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 tanh ( a cos - 1 ( z ) ) z 1 2 ( 4 a 2 + 1 ) ( - cos - 1 ( z ) ( ( 1 + 2 a ) ( ( 2 a + ) ( 2 cos - 1 ( z ) 2 F 1 ( 2 a , 1 ; 1 + 2 a ; - 2 a cos - 1 ( z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List[SuperscriptBox["cos", RowBox[List["-", "1"]]], "(", "z", ")"]]]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 2 F 1 ( - 2 a , 1 ; 1 - 2 a ; - 2 a cos - 1 ( z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List[SuperscriptBox["cos", RowBox[List["-", "1"]]], "(", "z", ")"]]]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) + 2 ( a + ) cos - 1 ( z ) 2 F 1 ( 1 + 2 a , 1 ; 2 + 2 a ; - 2 a cos - 1 ( z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List[SuperscriptBox["cos", RowBox[List["-", "1"]]], "(", "z", ")"]]]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) + ( 1 - 2 a ) 2 a cos - 1 ( z ) 2 F 1 ( 1 - 2 a , 1 ; 2 - 2 a ; - 2 a cos - 1 ( z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List[SuperscriptBox["cos", RowBox[List["-", "1"]]], "(", "z", ")"]]]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) ) z a z 1 2 4 a 2 1 -1 -1 z 1 2 a 2 a 2 z Hypergeometric2F1 2 a -1 1 1 2 a -1 -1 2 a z Hypergeometric2F1 -1 2 a -1 1 1 -1 2 a -1 -1 2 a z 2 a z Hypergeometric2F1 1 2 a -1 1 2 2 a -1 -1 2 a z 1 -1 2 a 2 a z Hypergeometric2F1 1 -1 2 a -1 1 2 -1 2 a -1 -1 2 a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["Tanh", "[", RowBox[List["a_", " ", RowBox[List["ArcCos", "[", "z_", "]"]]]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", "a"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]], ")"]]]]]]]]]

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

 2002-12-18