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 Coth

 http://functions.wolfram.com/01.22.21.0030.01

 Input Form

 Integrate[Coth[a ArcSinh[z]], z] == (-(1/(-2 + 8 a^2))) (((1 + 2 a) E^(ArcSinh[z] + 2 a ArcSinh[z]) Hypergeometric2F1[1 - 1/(2 a), 1, 2 - 1/(2 a), E^(2 a ArcSinh[z])] + (-1 + 2 a) (E^((3 + 2 a) ArcSinh[z]) Hypergeometric2F1[1 + 1/(2 a), 1, 2 + 1/(2 a), E^(2 a ArcSinh[z])] + (1 + 2 a) E^ArcSinh[z] (-Hypergeometric2F1[-(1/(2 a)), 1, 1 - 1/(2 a), E^(2 a ArcSinh[z])] + E^(2 ArcSinh[z]) Hypergeometric2F1[1/(2 a), 1, 1 + 1/(2 a), E^(2 a ArcSinh[z])])))/E^(2 ArcSinh[z]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Coth", "[", RowBox[List["a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["8", " ", SuperscriptBox["a", "2"]]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["ArcSinh", "[", "z", "]"]], "+", RowBox[List["2", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSinh", "[", "z", "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["1", "-", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", RowBox[List["2", " ", "a"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 coth ( a sinh - 1 ( z ) ) z - 1 8 a 2 - 2 ( - 2 sinh - 1 ( z ) ( ( 2 a - 1 ) ( ( 2 a + 1 ) sinh - 1 ( z ) ( 2 sinh - 1 ( z ) 2 F 1 ( 1 2 a , 1 ; 1 + 1 2 a ; 2 a sinh - 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["1", 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["1", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List[SuperscriptBox["sinh", RowBox[List["-", "1"]]], "(", "z", ")"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - 2 F 1 ( - 1 2 a , 1 ; 1 - 1 2 a ; 2 a sinh - 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["1", 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["1", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List[SuperscriptBox["sinh", RowBox[List["-", "1"]]], "(", "z", ")"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) + ( 2 a + 3 ) sinh - 1 ( z ) 2 F 1 ( 1 + 1 2 a , 1 ; 2 + 1 2 a ; 2 a sinh - 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["1", 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["1", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List[SuperscriptBox["sinh", RowBox[List["-", "1"]]], "(", "z", ")"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) + ( 2 a + 1 ) 2 a sinh - 1 ( z ) + sinh - 1 ( z ) 2 F 1 ( 1 - 1 2 a , 1 ; 2 - 1 2 a ; 2 a sinh - 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["1", 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["1", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List[SuperscriptBox["sinh", 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 1 8 a 2 -2 -1 -2 z 2 a -1 2 a 1 z 2 z Hypergeometric2F1 1 2 a -1 1 1 1 2 a -1 2 a z -1 Hypergeometric2F1 -1 1 2 a -1 1 1 -1 1 2 a -1 2 a z 2 a 3 z Hypergeometric2F1 1 1 2 a -1 1 2 1 2 a -1 2 a z 2 a 1 2 a z z Hypergeometric2F1 1 -1 1 2 a -1 1 2 -1 1 2 a -1 2 a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["Coth", "[", RowBox[List["a_", " ", RowBox[List["ArcSinh", "[", "z_", "]"]]]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["ArcSinh", "[", "z", "]"]], "+", RowBox[List["2", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSinh", "[", "z", "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["1", "-", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", RowBox[List["2", " ", "a"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox["1", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["8", " ", SuperscriptBox["a", "2"]]]]]]]]]]]]

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

 2002-12-18