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 Cosh

 http://functions.wolfram.com/01.20.21.0430.01

 Input Form

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

 Standard Form

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 MathML Form

 cosh ( a tan - 1 ( z ) ) z - 1 2 ( a 2 + 4 ) ( - a tan - 1 ( z ) ( ( a + 2 ) ( ( a - 2 ) ( ( 1 + 2 a tan - 1 ( z ) ) z + 2 F 1 ( a 2 , 1 ; 1 + a 2 ; - 2 tan - 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[RowBox[List["\[ImaginaryI]", " ", "a"]], "2"], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "2"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List[SuperscriptBox["tan", RowBox[List["-", "1"]]], "(", "z", ")"]]]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + 2 a tan - 1 ( z ) 2 F 1 ( - 1 2 ( a ) , 1 ; 1 - a 2 ; - 2 tan - 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[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a"]], ")"]]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "2"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List[SuperscriptBox["tan", RowBox[List["-", "1"]]], "(", "z", ")"]]]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) - a 2 tan - 1 ( z ) 2 F 1 ( 1 + a 2 , 1 ; 2 + a 2 ; - 2 tan - 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[RowBox[List["\[ImaginaryI]", " ", "a"]], "2"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "2"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List[SuperscriptBox["tan", RowBox[List["-", "1"]]], "(", "z", ")"]]]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) - a ( a - 2 ) 2 ( a + ) tan - 1 ( z ) 2 F 1 ( 1 - a 2 , 1 ; 2 - a 2 ; - 2 tan - 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[RowBox[List["\[ImaginaryI]", " ", "a"]], "2"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "2"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List[SuperscriptBox["tan", RowBox[List["-", "1"]]], "(", "z", ")"]]]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ) z a z -1 1 2 a 2 4 -1 -1 a z a 2 a -1 2 1 2 a z z Hypergeometric2F1 a 2 -1 1 1 a 2 -1 -1 2 z 2 a z Hypergeometric2F1 -1 1 2 a 1 1 -1 a 2 -1 -1 2 z -1 a 2 z Hypergeometric2F1 1 a 2 -1 1 2 a 2 -1 -1 2 z -1 a a -1 2 2 a z Hypergeometric2F1 1 -1 a 2 -1 1 2 -1 a 2 -1 -1 2 z [/itex]

 Rule Form

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

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

 2002-12-18