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 Cosh

 http://functions.wolfram.com/01.20.21.4708.01

 Input Form

 Integrate[E^(p z)/(a Sinh[e z]^2 + b Cosh[e z]^2)^2, z] == -(I E^((2 e + p) z) ((Sqrt[a] + I Sqrt[b])^2 (a - b) Hypergeometric2F1[1 + p/(2 e), 1, 2 + p/(2 e), ((a + b) E^(2 e z))/ (Sqrt[a] - I Sqrt[b])^2] - (Sqrt[a] - I Sqrt[b])^2 (a - b) Hypergeometric2F1[1 + p/(2 e), 1, 2 + p/(2 e), ((a + b) E^(2 e z))/ (Sqrt[a] + I Sqrt[b])^2] - 2 I Sqrt[a] Sqrt[b] ((Sqrt[a] + I Sqrt[b])^2 Hypergeometric2F1[1 + p/(2 e), 2, 2 + p/(2 e), ((a + b) E^(2 e z))/(Sqrt[a] - I Sqrt[b])^2] + (Sqrt[a] - I Sqrt[b])^2 Hypergeometric2F1[1 + p/(2 e), 2, 2 + p/(2 e), ((a + b) E^(2 e z))/(Sqrt[a] + I Sqrt[b])^2])))/ (2 a^(3/2) b^(3/2) (a + b) (2 e + p))

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18