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 Cosh

 http://functions.wolfram.com/01.20.21.4291.01

 Input Form

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

 Standard Form

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 MathML Form

 sinh ( d z ) ( a sinh 2 ( e z ) + b cosh 2 ( e z ) ) 2 z 1 2 ( 1 2 a 3 / 2 b 3 / 2 ( a + b ) ( 2 e - d ) ( ( 2 e - d ) z ( ( a - b ) 2 F 1 ( 1 - d 2 e , 1 ; 2 - d 2 e ; ( 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["1", "-", FractionBox["d", RowBox[List["2", " ", "e"]]]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", FractionBox["d", RowBox[List["2", " ", "e"]]]]], 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 ( 1 - d 2 e , 1 ; 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d 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["d", RowBox[List["2", " ", "e"]]], "+", "1"]], Hypergeometric2F1], ",", TagBox["2", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["d", 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 d z a e z 2 b e z 2 2 -1 1 2 1 2 a 3 2 b 3 2 a b 2 e -1 d -1 2 e -1 d z a -1 b Hypergeometric2F1 1 -1 d 2 e -1 1 2 -1 d 2 e -1 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 1 -1 d 2 e -1 1 2 -1 d 2 e -1 a b 2 e z a 1 2 b 1 2 2 -1 -1 2 a 1 2 b 1 2 Hypergeometric2F1 1 -1 d 2 e -1 2 2 -1 d 2 e -1 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 1 -1 d 2 e -1 2 2 -1 d 2 e -1 a b 2 e z a 1 2 b 1 2 2 -1 -1 1 2 a 3 2 b 3 2 a b d 2 e -1 d 2 e z a -1 b Hypergeometric2F1 d 2 e -1 1 1 d 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 d 2 e -1 1 1 d 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 d 2 e -1 1 2 d 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 d 2 e -1 1 2 d 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