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 Cosh

 http://functions.wolfram.com/01.20.21.4770.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List["d", " ", "z"]], "]"]], RowBox[List[RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", RowBox[List["2", " ", "e"]]]], ")"]], " ", "z"]]], " ", RowBox[List["(", 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 MathML Form

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

 Rule Form

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

 2002-12-18