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 Cosh

 http://functions.wolfram.com/01.20.21.4297.01

 Input Form

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

 Standard Form

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")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "c"]], ")"]]]]], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "e"]]]], ")"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18