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 Cosh

 http://functions.wolfram.com/01.20.21.4254.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sinh", "[", RowBox[List["d", " ", "z"]], "]"]], RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "d"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "1", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "1", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["b", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], " ", RowBox[List["(", RowBox[List["c", "-", "d"]], ")"]]]], ")"]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "1", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "1", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["b", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], " ", RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18