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 Cosh

 http://functions.wolfram.com/01.20.21.4402.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List["d_", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a_", " ", RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]], ")"]], "\[Beta]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], ")"]]]]]], ")"]]]], ")"]], "\[Beta]"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["e", " ", "\[Beta]"]]]], RowBox[List["2", " ", "e"]]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", FractionBox["d", "e"], "-", "\[Beta]"]], ")"]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "b"]]]]], "]"]]]], RowBox[List["d", "+", RowBox[List["e", " ", "\[Beta]"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], ")"]]]]]], ")"]]]], ")"]], "\[Beta]"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["d", "-", RowBox[List["e", " ", "\[Beta]"]]]], RowBox[List["2", " ", "e"]]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox["d", "e"], "-", "\[Beta]"]], ")"]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "b"]]]]], "]"]]]], RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["e", " ", "\[Beta]"]]]]]]], ")"]]]]]]]]

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

 2002-12-18