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 Cosh

 http://functions.wolfram.com/01.20.21.4876.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], RowBox[List["Sin", "[", RowBox[List["d", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["a", "-", "b"]]]]], RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "e", "+", "p"]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "e", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "e", "+", "p"]], RowBox[List["2", " ", "e"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", RowBox[List["3", " ", "e"]], "+", "p"]], RowBox[List["2", " ", "e"]]], ",", 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["\[ImaginaryI]", " ", "d"]], "+", "e", "+", "p"]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "e", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "e", "+", "p"]], RowBox[List["2", " ", "e"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", RowBox[List["3", " ", "e"]], "+", "p"]], RowBox[List["2", " ", "e"]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "b"]]]]], "]"]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Sin", "[", RowBox[List["d_", " ", "z_"]], "]"]]]], RowBox[List[RowBox[List["a_", " ", RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "e", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "e", "+", "p"]], RowBox[List["2", " ", "e"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", RowBox[List["3", " ", "e"]], "+", "p"]], RowBox[List["2", " ", "e"]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "b"]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "e", "+", "p"]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "e", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "e", "+", "p"]], RowBox[List["2", " ", "e"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", RowBox[List["3", " ", "e"]], "+", "p"]], RowBox[List["2", " ", "e"]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "b"]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "e", "+", "p"]]]]], ")"]]]], RowBox[List["a", "-", "b"]]]]]]]]]

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

 2002-12-18