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 Cosh

 http://functions.wolfram.com/01.20.21.4700.01

 Input Form

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

 Standard Form

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 MathML Form

 p z cosh ( d z ) ( a sinh ( e z ) + b cosh ( e z ) ) 2 z 2 ( a - b ) 2 ( ( 2 e - d + p ) z 2 e - d + p 2 F 1 ( p - d 2 e + 1 , 2 ; p - d 2 e + 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[RowBox[List[FractionBox[RowBox[List["p", "-", "d"]], RowBox[List["2", " ", "e"]]], "+", "1"]], Hypergeometric2F1], ",", TagBox["2", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["p", "-", "d"]], RowBox[List["2", " ", "e"]]], "+", "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 e + d + p ) z 2 e + d + p 2 F 1 ( d + p 2 e + 1 , 2 ; d + p 2 e + 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[RowBox[List[FractionBox[RowBox[List["d", "+", "p"]], RowBox[List["2", " ", "e"]]], "+", "1"]], Hypergeometric2F1], ",", TagBox["2", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["d", "+", "p"]], RowBox[List["2", " ", "e"]]], "+", "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 p z d z a e z b e z 2 -1 2 a -1 b 2 -1 2 e -1 d p z 2 e -1 d p -1 Hypergeometric2F1 p -1 d 2 e -1 1 2 p -1 d 2 e -1 2 a b 2 e z a -1 b -1 2 e d p z 2 e d p -1 Hypergeometric2F1 d p 2 e -1 1 2 d p 2 e -1 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[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Cosh", "[", 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_"]], "]"]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "e"]], "-", "d", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "d"]], "+", "p"]], RowBox[List["2", " ", "e"]]]]], ",", "2", ",", RowBox[List["2", "+", FractionBox[RowBox[List[RowBox[List["-", "d"]], "+", "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["2", " ", "e"]], "-", "d", "+", "p"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "e"]], "+", "d", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["d", "+", "p"]], RowBox[List["2", " ", "e"]]]]], ",", "2", ",", RowBox[List["2", "+", FractionBox[RowBox[List["d", "+", "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["2", " ", "e"]], "+", "d", "+", "p"]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"]]]]]]

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

 2002-12-18