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 Cosh

 http://functions.wolfram.com/01.20.21.4897.01

 Input Form

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

 Standard Form

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RowBox[List["(", RowBox[List["a", "+", "c"]], ")"]]]]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "c"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", RowBox[List["2", " ", "e"]], "+", "p"]], ")"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 p z cos ( d z ) ( a + b sinh 2 ( e z ) + c cosh 2 ( e z ) ) 2 z 1 2 ( - ( ( b + c ) ( 2 e - d + p ) z ( 2 ( a - b ) ( a + c ) ( 2 F 1 ( - d + p 2 e + 1 , 2 ; - d + p 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[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "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[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "p"]], 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 + p 2 e + 1 , 2 ; 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 Rule Form

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

 2002-12-18