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 Cosh

 http://functions.wolfram.com/01.20.21.4714.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18