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 Cosh

 http://functions.wolfram.com/01.20.21.1335.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18