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 Cosh

 http://functions.wolfram.com/01.20.21.0863.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18