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 Cosh

 http://functions.wolfram.com/01.20.21.4909.01

 Input Form

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

 Standard Form

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 MathML Form

 p z cos ( d z ) ( a sinh ( e z ) + b cosh ( e z ) ) β z 1 2 ( - 1 d - p + e β ( 2 - β ( - d + p ) z ( 2 e z ( a + b ) b - a + 1 ) - β ( - e z ( a ( - 1 + 2 e z ) + b ( 1 + 2 e z ) ) ) β 2 F 1 ( - d + p - e β 2 e , - β ; 1 2 ( - d + p e - β + 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[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "p", "-", RowBox[List["e", " ", "\[Beta]"]]]], RowBox[List["2", " ", "e"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "p"]], "e"], "-", "\[Beta]", "+", "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] ) - 1 - d - p + e β ( 2 - β ( d + p ) z ( 2 e z ( a + b ) b - a + 1 ) - β ( - e z ( a ( - 1 + 2 e z ) + b ( 1 + 2 e z ) ) ) β 2 F 1 ( d + p - e β 2 e , - β ; 1 2 ( d + p e - β + 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[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p", "-", RowBox[List["e", " ", "\[Beta]"]]]], RowBox[List["2", " ", "e"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], "e"], "-", "\[Beta]", "+", "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 β 1 2 -1 1 d -1 p e β -1 2 -1 β -1 d p z 2 e z a b b -1 a -1 1 -1 β -1 e z a -1 2 e z b 1 2 e z β Hypergeometric2F1 -1 d p -1 e β 2 e -1 -1 β 1 2 -1 d p e -1 -1 β 2 a b 2 e z a -1 b -1 -1 1 -1 d -1 p e β -1 2 -1 β d p z 2 e z a b b -1 a -1 1 -1 β -1 e z a -1 2 e z b 1 2 e z β Hypergeometric2F1 d p -1 e β 2 e -1 -1 β 1 2 d p e -1 -1 β 2 a b 2 e z a -1 b -1 [/itex]

 Rule Form

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

 2002-12-18