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 Cos

 http://functions.wolfram.com/01.07.21.2698.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18