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 Cosh

 http://functions.wolfram.com/01.20.21.1427.01

 Input Form

 Integrate[z^n E^(p z) Cos[c z]^\[Mu] Cosh[a z], z] == ((1/2) E^(p z) Cos[c z]^\[Mu] n! (Sum[(1/(n - j)!) ((-1)^j z^(n - j) (-a + p - I c \[Mu])^(-1 - j) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, 1 + j], -\[Mu]}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, 1 + j]}, -E^(2 I c z)]), {j, 0, n}]/E^(a z) + E^(a z) Sum[(1/(n - j)!) ((-1)^j z^(n - j) (a + p - I c \[Mu])^(-1 - j) HypergeometricPFQ[{Subscript[d, 1], \[Ellipsis], Subscript[d, 1 + j], -\[Mu]}, {1 + Subscript[d, 1], \[Ellipsis], 1 + Subscript[d, 1 + j]}, -E^(2 I c z)]), {j, 0, n}]))/ (1 + E^(2 I c z))^\[Mu] /; Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == -((I (p - a - I c \[Mu]))/(2 c)) && Subscript[d, 1] == Subscript[d, 2] == \[Ellipsis] == Subscript[d, n + 1] == -((I (p + a - I c \[Mu]))/(2 c)) && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z cos μ ( c z ) cosh ( a z ) z 1 2 p z ( 1 + 2 c z ) - μ cos μ ( c z ) n ! ( - a z j = 0 n ( - 1 ) j z n - j ( - a + p - c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - ( - a + p - c μ ) 2 c , , - ( - a + p - c μ ) 2 c , - μ ; 1 - ( - a + p - c μ ) 2 c , , 1 - ( - a + p - c μ ) 2 c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Mu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + a z j = 0 n ( - 1 ) j z n - j ( a + p - c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - ( a + p - c μ ) 2 c , , - ( a + p - c μ ) 2 c , - μ ; 1 - ( a + p - c μ ) 2 c , , 1 - ( a + p - c μ ) 2 c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Mu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) /; n Condition z z n p z c z μ a z 1 2 p z 1 2 c z -1 μ c z μ n -1 a z j 0 n -1 j z n -1 j -1 a p -1 c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 -1 a p -1 c μ 2 c -1 -1 -1 a p -1 c μ 2 c -1 -1 μ 1 -1 -1 a p -1 c μ 2 c -1 1 -1 -1 a p -1 c μ 2 c -1 -1 2 c z a z j 0 n -1 j z n -1 j a p -1 c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 a p -1 c μ 2 c -1 -1 a p -1 c μ 2 c -1 -1 μ 1 -1 a p -1 c μ 2 c -1 1 -1 a p -1 c μ 2 c -1 -1 2 c z n [/itex]

 Rule Form

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

 2002-12-18