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 Cosh

 http://functions.wolfram.com/01.20.21.0955.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18