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 Cosh

 http://functions.wolfram.com/01.20.21.0961.01

 Input Form

 Integrate[z^n Sin[c z + d]^\[Mu] Cosh[a z + b], z] == ((1/2) n! Sin[d + c z]^\[Mu] (E^(b + a z) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (a - I c \[Mu])^ (-1 - j) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, 1 + j], -\[Mu]}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, 1 + j]}, E^(2 I (d + c z))]), {j, 0, n}] + E^(-b - a z) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (-a - 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 (d + c z))]), {j, 0, n}]))/ (1 - E^(2 I (d + c z)))^\[Mu] /; Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == -((I a + c \[Mu])/(2 c)) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, 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 ( d + c z ) cosh ( b + a z ) z 1 2 ( 1 - 2 ( d + c z ) ) - μ n ! sin μ ( d + c z ) ( - b - a z j = 0 n ( - 1 ) j z n - j ( - a - c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - c μ - a 2 c , , - c μ - a 2 c , - μ ; 1 - c μ - a 2 c , , 1 - c μ - a 2 c ; 2 ( d + 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[RowBox[List["c", " ", "\[Mu]"]], "-", RowBox[List["\[ImaginaryI]", " ", "a"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "-", RowBox[List["\[ImaginaryI]", " ", "a"]]]], 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["c", " ", "\[Mu]"]], "-", RowBox[List["\[ImaginaryI]", " ", "a"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "-", RowBox[List["\[ImaginaryI]", " ", "a"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + b + a z j = 0 n ( - 1 ) j z n - j ( a - c μ ) - j - 1 ( n - j ) ! j + 2 F j + 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 ( d + 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[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]", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) /; n Condition z z n d c z μ b a z 1 2 1 -1 2 d c z -1 μ n d c z μ -1 b -1 a z j 0 n -1 j z n -1 j -1 a -1 c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 c μ -1 a 2 c -1 -1 c μ -1 a 2 c -1 -1 μ 1 -1 c μ -1 a 2 c -1 1 -1 c μ -1 a 2 c -1 2 d c z b a z j 0 n -1 j z n -1 j a -1 c μ -1 j -1 n -1 j -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 d c z n [/itex]

 Rule Form

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

 2002-12-18