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 Cosh

 http://functions.wolfram.com/01.20.21.0959.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", " ", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["n", "!"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["a", " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "n"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "p"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", SuperscriptBox["z", RowBox[List["n", "-", "p"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", " ", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "p"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["p", "+", "1"]]], ",", RowBox[List["-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["p", "+", "1"]]]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "n"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "p"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", SuperscriptBox["z", RowBox[List["n", "-", "p"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", " ", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "p"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", RowBox[List["p", "+", "1"]]], ",", RowBox[List["-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", RowBox[List["p", "+", "1"]]]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], ")"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["b", "1"], "\[Equal]", SubscriptBox["b", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["b", RowBox[List["n", "+", "1"]]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n sin μ ( c z ) cosh ( a z + b ) z 1 2 ( 1 - 2 c z ) - μ n ! sin μ ( c z ) ( - b - a z p = 0 n ( - 1 ) p z n - p ( - a - c μ ) - p - 1 ( n - p ) ! p + 2 F p + 1 ( a - c μ 2 c , , a - c μ 2 c , - μ ; a - c μ 2 c + 1 , , a - c μ 2 c + 1 ; 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[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], 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]]]]] + b + a z p = 0 n ( - 1 ) p z n - p ( a - c μ ) - p - 1 ( n - p ) ! p + 2 F p + 1 ( - a + c μ 2 c , , - a + c μ 2 c , - μ ; - a + c μ 2 c + 1 , , - a + c μ 2 c + 1 ; 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[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], "+", "1"]], 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 b 1 2 1 -1 2 c z -1 μ n c z μ -1 b -1 a z p 0 n -1 p z n -1 p -1 a -1 c μ -1 p -1 n -1 p -1 HypergeometricPFQ a -1 c μ 2 c -1 a -1 c μ 2 c -1 -1 μ a -1 c μ 2 c -1 1 a -1 c μ 2 c -1 1 2 c z b a z p 0 n -1 p z n -1 p a -1 c μ -1 p -1 n -1 p -1 HypergeometricPFQ -1 a c μ 2 c -1 -1 a c μ 2 c -1 -1 μ -1 a c μ 2 c -1 1 -1 a c μ 2 c -1 1 2 c z n SuperPlus [/itex]

 Rule Form

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

 2002-12-18