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 Cosh

 http://functions.wolfram.com/01.20.21.4002.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z cos μ ( d + c z ) cosh v ( b + a z ) z 2 - v p z ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] cos μ ( d + c z ) n ! ( 1 - v mod 2 \$CellContext`v 2 ) ( j = 0 n ( - 1 ) j z n - j ( p - c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - p + c μ 2 c , , - p + c μ 2 c , - μ ; 1 - p + c μ 2 c , , 1 - p + 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]", " ", "p"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", 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]", " ", "p"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", 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] ) ( 1 + 2 ( d + c z ) ) - μ + 2 - v cos μ ( d + c z ) n ! ( s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( p - a ( v - 2 s ) ) z - b ( v - 2 s ) j = 0 n ( - 1 ) j z n - j ( p - a ( v - 2 s ) - c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - p - a ( v - 2 s ) + c μ 2 c , , - p - a ( v - 2 s ) + c μ 2 c , - μ ; 1 - p - a ( v - 2 s ) + c μ 2 c , , 1 - p - a ( v - 2 s ) + 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]", " ", "p"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", 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]", " ", "p"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", 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 ( v - 2 s ) + ( p + a ( v - 2 s ) ) z j = 0 n ( - 1 ) j z n - j ( p + a ( v - 2 s ) - c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - p + a ( v - 2 s ) + c μ 2 c , , - p + a ( v - 2 s ) + c μ 2 c , - μ ; 1 - p + a ( v - 2 s ) + c μ 2 c , , 1 - p + a ( v - 2 s ) + 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]", " ", "p"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", 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]", " ", "p"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", 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] ) ) ( 1 + 2 ( d + c z ) ) - μ /; n v + Condition z z n p z d c z μ b a z v 2 -1 v p z Binomial v v 2 -1 d c z μ n 1 -1 \$CellContext`v 2 j 0 n -1 j z n -1 j p -1 c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 p c μ 2 c -1 -1 p c μ 2 c -1 -1 μ 1 -1 p c μ 2 c -1 1 -1 p c μ 2 c -1 -1 2 d c z 1 2 d c z -1 μ 2 -1 v d c z μ n s 0 v -1 2 -1 Binomial v s p -1 a v -1 2 s z -1 b v -1 2 s j 0 n -1 j z n -1 j p -1 a v -1 2 s -1 c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 p -1 a v -1 2 s c μ 2 c -1 -1 p -1 a v -1 2 s c μ 2 c -1 -1 μ 1 -1 p -1 a v -1 2 s c μ 2 c -1 1 -1 p -1 a v -1 2 s c μ 2 c -1 -1 2 d c z b v -1 2 s p a v -1 2 s z j 0 n -1 j z n -1 j p a v -1 2 s -1 c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 p a v -1 2 s c μ 2 c -1 -1 p a v -1 2 s c μ 2 c -1 -1 μ 1 -1 p a v -1 2 s c μ 2 c -1 1 -1 p a v -1 2 s c μ 2 c -1 -1 2 d c z 1 2 d c z -1 μ n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18