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 Cosh

 http://functions.wolfram.com/01.20.21.4835.01

 Input Form

 Integrate[z^n E^(p z) Sinh[c z]^\[Mu] Cosh[a z + b]^v, z] == (E^(p z) Binomial[v, v/2] n! (1 - Mod[v, 2]) Sinh[c z]^\[Mu] Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (p + 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 c z)]), {j, 0, n}])/ (2^v (1 - E^(-2 c z))^\[Mu]) + (n! Sinh[c z]^\[Mu] Sum[Binomial[v, k] (E^(b (-2 k + v) + (p + a (-2 k + v)) z) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (p + a (-2 k + v) + 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 c z)]), {j, 0, n}] + E^((-b) (-2 k + v) + (p - a (-2 k + v)) z) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (p - a (-2 k + v) + 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 c z)]), {j, 0, n}]), {k, 0, Floor[(1/2) (-1 + v)]}])/ (2^v (1 - E^(-2 c z))^\[Mu]) /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == -((p + c \[Mu])/(2 c)) && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == -((p + a (-2 k + v) + c \[Mu])/(2 c)) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == -((p - a (-2 k + v) + 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 sinh μ ( 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]] n ! ( 1 - v mod 2 \$CellContext`v 2 ) sinh μ ( c z ) ( 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 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["p", "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List["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["p", "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) ( 1 - - 2 c z ) - μ + 2 - v n ! sinh μ ( c z ) ( k = 0 v - 1 2 ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( p - a ( v - 2 k ) ) z - b ( v - 2 k ) j = 0 n ( - 1 ) j z n - j ( p - a ( v - 2 k ) + c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - p - a ( v - 2 k ) + c μ 2 c , , - p - a ( v - 2 k ) + c μ 2 c , - μ ; 1 - p - a ( v - 2 k ) + c μ 2 c , , 1 - p - a ( v - 2 k ) + 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["p", "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List["p", "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", 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["p", "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["p", "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + b ( v - 2 k ) + ( p + a ( v - 2 k ) ) z j = 0 n ( - 1 ) j z n - j ( p + a ( v - 2 k ) + c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - p + a ( v - 2 k ) + c μ 2 c , , - p + a ( v - 2 k ) + c μ 2 c , - μ ; 1 - p + a ( v - 2 k ) + c μ 2 c , , 1 - p + a ( v - 2 k ) + 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["p", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List["p", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", 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["p", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["p", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) ) ( 1 - - 2 c z ) - μ /; n v + Condition z z n p z c z μ b a z v 2 -1 v p z Binomial v v 2 -1 n 1 -1 \$CellContext`v 2 c z μ j 0 n -1 j z n -1 j p 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 -2 c z 1 -1 -2 c z -1 μ 2 -1 v n c z μ k 0 v -1 2 -1 Binomial v k p -1 a v -1 2 k z -1 b v -1 2 k j 0 n -1 j z n -1 j p -1 a v -1 2 k c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 p -1 a v -1 2 k c μ 2 c -1 -1 p -1 a v -1 2 k c μ 2 c -1 -1 μ 1 -1 p -1 a v -1 2 k c μ 2 c -1 1 -1 p -1 a v -1 2 k c μ 2 c -1 -2 c z b v -1 2 k p a v -1 2 k z j 0 n -1 j z n -1 j p a v -1 2 k c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 p a v -1 2 k c μ 2 c -1 -1 p a v -1 2 k c μ 2 c -1 -1 μ 1 -1 p a v -1 2 k c μ 2 c -1 1 -1 p a v -1 2 k c μ 2 c -1 -2 c z 1 -1 -2 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