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 Cosh

 http://functions.wolfram.com/01.20.21.4523.01

 Input Form

 Integrate[z^n Sinh[c z + d]^\[Mu] Cosh[a z]^v, z] == ((-2^(-v)) n! Sinh[d + c z]^\[Mu] (Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (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 (d + c z))]), {j, 0, n}] - Sum[(Binomial[v, k] (E^(2 v a z) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (-2 a k + a 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 (d + c z))]), {j, 0, n}] + E^(4 k a z) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (2 a k - a 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 (d + c z))]), {j, 0, n}]))/E^((2 k + v) a z), {k, 0, Floor[(1/2) (-1 + v)]}]))/ (1 - E^(-2 (d + c z)))^\[Mu] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == -(\[Mu]/2) && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == -((a (-2 k + v) + c \[Mu])/(2 c)) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == -(((-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 sinh μ ( d + c z ) cosh v ( a z ) z - 2 - v ( 1 - - 2 ( d + c z ) ) - μ n ! sinh μ ( d + c z ) ( ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) j = 0 n ( - 1 ) j z n - j ( c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - μ 2 , , - μ 2 , - μ ; 1 - μ 2 , , 1 - μ 2 ; - 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["\[Mu]", "2"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox["\[Mu]", "2"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Mu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] - k = 0 v - 1 2 - ( 2 k + v ) a z ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 4 k a z j = 0 n ( - 1 ) j z n - j ( 2 a k - a v + c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - c μ - a ( v - 2 k ) 2 c , , - c μ - a ( v - 2 k ) 2 c , - μ ; 1 - c μ - a ( v - 2 k ) 2 c , , 1 - c μ - a ( v - 2 k ) 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["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], 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["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + 2 v a z j = 0 n ( - 1 ) j z n - j ( - 2 a k + a v + c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - a ( v - 2 k ) + c μ 2 c , , - a ( v - 2 k ) + c μ 2 c , - μ ; 1 - a ( v - 2 k ) + c μ 2 c , , 1 - a ( v - 2 k ) + 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["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[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[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[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"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) ) /; n v + Condition z z n d c z μ a z v -1 2 -1 v 1 -1 -2 d c z -1 μ n d c z μ Binomial v v 2 -1 \$CellContext`v 2 -1 j 0 n -1 j z n -1 j c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 μ 2 -1 -1 μ 2 -1 -1 μ 1 -1 μ 2 -1 1 -1 μ 2 -1 -2 d c z -1 k 0 v -1 2 -1 -1 2 k v a z Binomial v k 4 k a z j 0 n -1 j z n -1 j 2 a k -1 a v c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 c μ -1 a v -1 2 k 2 c -1 -1 c μ -1 a v -1 2 k 2 c -1 -1 μ 1 -1 c μ -1 a v -1 2 k 2 c -1 1 -1 c μ -1 a v -1 2 k 2 c -1 -2 d c z 2 v a z j 0 n -1 j z n -1 j -2 a k a v c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 a v -1 2 k c μ 2 c -1 -1 a v -1 2 k c μ 2 c -1 -1 μ 1 -1 a v -1 2 k c μ 2 c -1 1 -1 a v -1 2 k c μ 2 c -1 -2 d c z n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18