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 Csch

 http://functions.wolfram.com/01.23.21.0074.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z cos m ( b z ) csch ( c z ) z - 2 1 - m ( c + p ) z ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] n ! ( 1 - m mod 2 \$CellContext`m 2 ) j = 0 n ( - 1 ) j z n - j ( c + p ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( c + p 2 c , , c + p 2 c , 1 ; c + p 2 c + 1 , , c + p 2 c + 1 ; 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[FractionBox[RowBox[List["c", "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] - 2 1 - m n ! c z k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( p - b ( m - 2 k ) ) z j = 0 n ( - 1 ) j z n - j ( c - b ( m - 2 k ) + p ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( c - b ( m - 2 k ) + p 2 c , , c - b ( m - 2 k ) + p 2 c , 1 ; c - b ( m - 2 k ) + p 2 c + 1 , , c - b ( m - 2 k ) + p 2 c + 1 ; 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[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + ( b ( m - 2 k ) + p ) z j = 0 n ( - 1 ) j z n - j ( c + b ( m - 2 k ) + p ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( c + b ( m - 2 k ) + p 2 c , , c + b ( m - 2 k ) + p 2 c , 1 ; c + b ( m - 2 k ) + p 2 c + 1 , , c + b ( m - 2 k ) + p 2 c + 1 ; 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[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) /; n m + Condition z z n p z b z m c z -1 2 1 -1 m c p z Binomial m m 2 -1 n 1 -1 \$CellContext`m 2 j 0 n -1 j z n -1 j c p -1 j -1 n -1 j -1 HypergeometricPFQ c p 2 c -1 c p 2 c -1 1 c p 2 c -1 1 c p 2 c -1 1 2 c z -1 2 1 -1 m n c z k 0 m -1 2 -1 Binomial m k p -1 b m -1 2 k z j 0 n -1 j z n -1 j c -1 b m -1 2 k p -1 j -1 n -1 j -1 HypergeometricPFQ c -1 b m -1 2 k p 2 c -1 c -1 b m -1 2 k p 2 c -1 1 c -1 b m -1 2 k p 2 c -1 1 c -1 b m -1 2 k p 2 c -1 1 2 c z b m -1 2 k p z j 0 n -1 j z n -1 j c b m -1 2 k p -1 j -1 n -1 j -1 HypergeometricPFQ c b m -1 2 k p 2 c -1 c b m -1 2 k p 2 c -1 1 c b m -1 2 k p 2 c -1 1 c b m -1 2 k p 2 c -1 1 2 c z n m SuperPlus [/itex]

 Rule Form

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

 2002-12-18