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http://functions.wolfram.com/01.24.21.0486.01
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Integrate[z^n Cosh[q \[Nu] c z] Sech[c z]^\[Nu], z] ==
(n!/2) Sech[c z]^\[Nu] (1 + E^(2 c z))^\[Nu]
(E^(q \[Nu] c z) Sum[(((-1)^j z^(-j + n))/((n - j)! (c \[Nu] (q + 1))^
(j + 1))) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis],
Subscript[a, j + 1], \[Nu]}, {1 + Subscript[a, 1], \[Ellipsis],
1 + Subscript[a, j + 1]}, -E^(2 c z)], {j, 0, n}] +
((-1)^((\[Nu] (q - 1))/2) Gamma[(\[Nu] (q + 1))/2] z^(n + 1))/
(E^(c \[Nu] z) (Gamma[1 + (\[Nu] (q - 1))/2] Gamma[\[Nu]] (n + 1)!)) -
Sum[(((-1)^k Pochhammer[\[Nu], k] z^(-j + n))/
((c (q \[Nu] - 2 k - \[Nu]))^(j + 1) (n - j)! k!))
E^(c z (2 k - q \[Nu])), {j, 0, n}, {k, 0, (\[Nu] (q - 1))/2 - 1}] +
(-1)^((\[Nu] (q - 1))/2)
Sum[((E^(c z (2 - \[Nu])) Pochhammer[\[Nu], 1 + (\[Nu] (q - 1))/2]
z^(-j + n))/((n - j)! (-2 c)^(1 + j) (1 + (\[Nu] (q - 1))/2)!))
HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, j + 2],
1 + (\[Nu] (q + 1))/2}, {1 + Subscript[b, 1], \[Ellipsis],
1 + Subscript[b, j + 1], 2 + (\[Nu] (q - 1))/2}, -E^(2 c z)],
{j, 0, n}]) /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] ==
Subscript[a, n + 1] == (\[Nu] (q + 1))/2 &&
Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 2] ==
1 && Element[n, Integers] && n >= 0 && Element[(\[Nu] (q - 1))/2,
Integers] && (\[Nu] (q - 1))/2 >= 0
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mo> ∫ </mo> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> q </mi> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mi> ν </mi> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mi> ν </mi> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> q </mi> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> j </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <msub> <mi> F </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mi> ν </mi> </mrow> <mo> ; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], 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TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mrow> <mfrac> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], RowBox[List[FractionBox[RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["q", "-", "1"]], ")"]]]], "2"], "+", "1"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> 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Date Added to functions.wolfram.com (modification date)
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