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http://functions.wolfram.com/01.20.21.3967.01
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Integrate[z^n E^(p z) Cos[c z + d] Cosh[a z]^\[Nu], z] ==
((n!/2) Cosh[a z]^\[Nu] (E^(I d + (p + I c) z)
Sum[(((-1)^j z^(n - j))/((n - j)! (p + I c - a \[Nu])^(j + 1)))
HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, j + 1],
-\[Nu]}, {1 + Subscript[c, 1], \[Ellipsis],
1 + Subscript[c, j + 1]}, -E^(2 a z)], {j, 0, n}] +
E^((-I) d + (p - I c) z) Sum[(((-1)^j z^(n - j))/
((n - j)! (p - I c - a \[Nu])^(j + 1))) HypergeometricPFQ[
{Subscript[d, 1], \[Ellipsis], Subscript[d, j + 1], -\[Nu]},
{1 + Subscript[d, 1], \[Ellipsis], 1 + Subscript[d, j + 1]},
-E^(2 a z)], {j, 0, n}]))/(1 + E^(2 a z))^\[Nu] /;
Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] ==
(p + I c - a \[Nu])/(2 a) && Subscript[d, 1] == Subscript[d, 2] ==
\[Ellipsis] == Subscript[d, n + 1] == (p - I c - a \[Nu])/(2 a) &&
Element[n, Integers] && n >= 0
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</mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", "p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", "p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], 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){kind=small}
