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http://functions.wolfram.com/01.23.21.0247.01
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Integrate[E^(p z) Sin[a z]^m Sinh[c z]^\[Mu] Csch[c z], z] ==
(1/(c + p - c \[Mu])) ((2^(1 - m - \[Mu]) E^((c + p) z)
(-E^((-c) z) + E^(c z))^\[Mu] Binomial[m, m/2]
Hypergeometric2F1[(c + p - c \[Mu])/(2 c), 1 - \[Mu],
(3 c + p - c \[Mu])/(2 c), E^(2 c z)] (-1 + Mod[m, 2]))/
(1 - E^(2 c z))^\[Mu]) - (2^(1 - m) E^(c z) Sinh[c z]^\[Mu]
Sum[(-1)^k Binomial[m, k] (E^((I m Pi)/2 + (I a (2 k - m) + p) z)
(Hypergeometric2F1[(I a (2 k - m) + p - c \[Mu] + c)/(2 c),
-\[Mu] + 1, (I a (2 k - m) + p + c (3 - \[Mu]))/(2 c), E^(2 c z)]/
(I a (2 k - m) + p + c (-\[Mu] + 1))) +
E^((-(1/2)) I m Pi + (I a (-2 k + m) + p) z)
(Hypergeometric2F1[(I a (-2 k + m) + p - c \[Mu] + c)/(2 c),
-\[Mu] + 1, (I a (-2 k + m) + p + c (3 - \[Mu]))/(2 c), E^(2 c z)]/
(I a (-2 k + m) + p + c (-\[Mu] + 1)))),
{k, 0, Floor[(1/2) (-1 + m)]}])/(1 - E^(2 c z))^\[Mu] /;
Element[m, Integers] && m > 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> ⅇ </mi> <mrow> <mi> p </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mi> m </mi> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mi> μ </mi> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <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> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mi> p </mi> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> μ </mi> </msup> <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> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> m </mi> </mtd> </mtr> <mtr> 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<mi> p </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> <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> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], " ", "c"]], "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], " ", "c"]], "+", RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], 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Date Added to functions.wolfram.com (modification date)
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