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http://functions.wolfram.com/01.19.21.3118.01
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Integrate[E^(p z) Sin[c z] Sinh[a z]^\[Nu], z] ==
(2^(-1 - \[Nu]) E^(((-I) c + p) z) (-E^((-a) z) + E^(a z))^\[Nu]
((((-I) c - p + a \[Nu]) Hypergeometric2F1[((-I) c + p - a \[Nu])/(2 a),
-\[Nu], (2 a - I c + p - a \[Nu])/(2 a), E^(2 a z)] +
E^(2 I c z) ((-I) c + p - a \[Nu]) Hypergeometric2F1[
(I c + p - a \[Nu])/(2 a), -\[Nu], (2 a + I c + p - a \[Nu])/(2 a),
E^(2 a z)])/((I c + p - a \[Nu]) (c + I (p - a \[Nu])))))/
(1 - E^(2 a z))^\[Nu]
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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> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> p </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mi> ν </mi> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> 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type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> c </ci> </apply> <ci> p </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> ν </ci> </apply> </apply> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> ν </ci> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <apply> <times /> <ci> a </ci> <ci> ν </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> c </ci> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> ν </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> 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Date Added to functions.wolfram.com (modification date)
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