Integrate[z^n E^(p Sqrt[z]) Sinh[b Sqrt[z]]^m Cosh[c Sqrt[z]], z] == (-(2 I)^(-m)) Binomial[m, m/2] (Gamma[2 (1 + n), (-c - p) Sqrt[z]]/ (-c - p)^(2 (1 + n)) + Gamma[2 (1 + n), (c - p) Sqrt[z]]/ (c - p)^(2 (1 + n))) (1 - Mod[m, 2]) - Sum[(-1)^k Binomial[m, k] ((E^((I m Pi)/2) Gamma[2 (1 + n), (-c - b (-2 k + m) - p) Sqrt[z]])/ (-c - b (-2 k + m) - p)^(2 (1 + n)) + (E^((I m Pi)/2) Gamma[2 (1 + n), (c - b (-2 k + m) - p) Sqrt[z]])/ (c - b (-2 k + m) - p)^(2 (1 + n)) + Gamma[2 (1 + n), (-c + b (-2 k + m) - p) Sqrt[z]]/ (E^((1/2) I m Pi) (-c + b (-2 k + m) - p)^(2 (1 + n))) + Gamma[2 (1 + n), (c + b (-2 k + m) - p) Sqrt[z]]/ (E^((1/2) I m Pi) (c + b (-2 k + m) - p)^(2 (1 + n)))), {k, 0, Floor[(1/2) (-1 + m)]}]/(2 I)^m /; Element[m, Integers] && m > 0 && Element[n, Integers] && n >= 0

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&#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> 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<mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> m </mi> </mtd> </mtr> <mtr> <mtd> <mfrac> <mi> m </mi> <mn> 2 </mn> </mfrac> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, 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2002-12-18