Integrate[z^n Cos[b z] Cosh[c z^2], z] == (1/8) (((-(-c)^(-1 - n)) Sum[2^(j - n) (I b)^(-j + n) ((-I) b - 2 c z)^(1 + j) (((-I) b - 2 c z)^2/c)^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, ((-I) b - 2 c z)^2/(4 c)], {j, 0, n}])/E^(b^2/(4 c)) - ((-c)^(-1 - n) Sum[2^(j - n) ((-I) b)^(-j + n) (I b - 2 c z)^(1 + j) ((I b - 2 c z)^2/c)^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, (I b - 2 c z)^2/(4 c)], {j, 0, n}])/E^(b^2/(4 c)) - c^(-1 - n) E^(b^2/(4 c)) Sum[2^(j - n) (I b)^(-j + n) ((-I) b + 2 c z)^(1 + j) (-(((-I) b + 2 c z)^2/c))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -(((-I) b + 2 c z)^2/(4 c))], {j, 0, n}] - c^(-1 - n) E^(b^2/(4 c)) Sum[2^(j - n) ((-I) b)^(-j + n) (I b + 2 c z)^(1 + j) (-((I b + 2 c z)^2/c))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((I b + 2 c z)^2/(4 c))], {j, 0, n}]) /; Element[n, Integers] && n >= 0

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</mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mi> c </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity]], List[TagBox[&quot;j&quot;, Identity]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <msup> <mrow> <mo> ( </mo> 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<munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> j </mi> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mi> c </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity]], List[TagBox[&quot;j&quot;, Identity]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> 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2002-12-18