Integrate[Cos[b z^2 + d z + e] Cos[c z^2 + g]^v, z] == (1/Sqrt[b]) (2^(-(1/2) - v) Sqrt[Pi] Binomial[v, v/2] (1 - Mod[v, 2]) (Cos[d^2/(4 b) - e] FresnelC[(d + 2 b z)/(Sqrt[b] Sqrt[2 Pi])] + FresnelS[(d + 2 b z)/(Sqrt[b] Sqrt[2 Pi])] Sin[d^2/(4 b) - e])) + 2^(-(1/2) - v) Sqrt[Pi] Sum[Binomial[v, s] ((Cos[e - 2 g s + g v + d^2/(4 (-b + 2 c s - c v))] FresnelC[(-d + 2 (-b + 2 c s - c v) z)/(Sqrt[2 Pi] Sqrt[-b + 2 c s - c v])] + FresnelS[(-d + 2 (-b + 2 c s - c v) z)/(Sqrt[2 Pi] Sqrt[-b + 2 c s - c v])] Sin[e - 2 g s + g v + d^2/(4 (-b + 2 c s - c v))])/Sqrt[-b + 2 c s - c v] + (Cos[e + 2 g s - g v + d^2/(4 (-b - 2 c s + c v))] FresnelC[(-d + 2 (-b - 2 c s + c v) z)/(Sqrt[2 Pi] Sqrt[-b - 2 c s + c v])] + FresnelS[(-d + 2 (-b - 2 c s + c v) z)/(Sqrt[2 Pi] Sqrt[-b - 2 c s + c v])] Sin[e + 2 g s - g v + d^2/(4 (-b - 2 c s + c v))])/Sqrt[-b - 2 c s + c v]), {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[v, Integers] && v > 0

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<mrow> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> d </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mfrac> <mo> - </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> v </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> s </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> v </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> 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<mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mi> e </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> g </mi> <mo> &#8290; </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> g </mi> <mo> &#8290; </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mi> d </mi> </mrow> 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2002-12-18