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http://functions.wolfram.com/06.33.21.0130.01
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Integrate[z^2 FresnelS[b z] FresnelC[a z], z] ==
(1/72) (((3 Sqrt[2] a)/(b Pi^2 z))
(-(Sqrt[I (a^2 - b^2) z^2]/(a^2 - b^2)^2) - Sqrt[I (-a^2 + b^2) z^2]/
(a^2 - b^2)^2 - Sqrt[(-I) (a^2 + b^2) z^2]/(a^2 + b^2)^2 -
Sqrt[I (a^2 + b^2) z^2]/(a^2 + b^2)^2) + 24 z^3 FresnelC[a z]
FresnelS[b z] + (12 (-2 a^4 + a^2 b^2 - 2 b^4)
FresnelS[Sqrt[a^2 - b^2] z])/(a^3 b^3 Sqrt[a^2 - b^2] Pi^2) +
(12 (2 a^4 + a^2 b^2 + 2 b^4) FresnelS[Sqrt[a^2 + b^2] z])/
(a^3 b^3 Sqrt[a^2 + b^2] Pi^2) - (24/(a^3 Pi^2)) FresnelS[b z]
(2 Cos[(1/2) a^2 Pi z^2] + a^2 Pi z^2 Sin[(1/2) a^2 Pi z^2]) +
(24/(b^3 Pi^2)) FresnelC[a z] (b^2 Pi z^2 Cos[(1/2) b^2 Pi z^2] -
2 Sin[(1/2) b^2 Pi z^2]) - (24 z Cos[(1/2) b^2 Pi z^2]
Sin[(1/2) a^2 Pi z^2])/(a b Pi^2))
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</msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> 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<mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 24 </mn> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> 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<apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <sin /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cos /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <ci> 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type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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