\[ \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx \]
Optimal antiderivative \[ -\frac {x^{3}}{6 b \pi }+\frac {2 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{b^{4} \pi ^{2}}+\frac {x^{2} \mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {x \sin \left (b^{2} \pi \,x^{2}\right )}{4 b^{3} \pi ^{2}}-\frac {5 \,\mathrm {S}\left (b x \sqrt {2}\right ) \sqrt {2}}{8 b^{4} \pi ^{2}} \]
command
integrate(x^3*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {4 \, \pi b^{4} x^{3} - 48 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 15 \, \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 12 \, {\left (2 \, \pi b^{3} x^{2} \operatorname {S}\left (b x\right ) + b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{24 \, \pi ^{2} b^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right ), x\right ) \]