\[ \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx \]
Optimal antiderivative \[ -\frac {x^{2}}{4 b \pi }-\frac {\mathrm {S}\left (b x \right )^{2}}{2 b^{3} \pi }+\frac {x \,\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\sin \left (b^{2} \pi \,x^{2}\right )}{4 b^{3} \pi ^{2}} \]
command
integrate(x^2*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
\[ -\frac {\pi b^{2} x^{2} + 2 \, \pi \operatorname {S}\left (b x\right )^{2} - 2 \, {\left (2 \, \pi b x \operatorname {S}\left (b x\right ) + \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{4 \, \pi ^{2} b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right ), x\right ) \]