Optimal antiderivative \[ \frac {2 x}{3 b^{2} \pi ^{2}}+\frac {x \cos \left (b^{2} \pi \,x^{2}\right )}{6 b^{2} \pi ^{2}}+\frac {2 x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{3 b \pi }+\frac {x^{3} \mathrm {S}\left (b x \right )^{2}}{3}-\frac {4 \,\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} \pi ^{2}}-\frac {5 \FresnelC \left (b x \sqrt {2}\right ) \sqrt {2}}{12 b^{3} \pi ^{2}} \]
command
integrate(x^2*fresnel_sin(b*x)^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {4 \, \pi ^{2} b^{4} x^{3} \operatorname {S}\left (b x\right )^{2} + 8 \, \pi b^{3} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 4 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 6 \, b^{2} x - 16 \, b \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 5 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{12 \, \pi ^{2} b^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{2} {\rm fresnels}\left (b x\right )^{2}, x\right ) \]