\[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx \]
Optimal antiderivative \[ \frac {15 x^{2}}{4 b^{5} \pi ^{3}}-\frac {x^{6}}{12 b \pi }+\frac {7 x^{2} \cos \left (b^{2} \pi \,x^{2}\right )}{4 b^{5} \pi ^{3}}+\frac {5 x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{b^{4} \pi ^{2}}+\frac {15 \mathrm {S}\left (b x \right )^{2}}{2 b^{7} \pi ^{3}}-\frac {15 x \,\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{6} \pi ^{3}}+\frac {x^{5} \mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {11 \sin \left (b^{2} \pi \,x^{2}\right )}{2 b^{7} \pi ^{4}}+\frac {x^{4} \sin \left (b^{2} \pi \,x^{2}\right )}{4 b^{3} \pi ^{2}} \]
command
integrate(x^6*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
\[ -\frac {\pi ^{3} b^{6} x^{6} - 60 \, \pi ^{2} b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 42 \, \pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 24 \, \pi b^{2} x^{2} - 90 \, \pi \operatorname {S}\left (b x\right )^{2} - 6 \, {\left ({\left (\pi ^{2} b^{4} x^{4} - 22\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, {\left (\pi ^{3} b^{5} x^{5} - 15 \, \pi b x\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{12 \, \pi ^{4} b^{7}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{6} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right ), x\right ) \]