\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^6} \, dx \]
Optimal antiderivative \[ \frac {b^{3} \pi }{60 x^{2}}-\frac {b^{3} \pi \cos \left (b^{2} \pi \,x^{2}\right )}{24 x^{2}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{5 x^{5}}+\frac {b^{4} \pi ^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{15 x}+\frac {b^{5} \pi ^{3} \mathrm {S}\left (b x \right )^{2}}{30}-\frac {7 b^{5} \pi ^{2} \sinIntegral \left (b^{2} \pi \,x^{2}\right )}{120}+\frac {b^{2} \pi \,\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{15 x^{3}}-\frac {b \sin \left (b^{2} \pi \,x^{2}\right )}{40 x^{4}} \]
command
integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^6,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {4 \, \pi ^{3} b^{5} x^{5} \operatorname {S}\left (b x\right )^{2} - 7 \, \pi ^{2} b^{5} x^{5} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 10 \, \pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 7 \, \pi b^{3} x^{3} + 8 \, {\left (\pi ^{2} b^{4} x^{4} - 3\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 2 \, {\left (4 \, \pi b^{2} x^{2} \operatorname {S}\left (b x\right ) - 3 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{120 \, x^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{6}}, x\right ) \]