\[ \int \frac {S(b x)^2}{x^9} \, dx \]
Optimal antiderivative \[ -\frac {b^{2}}{336 x^{6}}+\frac {b^{6} \pi ^{2}}{1680 x^{2}}+\frac {b^{2} \cos \left (b^{2} \pi \,x^{2}\right )}{336 x^{6}}-\frac {b^{6} \pi ^{2} \cos \left (b^{2} \pi \,x^{2}\right )}{336 x^{2}}-\frac {b^{3} \pi \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{140 x^{5}}+\frac {b^{7} \pi ^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{420 x}+\frac {b^{8} \pi ^{4} \mathrm {S}\left (b x \right )^{2}}{840}-\frac {\mathrm {S}\left (b x \right )^{2}}{8 x^{8}}-\frac {b^{8} \pi ^{3} \sinIntegral \left (b^{2} \pi \,x^{2}\right )}{280}-\frac {b \,\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{28 x^{7}}+\frac {b^{5} \pi ^{2} \mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{420 x^{3}}-\frac {b^{4} \pi \sin \left (b^{2} \pi \,x^{2}\right )}{420 x^{4}} \]
command
integrate(fresnel_sin(b*x)^2/x^9,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, \pi ^{3} b^{8} x^{8} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 3 \, \pi ^{2} b^{6} x^{6} + 5 \, b^{2} x^{2} + 5 \, {\left (\pi ^{2} b^{6} x^{6} - b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 2 \, {\left (\pi ^{3} b^{7} x^{7} - 3 \, \pi b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {S}\left (b x\right )^{2} + 2 \, {\left (2 \, \pi b^{4} x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{840 \, x^{8}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right )^{2}}{x^{9}}, x\right ) \]