\[ \int \frac {S(b x)}{x^9} \, dx \]
Optimal antiderivative \[ -\frac {b^{3} \pi \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{280 x^{5}}+\frac {b^{7} \pi ^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{840 x}+\frac {b^{8} \pi ^{4} \mathrm {S}\left (b x \right )}{840}-\frac {\mathrm {S}\left (b x \right )}{8 x^{8}}-\frac {b \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{56 x^{7}}+\frac {b^{5} \pi ^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{840 x^{3}} \]
command
integrate(fresnel_sin(b*x)/x^9,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {7}{2}} {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{8}}{512 \, x^{7}} - \frac {\operatorname {S}\left (b x\right )}{8 \, x^{8}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\rm fresnels}\left (b x\right )}{x^{9}}\,{d x} \]________________________________________________________________________________________