\[ \int \frac {S(b x)}{x^{10}} \, dx \]
Optimal antiderivative \[ -\frac {b^{3} \pi \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{432 x^{6}}+\frac {b^{7} \pi ^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3456 x^{2}}-\frac {\mathrm {S}\left (b x \right )}{9 x^{9}}+\frac {b^{9} \pi ^{4} \sinIntegral \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6912}-\frac {b \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{72 x^{8}}+\frac {b^{5} \pi ^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{1728 x^{4}} \]
command
integrate(fresnel_sin(b*x)/x^10,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {1}{576} \, {\left (i \, \pi ^{4} \Gamma \left (-4, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) - i \, \pi ^{4} \Gamma \left (-4, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{9} - \frac {\operatorname {S}\left (b x\right )}{9 \, x^{9}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\rm fresnels}\left (b x\right )}{x^{10}}\,{d x} \]________________________________________________________________________________________