\[ \int \frac {\text {FresnelC}(b x)}{x^8} \, dx \]
Optimal antiderivative \[ -\frac {b \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{42 x^{6}}+\frac {b^{5} \pi ^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{336 x^{2}}-\frac {\FresnelC \left (b x \right )}{7 x^{7}}+\frac {b^{7} \pi ^{3} \sinIntegral \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{672}+\frac {b^{3} \pi \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{168 x^{4}} \]
command
integrate(fresnel_cos(b*x)/x^8,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\pi ^{3} b^{7} x^{7} \operatorname {Si}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 4 \, \pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, {\left (\pi ^{2} b^{5} x^{5} - 8 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 96 \, \operatorname {C}\left (b x\right )}{672 \, x^{7}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\rm fresnelc}\left (b x\right )}{x^{8}}, x\right ) \]