\[ \int \frac {\text {FresnelC}(b x)}{x^9} \, dx \]
Optimal antiderivative \[ -\frac {b \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{56 x^{7}}+\frac {b^{5} \pi ^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{840 x^{3}}+\frac {b^{8} \pi ^{4} \FresnelC \left (b x \right )}{840}-\frac {\FresnelC \left (b x \right )}{8 x^{8}}+\frac {b^{3} \pi \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{280 x^{5}}-\frac {b^{7} \pi ^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{840 x} \]
command
integrate(fresnel_cos(b*x)/x^9,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {C}\left (b x\right ) - {\left (\pi ^{3} b^{7} x^{7} - 3 \, \pi b^{3} x^{3}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{840 \, x^{8}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\rm fresnelc}\left (b x\right )}{x^{9}}, x\right ) \]