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