\[ \int \frac {\text {FresnelC}(b x)^2}{x^5} \, dx \]
Optimal antiderivative \[ -\frac {b^{2}}{24 x^{2}}-\frac {b^{2} \cos \left (b^{2} \pi \,x^{2}\right )}{24 x^{2}}-\frac {b \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{6 x^{3}}-\frac {b^{4} \pi ^{2} \FresnelC \left (b x \right )^{2}}{12}-\frac {\FresnelC \left (b x \right )^{2}}{4 x^{4}}-\frac {b^{4} \pi \sinIntegral \left (b^{2} \pi \,x^{2}\right )}{12}+\frac {b^{3} \pi \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 x} \]
command
integrate(fresnel_cos(b*x)^2/x^5,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {\pi b^{4} x^{4} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 2 \, \pi b^{3} x^{3} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 2 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {C}\left (b x\right )^{2}}{12 \, x^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\rm fresnelc}\left (b x\right )^{2}}{x^{5}}, x\right ) \]