\[ \int \frac {\text {FresnelC}(b x)}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {b \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 x}-\frac {\FresnelC \left (b x \right )}{2 x^{2}}-\frac {b^{2} \pi \,\mathrm {S}\left (b x \right )}{2} \]
command
integrate(fresnel_cos(b*x)/x^3,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {\sqrt {\frac {1}{2}} \sqrt {\pi x^{2}} {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{2}}{16 \, x} - \frac {\operatorname {C}\left (b x\right )}{2 \, x^{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\rm fresnelc}\left (b x\right )}{x^{3}}\,{d x} \]________________________________________________________________________________________