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