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