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