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