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