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