\[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x) \, dx \]
Optimal antiderivative \[ \frac {\FresnelC \left (b x \right )^{2}}{2 b} \]
command
integrate(cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {\operatorname {C}\left (b x\right )^{2}}{2 \, b} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right )\,{d x} \]________________________________________________________________________________________