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