\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{\text {FresnelC}(b x)^3} \, dx \]
Optimal antiderivative \[ -\frac {1}{2 b \FresnelC \left (b x \right )^{2}} \]
command
integrate(cos(1/2*b^2*pi*x^2)/fresnel_cos(b*x)^3,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {1}{2 \, b \operatorname {C}\left (b x\right )^{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnelc}\left (b x\right )^{3}}\,{d x} \]________________________________________________________________________________________