\[ \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{S(b x)^2} \, dx \]
Optimal antiderivative \[ -\frac {1}{b \,\mathrm {S}\left (b x \right )} \]
command
integrate(sin(1/2*b^2*pi*x^2)/fresnel_sin(b*x)^2,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {1}{b \operatorname {S}\left (b x\right )} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {\sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnels}\left (b x\right )^{2}}\,{d x} \]________________________________________________________________________________________