Optimal antiderivative \[ \frac {x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b \pi }-\frac {\FresnelC \left (b x \right )}{2 b^{2} \pi }+\frac {x^{2} \mathrm {S}\left (b x \right )}{2} \]
command
integrate(x*fresnel_sin(b*x),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{2} \, x^{2} \operatorname {S}\left (b x\right ) + \frac {\sqrt {\frac {1}{2}} {\left (4 \, \sqrt {\frac {1}{2}} \pi b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \left (i - 1\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) - \left (i + 1\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right )\right )}}{4 \, \pi ^{2} b^{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int x {\rm fresnels}\left (b x\right )\,{d x} \]________________________________________________________________________________________