\[ \int x^2 \text {CosIntegral}(b x) \sin (b x) \, dx \]
Optimal antiderivative \[ \frac {x^{2}}{4 b}-\frac {\cosineIntegral \! \left (2 b x \right )}{b^{3}}+\frac {2 \cosineIntegral \! \left (b x \right ) \cos \! \left (b x \right )}{b^{3}}-\frac {x^{2} \cosineIntegral \! \left (b x \right ) \cos \! \left (b x \right )}{b}+\frac {\cos ^{2}\left (b x \right )}{4 b^{3}}-\frac {\ln \! \left (x \right )}{b^{3}}+\frac {2 x \cosineIntegral \! \left (b x \right ) \sin \! \left (b x \right )}{b^{2}}+\frac {x \cos \! \left (b x \right ) \sin \! \left (b x \right )}{2 b^{2}}-\frac {\sin ^{2}\left (b x \right )}{b^{3}} \]
command
integrate(x^2*fresnel_cos(b*x)*sin(b*x),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ {\left (\frac {2 \, x \sin \left (b x\right )}{b^{2}} - \frac {{\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{b^{3}}\right )} \operatorname {Ci}\left (b x\right ) + \frac {2 \, b^{2} x^{2} + 2 \, b x \sin \left (2 \, b x\right ) + 5 \, \cos \left (2 \, b x\right ) - 4 \, \operatorname {Ci}\left (2 \, b x\right ) - 4 \, \operatorname {Ci}\left (-2 \, b x\right ) - 8 \, \log \left (x\right )}{8 \, b^{3}} \]