\[ \int x \text {CosIntegral}(b x) \sin (b x) \, dx \]
Optimal antiderivative \[ \frac {x}{2 b}-\frac {x \cosineIntegral \! \left (b x \right ) \cos \! \left (b x \right )}{b}-\frac {\sinIntegral \! \left (2 b x \right )}{2 b^{2}}+\frac {\cosineIntegral \! \left (b x \right ) \sin \! \left (b x \right )}{b^{2}}+\frac {\cos \! \left (b x \right ) \sin \! \left (b x \right )}{2 b^{2}} \]
command
integrate(x*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
\[ -{\left (\frac {x \cos \left (b x\right )}{b} - \frac {\sin \left (b x\right )}{b^{2}}\right )} \operatorname {Ci}\left (b x\right ) + \frac {2 \, b x \tan \left (b x\right )^{2} - \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) \tan \left (b x\right )^{2} + \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) \tan \left (b x\right )^{2} - 2 \, \operatorname {Si}\left (2 \, b x\right ) \tan \left (b x\right )^{2} + 2 \, b x - \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) + \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) - 2 \, \operatorname {Si}\left (2 \, b x\right ) + 2 \, \tan \left (b x\right )}{4 \, {\left (b^{2} \tan \left (b x\right )^{2} + b^{2}\right )}} \]