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