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