\[ \int \text {CosIntegral}(b x) \sin (b x) \, dx \]
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="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, b \cos \left (b x\right ) \operatorname {C}\left (b x\right ) - \sqrt {b^{2}} \cos \left (\frac {1}{2 \, \pi }\right ) \operatorname {C}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} \cos \left (\frac {1}{2 \, \pi }\right ) \operatorname {C}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) \sin \left (\frac {1}{2 \, \pi }\right ) - \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) \sin \left (\frac {1}{2 \, \pi }\right )}{2 \, b^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\operatorname {Ci}\left (b x\right ) \sin \left (b x\right ), x\right ) \]