\[ \int x^3 \text {CosIntegral}(b x) \sin (b x) \, dx \]
Optimal antiderivative \[ -\frac {5 x}{2 b^{3}}+\frac {x^{3}}{6 b}+\frac {6 x \cosineIntegral \! \left (b x \right ) \cos \! \left (b x \right )}{b^{3}}-\frac {x^{3} \cosineIntegral \! \left (b x \right ) \cos \! \left (b x \right )}{b}+\frac {x \left (\cos ^{2}\left (b x \right )\right )}{2 b^{3}}+\frac {3 \sinIntegral \! \left (2 b x \right )}{b^{4}}-\frac {6 \cosineIntegral \! \left (b x \right ) \sin \! \left (b x \right )}{b^{4}}+\frac {3 x^{2} \cosineIntegral \! \left (b x \right ) \sin \! \left (b x \right )}{b^{2}}-\frac {4 \cos \! \left (b x \right ) \sin \! \left (b x \right )}{b^{4}}+\frac {x^{2} \cos \! \left (b x \right ) \sin \! \left (b x \right )}{2 b^{2}}-\frac {3 x \left (\sin ^{2}\left (b x \right )\right )}{2 b^{3}} \]
command
integrate(x^3*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
\[ -{\left (\frac {{\left (b^{3} x^{3} - 6 \, b x\right )} \cos \left (b x\right )}{b^{4}} - \frac {3 \, {\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname {Ci}\left (b x\right ) + \frac {b^{3} x^{3} \tan \left (b x\right )^{2} + b^{3} x^{3} + 3 \, b^{2} x^{2} \tan \left (b x\right ) - 24 \, b x \tan \left (b x\right )^{2} + 9 \, \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) \tan \left (b x\right )^{2} - 9 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) \tan \left (b x\right )^{2} + 18 \, \operatorname {Si}\left (2 \, b x\right ) \tan \left (b x\right )^{2} - 12 \, b x + 9 \, \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) - 9 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) + 18 \, \operatorname {Si}\left (2 \, b x\right ) - 24 \, \tan \left (b x\right )}{6 \, {\left (b^{4} \tan \left (b x\right )^{2} + b^{4}\right )}} \]