\[ \int x^3 \cos (b x) \text {CosIntegral}(b x) \, dx \]
Optimal antiderivative \[ -\frac {x^{2}}{2 b^{2}}+\frac {3 \cosineIntegral \! \left (2 b x \right )}{b^{4}}-\frac {6 \cosineIntegral \! \left (b x \right ) \cos \! \left (b x \right )}{b^{4}}+\frac {3 x^{2} \cosineIntegral \! \left (b x \right ) \cos \! \left (b x \right )}{b^{2}}-\frac {3 \left (\cos ^{2}\left (b x \right )\right )}{4 b^{4}}+\frac {3 \ln \! \left (x \right )}{b^{4}}-\frac {6 x \cosineIntegral \! \left (b x \right ) \sin \! \left (b x \right )}{b^{3}}+\frac {x^{3} \cosineIntegral \! \left (b x \right ) \sin \! \left (b x \right )}{b}-\frac {2 x \cos \! \left (b x \right ) \sin \! \left (b x \right )}{b^{3}}+\frac {13 \left (\sin ^{2}\left (b x \right )\right )}{4 b^{4}}-\frac {x^{2} \left (\sin ^{2}\left (b x \right )\right )}{2 b^{2}} \]
command
integrate(x^3*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
\[ {\left (\frac {3 \, {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{b^{4}} + \frac {{\left (b^{3} x^{3} - 6 \, b x\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname {Ci}\left (b x\right ) + \frac {b^{2} x^{2} \cos \left (2 \, b x\right ) - 3 \, b^{2} x^{2} - 4 \, b x \sin \left (2 \, b x\right ) - 8 \, \cos \left (2 \, b x\right ) + 6 \, \operatorname {Ci}\left (2 \, b x\right ) + 6 \, \operatorname {Ci}\left (-2 \, b x\right ) + 12 \, \log \left (x\right )}{4 \, b^{4}} \]