\[ \int x^3 \text {CosIntegral}(b x)^2 \, dx \]
Optimal antiderivative \[ \frac {x^{2}}{4 b^{2}}+\frac {x^{4} \cosineIntegral \! \left (b x \right )^{2}}{4}-\frac {3 \cosineIntegral \! \left (2 b x \right )}{2 b^{4}}+\frac {3 \cosineIntegral \! \left (b x \right ) \cos \! \left (b x \right )}{b^{4}}-\frac {3 x^{2} \cosineIntegral \! \left (b x \right ) \cos \! \left (b x \right )}{2 b^{2}}+\frac {3 \left (\cos ^{2}\left (b x \right )\right )}{8 b^{4}}-\frac {3 \ln \! \left (x \right )}{2 b^{4}}+\frac {3 x \cosineIntegral \! \left (b x \right ) \sin \! \left (b x \right )}{b^{3}}-\frac {x^{3} \cosineIntegral \! \left (b x \right ) \sin \! \left (b x \right )}{2 b}+\frac {x \cos \! \left (b x \right ) \sin \! \left (b x \right )}{b^{3}}-\frac {13 \left (\sin ^{2}\left (b x \right )\right )}{8 b^{4}}+\frac {x^{2} \left (\sin ^{2}\left (b x \right )\right )}{4 b^{2}} \]
command
integrate(x^3*fresnel_cos(b*x)^2,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
\[ \frac {1}{4} \, x^{4} \operatorname {Ci}\left (b x\right )^{2} - \frac {1}{2} \, {\left (\frac {3 \, {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{b^{4}} + \frac {{\left (b^{3} x^{3} - 6 \, b x\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname {Ci}\left (b x\right ) - \frac {b^{2} x^{2} \cos \left (2 \, b x\right ) - 3 \, b^{2} x^{2} - 4 \, b x \sin \left (2 \, b x\right ) - 8 \, \cos \left (2 \, b x\right ) + 6 \, \operatorname {Ci}\left (2 \, b x\right ) + 6 \, \operatorname {Ci}\left (-2 \, b x\right ) + 12 \, \log \left (x\right )}{8 \, b^{4}} \]