\[ \int x^3 \text {CosIntegral}(b x)^2 \, dx \]
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="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {\pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 2 \, \pi b^{2} x^{2} + 6 \, \pi b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - {\left (3 \, \pi + \pi ^{3} b^{4} x^{4}\right )} \operatorname {C}\left (b x\right )^{2} + 2 \, {\left (\pi ^{2} b^{3} x^{3} \operatorname {C}\left (b x\right ) - 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{4 \, \pi ^{3} b^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{3} \operatorname {Ci}\left (b x\right )^{2}, x\right ) \]