\[ \int x^3 \cos (b x) \text {CosIntegral}(b x) \, dx \]
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="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, \pi b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \cos \left (b x\right ) - 6 \, {\left (\pi ^{3} b^{3} x^{2} - 2 \, \pi ^{3} b\right )} \cos \left (b x\right ) \operatorname {C}\left (b x\right ) - {\left (6 \, \pi ^{3} \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (3 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - {\left (6 \, \pi ^{3} \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (3 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - {\left (6 \, \pi ^{3} \sin \left (\frac {1}{2 \, \pi }\right ) - {\left (3 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - {\left (6 \, \pi ^{3} \sin \left (\frac {1}{2 \, \pi }\right ) - {\left (3 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + 2 \, {\left (3 \, \pi ^{2} b^{2} x \cos \left (b x\right ) + {\left (\pi ^{2} b^{3} x^{2} - 6 \, \pi ^{2} b + b\right )} \sin \left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, {\left (\pi b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{3} b^{4} x^{3} - 6 \, \pi ^{3} b^{2} x\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (b x\right )}{2 \, \pi ^{3} b^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{3} \cos \left (b x\right ) \operatorname {Ci}\left (b x\right ), x\right ) \]