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