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