\[ \int x \text {CosIntegral}(a+b x) \, dx \]
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="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{2} \, x^{2} \operatorname {C}\left (b x + a\right ) + \frac {{\left (8 \, {\left (-i \, \pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a b x + 8 \, {\left (-i \, \pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{2} - \sqrt {2 \, \pi b^{2} x^{2} + 4 \, \pi a b x + 2 \, \pi a^{2}} {\left ({\left (-\left (i - 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )} + \left (i + 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )}\right )} a^{2} + \left (2 i + 2\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) - \left (2 i - 2\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )}\right )} b}{16 \, {\left (\pi ^{2} b^{4} x + \pi ^{2} a b^{3}\right )}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int x {\rm Ci}\left (b x + a\right )\,{d x} \]________________________________________________________________________________________