\[ \int x \cos (a+b x) \text {CosIntegral}(a+b x) \, dx \]
Optimal antiderivative \[ -\frac {\cosineIntegral \left (2 b x +2 a \right )}{2 b^{2}}+\frac {\cosineIntegral \left (b x +a \right ) \cos \left (b x +a \right )}{b^{2}}+\frac {\cos \left (2 b x +2 a \right )}{4 b^{2}}-\frac {\ln \left (b x +a \right )}{2 b^{2}}+\frac {a \sinIntegral \left (2 b x +2 a \right )}{2 b^{2}}+\frac {x \cosineIntegral \left (b x +a \right ) \sin \left (b x +a \right )}{b} \]
command
integrate(x*fresnel_cos(b*x+a)*cos(b*x+a),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, \pi b^{2} x \operatorname {C}\left (b x + a\right ) \sin \left (b x + a\right ) + 2 \, \pi b \cos \left (b x + a\right ) \operatorname {C}\left (b x + a\right ) - 2 \, b \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) \sin \left (b x + a\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (\pi a + 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) - {\left (\pi a - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} {\left ({\left (\pi a + 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) - \pi \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left ({\left (\pi a - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) + \pi \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right )}{2 \, \pi b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x \cos \left (b x + a\right ) \operatorname {Ci}\left (b x + a\right ), x\right ) \]