\[ \int \cos (a+b x) \text {CosIntegral}(a+b x) \, dx \]
Optimal antiderivative \[ -\frac {\sinIntegral \left (2 b x +2 a \right )}{2 b}+\frac {\cosineIntegral \left (b x +a \right ) \sin \left (b x +a \right )}{b} \]
command
integrate(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
\[ \frac {2 \, b \operatorname {C}\left (b x + a\right ) \sin \left (b x + a\right ) - \sqrt {b^{2}} \cos \left (\frac {1}{2 \, \pi }\right ) \operatorname {S}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} \cos \left (\frac {1}{2 \, \pi }\right ) \operatorname {S}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) \sin \left (\frac {1}{2 \, \pi }\right ) - \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) \sin \left (\frac {1}{2 \, \pi }\right )}{2 \, b^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\cos \left (b x + a\right ) \operatorname {Ci}\left (b x + a\right ), x\right ) \]