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