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