\[ \int \cos (a+b x) \text {Si}(c+d x) \, dx \]
Optimal antiderivative \[ -\frac {\cosineIntegral \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right ) \cos \left (a -\frac {b c}{d}\right )}{2 b}+\frac {\cosineIntegral \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right ) \cos \left (a -\frac {b c}{d}\right )}{2 b}+\frac {\sinIntegral \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right ) \sin \left (a -\frac {b c}{d}\right )}{2 b}-\frac {\sinIntegral \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right ) \sin \left (a -\frac {b c}{d}\right )}{2 b}+\frac {\sinIntegral \left (d x +c \right ) \sin \left (b x +a \right )}{b} \]
command
integrate(cos(b*x+a)*sin_integral(d*x+c),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (\operatorname {Ci}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) + \operatorname {Ci}\left (-\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - \operatorname {Ci}\left (\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) - \operatorname {Ci}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) - 2 \, {\left (\operatorname {Si}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) + \operatorname {Si}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + 4 \, \sin \left (b x + a\right ) \operatorname {Si}\left (d x + c\right )}{4 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\cos \left (b x + a\right ) \operatorname {Si}\left (d x + c\right ), x\right ) \]