\[ \int x \cos (a+b x) \text {Si}(c+d x) \, dx \]
Optimal antiderivative \[ \frac {c \cosineIntegral \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right ) \cos \left (a -\frac {b c}{d}\right )}{2 b d}-\frac {c \cosineIntegral \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right ) \cos \left (a -\frac {b c}{d}\right )}{2 b d}+\frac {\cos \left (a -\frac {b c}{d}\right ) \sinIntegral \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right )}{2 b^{2}}+\frac {\cos \left (b x +a \right ) \sinIntegral \left (d x +c \right )}{b^{2}}-\frac {\cos \left (a -\frac {b c}{d}\right ) \sinIntegral \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right )}{2 b^{2}}+\frac {\cosineIntegral \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right ) \sin \left (a -\frac {b c}{d}\right )}{2 b^{2}}-\frac {\cosineIntegral \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right ) \sin \left (a -\frac {b c}{d}\right )}{2 b^{2}}-\frac {c \sinIntegral \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right ) \sin \left (a -\frac {b c}{d}\right )}{2 b d}+\frac {c \sinIntegral \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right ) \sin \left (a -\frac {b c}{d}\right )}{2 b d}+\frac {x \sinIntegral \left (d x +c \right ) \sin \left (b x +a \right )}{b}-\frac {\sin \left (a -c +\left (b -d \right ) x \right )}{2 b \left (b -d \right )}+\frac {\sin \left (a +c +\left (b +d \right ) x \right )}{2 b \left (b +d \right )} \]
command
integrate(x*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 {4 \, b^{2} d \cos \left (b x + a\right ) \sin \left (d x + c\right ) + 4 \, {\left (b^{2} d - d^{3}\right )} \cos \left (b x + a\right ) \operatorname {Si}\left (d x + c\right ) - {\left ({\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) + {\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (-\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - {\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) - {\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) + 2 \, {\left (b^{2} d - d^{3}\right )} \operatorname {Si}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) + 2 \, {\left (b^{2} d - d^{3}\right )} \operatorname {Si}\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 ) - 4 \, {\left (b d^{2} \cos \left (d x + c\right ) - {\left (b^{3} d - b d^{3}\right )} x \operatorname {Si}\left (d x + c\right )\right )} \sin \left (b x + a\right ) - {\left ({\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) + {\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (-\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - {\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) - {\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) - 2 \, {\left (b^{3} c - b c d^{2}\right )} \operatorname {Si}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - 2 \, {\left (b^{3} c - b c d^{2}\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 \, {\left (b^{4} d - b^{2} d^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x \cos \left (b x + a\right ) \operatorname {Si}\left (d x + c\right ), x\right ) \]