\[ \int x \text {CosIntegral}(c+d x) \sin (a+b x) \, dx \]
Optimal antiderivative \[ -\frac {c \cosineIntegral \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right ) \cos \left (a -\frac {b c}{d}\right )}{2 b d}-\frac {c \cosineIntegral \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right ) \cos \left (a -\frac {b c}{d}\right )}{2 b d}-\frac {x \cosineIntegral \left (d x +c \right ) \cos \left (b x +a \right )}{b}-\frac {\cos \left (a -\frac {b c}{d}\right ) \sinIntegral \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right )}{2 b^{2}}-\frac {\cos \left (a -\frac {b c}{d}\right ) \sinIntegral \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right )}{2 b^{2}}-\frac {\cosineIntegral \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right ) \sin \left (a -\frac {b c}{d}\right )}{2 b^{2}}-\frac {\cosineIntegral \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right ) \sin \left (a -\frac {b c}{d}\right )}{2 b^{2}}+\frac {c \sinIntegral \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right ) \sin \left (a -\frac {b c}{d}\right )}{2 b d}+\frac {c \sinIntegral \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right ) \sin \left (a -\frac {b c}{d}\right )}{2 b d}+\frac {\cosineIntegral \left (d x +c \right ) \sin \left (b x +a \right )}{b^{2}}+\frac {\sin \left (a -c +\left (b -d \right ) x \right )}{2 b \left (b -d \right )}+\frac {\sin \left (a +c +\left (b +d \right ) x \right )}{2 b \left (b +d \right )} \]
command
integrate(x*fresnel_cos(d*x+c)*sin(b*x+a),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, \pi b d^{3} x \cos \left (b x + a\right ) \operatorname {C}\left (d x + c\right ) - 2 \, \pi d^{3} \operatorname {C}\left (d x + c\right ) \sin \left (b x + a\right ) - 2 \, b d^{2} \cos \left (b x + a\right ) \sin \left (\frac {1}{2} \, \pi d^{2} x^{2} + \pi c d x + \frac {1}{2} \, \pi c^{2}\right ) + {\left (\pi d^{2} \sin \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right ) + {\left (\pi b c d + b^{2}\right )} \cos \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right )\right )} \sqrt {d^{2}} \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) + {\left (\pi d^{2} \sin \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) + {\left (\pi b c d - b^{2}\right )} \cos \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right )\right )} \sqrt {d^{2}} \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) + {\left (\pi d^{2} \cos \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right ) - {\left (\pi b c d + b^{2}\right )} \sin \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right )\right )} \sqrt {d^{2}} \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) - {\left (\pi d^{2} \cos \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) - {\left (\pi b c d - b^{2}\right )} \sin \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right )\right )} \sqrt {d^{2}} \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right )}{2 \, \pi b^{2} d^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x \operatorname {Ci}\left (d x + c\right ) \sin \left (b x + a\right ), x\right ) \]