\[ \int \text {CosIntegral}(c+d x) \sin (a+b x) \, dx \]
Optimal antiderivative \[ \frac {\cosineIntegral \! \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right ) \cos \! \left (a -\frac {b c}{d}\right )}{2 b}+\frac {\cosineIntegral \! \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right ) \cos \! \left (a -\frac {b c}{d}\right )}{2 b}-\frac {\cosineIntegral \! \left (d x +c \right ) \cos \! \left (b x +a \right )}{b}-\frac {\sinIntegral \! \left (\frac {c \left (b -d \right )}{d}+\left (b -d \right ) x \right ) \sin \! \left (a -\frac {b c}{d}\right )}{2 b}-\frac {\sinIntegral \! \left (\frac {c \left (b +d \right )}{d}+\left (b +d \right ) x \right ) \sin \! \left (a -\frac {b c}{d}\right )}{2 b} \]
command
integrate(fresnel_cos(d*x+c)*sin(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
\[ \text {output too large to display} \]