\[ \int \text {CosIntegral}(a+b x) \sin (a+b x) \, dx \]
Optimal antiderivative \[ \frac {\cosineIntegral \! \left (2 b x +2 a \right )}{2 b}-\frac {\cosineIntegral \! \left (b x +a \right ) \cos \! \left (b x +a \right )}{b}+\frac {\ln \! \left (b x +a \right )}{2 b} \]
command
integrate(fresnel_cos(b*x+a)*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
\[ -\frac {\cos \left (b x + a\right ) \operatorname {Ci}\left (b x + a\right )}{b} + \frac {\cos \left (2 \, a\right )^{2} \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, a\right )^{2} \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) + \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \sin \left (2 \, a\right )^{2} + \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \sin \left (2 \, a\right )^{2} + 2 \, \log \left (b x + a\right )}{4 \, b} \]