\[ \int \cos (a+b x) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \, dx \]
Optimal antiderivative \[ -\frac {\sin \! \left (b x +a \right )}{b}+\frac {\ln \! \left (\cos \! \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \! \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \sin \! \left (b x +a \right )}{b} \]
command
integrate(cos(b*x+a)*log(cos(1/2*a+1/2*b*x)*sin(1/2*a+1/2*b*x)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {\log \left (\cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right ) \sin \left (b x + a\right )}{b} - \frac {\sin \left (b x + a\right )}{b} \]