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