\[ \int \frac {a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)}{(b \sec (c+d x)+a \sin (c+d x))^3} \, dx \]
Optimal antiderivative \[ -\frac {1}{2 d \left (b \sec \left (d x +c \right )+a \sin \left (d x +c \right )\right )^{2}} \]
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))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\tan \left (d x + c\right )^{2} + 1}{2 \, {\left (b \tan \left (d x + c\right )^{2} + a \tan \left (d x + c\right ) + b\right )}^{2} d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________