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