\[ \int \sec (e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx \]
Optimal antiderivative \[ \frac {\left (2 a +b \right ) \arctanh \left (\sin \left (f x +e \right )\right )}{2 f}+\frac {b \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f} \]
command
integrate(sec(f*x+e)*(a+b*sec(f*x+e)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (2 \, a + b\right )} \log \left ({\left | \sin \left (f x + e\right ) + 1 \right |}\right ) - {\left (2 \, a + b\right )} \log \left ({\left | \sin \left (f x + e\right ) - 1 \right |}\right ) - \frac {2 \, b \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1}}{4 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________