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