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