\[ \int \frac {\sec ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\sin \left (f x +e \right )\right )}{b f}-\frac {\arctanh \left (\frac {\sin \left (f x +e \right ) \sqrt {a}}{\sqrt {a +b}}\right ) \sqrt {a}}{b f \sqrt {a +b}} \]
command
integrate(sec(f*x+e)^3/(a+b*sec(f*x+e)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {2 \, a \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} b} + \frac {\log \left ({\left | \sin \left (f x + e\right ) + 1 \right |}\right )}{b} - \frac {\log \left ({\left | \sin \left (f x + e\right ) - 1 \right |}\right )}{b}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________