\[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\sin \left (f x +e \right )\right )}{b^{2} f}-\frac {a \sin \left (f x +e \right )}{2 b \left (a +b \right ) f \left (a +b -a \left (\sin ^{2}\left (f x +e \right )\right )\right )}-\frac {\left (2 a +3 b \right ) \arctanh \left (\frac {\sin \left (f x +e \right ) \sqrt {a}}{\sqrt {a +b}}\right ) \sqrt {a}}{2 b^{2} \left (a +b \right )^{\frac {3}{2}} f} \]
command
integrate(sec(f*x+e)^5/(a+b*sec(f*x+e)^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {{\left (2 \, a^{2} + 3 \, a b\right )} \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{{\left (a b^{2} + b^{3}\right )} \sqrt {-a^{2} - a b}} + \frac {a \sin \left (f x + e\right )}{{\left (a \sin \left (f x + e\right )^{2} - a - b\right )} {\left (a b + b^{2}\right )}} + \frac {\log \left ({\left | \sin \left (f x + e\right ) + 1 \right |}\right )}{b^{2}} - \frac {\log \left ({\left | \sin \left (f x + e\right ) - 1 \right |}\right )}{b^{2}}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________