\[ \int \frac {\sec ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (2 a -b \right ) \arctanh \left (\sin \left (f x +e \right )\right )}{2 b^{2} f}+\frac {a^{\frac {3}{2}} \arctanh \left (\frac {\sin \left (f x +e \right ) \sqrt {a}}{\sqrt {a +b}}\right )}{b^{2} f \sqrt {a +b}}+\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 b f} \]
command
integrate(sec(f*x+e)^5/(a+b*sec(f*x+e)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {4 \, a^{2} \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} b^{2}} + \frac {{\left (2 \, a - b\right )} \log \left ({\left | \sin \left (f x + e\right ) + 1 \right |}\right )}{b^{2}} - \frac {{\left (2 \, a - b\right )} \log \left ({\left | \sin \left (f x + e\right ) - 1 \right |}\right )}{b^{2}} + \frac {2 \, \sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{2} - 1\right )} b}}{4 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________