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