\[ \int \frac {\tan ^2(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\sin \left (f x +e \right )\right ) \cos \left (f x +e \right )}{8 a f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {\tan \left (f x +e \right )}{8 a f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {\left (\sec ^{2}\left (f x +e \right )\right ) \tan \left (f x +e \right )}{4 a f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}} \]
command
integrate(tan(f*x+e)^2/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {a \tan \left (f x + e\right )^{2} + a} {\left (\frac {2 \, \tan \left (f x + e\right )^{2}}{a} + \frac {1}{a}\right )} \tan \left (f x + e\right ) + \frac {\log \left ({\left | -\sqrt {a} \tan \left (f x + e\right ) + \sqrt {a \tan \left (f x + e\right )^{2} + a} \right |}\right )}{\sqrt {a}}}{8 \, a f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________