\[ \int \frac {\tan ^4(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ \frac {3 \arctanh \left (\sin \left (f x +e \right )\right ) \cos \left (f x +e \right )}{8 f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {3 \tan \left (f x +e \right )}{8 f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {\tan ^{3}\left (f x +e \right )}{4 f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}} \]
command
integrate(tan(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {3 \, \log \left ({\left | \frac {1}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \right |}\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )} - \frac {3 \, \log \left ({\left | \frac {1}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \right |}\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )} - \frac {4 \, {\left (3 \, {\left (\frac {1}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}^{3} - \frac {20}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} - 20 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left ({\left (\frac {1}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}^{2} - 4\right )}^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}}{16 \, \sqrt {a} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________