\[ \int \frac {\tan ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {7 \cos \left (f x +e \right )}{256 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {\sec \left (f x +e \right ) \left (65+87 \sin \left (f x +e \right )\right )}{192 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {7 \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{512 a^{\frac {3}{2}} f}+\frac {a \sin \left (f x +e \right ) \tan \left (f x +e \right )}{12 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {\tan ^{3}\left (f x +e \right )}{3 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}} \]
command
integrate(tan(f*x+e)^4/(a+a*sin(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {21 \, \sqrt {2} \log \left (\sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {21 \, \sqrt {2} \log \left (-\sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {2 \, \sqrt {2} {\left (21 \, \sqrt {a} \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 312 \, \sqrt {a} \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 507 \, \sqrt {a} \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 240 \, \sqrt {a} \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 16 \, \sqrt {a}\right )}}{{\left (\sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}^{3} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3072 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________