\[ \int \frac {\tan ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {317 \cos \left (f x +e \right )}{3072 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {\sec \left (f x +e \right ) \left (115+129 \sin \left (f x +e \right )\right )}{384 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {317 \cos \left (f x +e \right )}{4096 a f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {317 \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{8192 a^{\frac {5}{2}} f}+\frac {5 a \sin \left (f x +e \right ) \tan \left (f x +e \right )}{48 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}}}+\frac {\tan ^{3}\left (f x +e \right )}{3 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}} \]
command
integrate(tan(f*x+e)^4/(a+a*sin(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {128 \, \sqrt {2} {\left (9 \, \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}} - \frac {\sqrt {2} {\left (201 \, \sqrt {a} \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 1249 \, \sqrt {a} \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 1567 \, \sqrt {a} \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 567 \, \sqrt {a} \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{4} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{24576 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________