\[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}}{\sqrt {a +a \sin \left (f x +e \right )}}\right )}{8 a^{\frac {3}{2}} f}-\frac {\cot \left (f x +e \right )}{8 a f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {11 \cot \left (f x +e \right ) \csc \left (f x +e \right )}{12 a f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {\cot \left (f x +e \right ) \left (\csc ^{2}\left (f x +e \right )\right ) \sqrt {a +a \sin \left (f x +e \right )}}{3 a^{2} f} \]
command
integrate(cot(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
\[ -\frac {\sqrt {2} \sqrt {a} {\left (\frac {3 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {4 \, {\left (12 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 16 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{96 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________