\[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\cot \left (f x +e \right ) \left (\csc ^{2}\left (f x +e \right )\right )}{3 a f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 \cot \left (f x +e \right ) \left (\csc ^{4}\left (f x +e \right )\right )}{5 a f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {\cot \left (f x +e \right ) \left (\csc ^{6}\left (f x +e \right )\right )}{7 a f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}} \]
command
integrate(cot(f*x+e)^8/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {525 \, \sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 35 \, \sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 63 \, \sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, \sqrt {a}}{a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7}} + \frac {15 \, a^{\frac {25}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 63 \, a^{\frac {25}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 35 \, a^{\frac {25}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 525 \, a^{\frac {25}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{14} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}}{13440 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________