\[ \int \frac {\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {8 \left (a -2 b \right ) b \tan \left (f x +e \right )}{3 a^{4} f \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}-\frac {\left (a -2 b \right ) \cot \left (f x +e \right )}{a^{2} f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {\cot ^{3}\left (f x +e \right )}{3 a f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {4 \left (a -2 b \right ) b \tan \left (f x +e \right )}{3 a^{3} f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}} \]
command
integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (2 \, {\left (a^{3} - 9 \, a^{2} b + 16 \, a b^{2} - 8 \, b^{3}\right )} \cos \left (f x + e\right )^{7} - 3 \, {\left (a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right )} \cos \left (f x + e\right )^{5} - 12 \, {\left (a^{2} b - 4 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - 8 \, {\left (a b^{2} - 2 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \, {\left ({\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{6} - a^{4} b^{2} f - {\left (a^{6} - 4 \, a^{5} b + 3 \, a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{4} - {\left (2 \, a^{5} b - 3 \, a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]