\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {3 \left (a -5 b \right ) \left (a -b \right ) \arctanh \left (\frac {\sec \left (f x +e \right ) \sqrt {a}}{\sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}\right )}{8 a^{\frac {7}{2}} f}-\frac {5 \left (a -b \right ) \cot \left (f x +e \right ) \csc \left (f x +e \right )}{8 a^{2} f \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}-\frac {\left (\cot ^{3}\left (f x +e \right )\right ) \csc \left (f x +e \right )}{4 a f \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}-\frac {\left (13 a -15 b \right ) b \sec \left (f x +e \right )}{8 a^{3} f \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}} \]
command
integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________