\[ \int \frac {\csc ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {3 \left (a -b \right )^{2} \arctanh \left (\frac {\sec \left (f x +e \right ) \sqrt {a}}{\sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}\right )}{8 a^{\frac {5}{2}} f}-\frac {\left (5 a -3 b \right ) \cot \left (f x +e \right ) \csc \left (f x +e \right ) \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}{8 a^{2} f}-\frac {\left (\cot ^{3}\left (f x +e \right )\right ) \csc \left (f x +e \right ) \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}{4 a f} \]
command
integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^(1/2),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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________