\[ \int \csc ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (3 a^{2}+12 a b +8 b^{2}\right ) \arctanh \left (\frac {\sec \left (f x +e \right ) \sqrt {a +b}}{\sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}\right )}{8 \left (a +b \right )^{\frac {3}{2}} f}+\frac {\arctanh \left (\frac {\sec \left (f x +e \right ) \sqrt {b}}{\sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}\right ) \sqrt {b}}{f}-\frac {\left (3 a +4 b \right ) \cot \left (f x +e \right ) \csc \left (f x +e \right ) \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{8 \left (a +b \right ) f}-\frac {\cot \left (f x +e \right ) \left (\csc ^{3}\left (f x +e \right )\right ) \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{4 f} \]
command
integrate(csc(f*x+e)^5*(a+b*sec(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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________