\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {3 a \left (a -4 b \right ) \arctanh \left (\frac {\sec \left (f x +e \right ) \sqrt {a +b}}{\sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}\right )}{8 \left (a +b \right )^{\frac {7}{2}} f}-\frac {5 a \cot \left (f x +e \right ) \csc \left (f x +e \right )}{8 \left (a +b \right )^{2} f \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}-\frac {\left (\cot ^{3}\left (f x +e \right )\right ) \csc \left (f x +e \right )}{4 \left (a +b \right ) f \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}-\frac {\left (13 a -2 b \right ) b \sec \left (f x +e \right )}{8 \left (a +b \right )^{3} f \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}} \]
command
integrate(csc(f*x+e)^5/(a+b*sec(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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________