\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {3 \left (a^{2}-10 a b +5 b^{2}\right ) \arctanh \left (\cos \left (f x +e \right )\right )}{8 \left (a +b \right )^{5} f}+\frac {\left (a^{2}-9 a b +2 b^{2}\right ) \cos \left (f x +e \right )}{8 \left (a +b \right )^{3} f \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {3 \left (a^{2}-6 a b +b^{2}\right ) \cos \left (f x +e \right )}{8 \left (a +b \right )^{4} f \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}-\frac {\left (a -7 b \right ) \cot \left (f x +e \right ) \csc \left (f x +e \right )}{8 \left (a +b \right )^{2} f \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {\left (\cot ^{3}\left (f x +e \right )\right ) \csc \left (f x +e \right )}{4 \left (a +b \right ) f \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {3 \left (5 a^{2}-10 a b +b^{2}\right ) \arctan \left (\frac {\cos \left (f x +e \right ) \sqrt {a}}{\sqrt {b}}\right ) \sqrt {b}}{8 \left (a +b \right )^{5} f \sqrt {a}} \]
command
integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________