\[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {\left (8 a^{2}+20 a b +15 b^{2}\right ) \arctanh \left (\frac {\sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{\sqrt {a +b}}\right )}{8 \left (a +b \right )^{\frac {5}{2}} f}+\frac {\arctanh \left (\frac {\sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{\sqrt {a}}\right )}{f \sqrt {a}}+\frac {\left (4 a +7 b \right ) \left (\cot ^{2}\left (f x +e \right )\right ) \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{8 \left (a +b \right )^{2} f}-\frac {\left (\cot ^{4}\left (f x +e \right )\right ) \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{4 \left (a +b \right ) f} \]
command
integrate(cot(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 \[ \text {Exception raised: TypeError} \]_______________________________________________________