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