\[ \int \sec ^6(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx \]
Optimal antiderivative \[ \frac {\tan \left (f x +e \right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}}{f \left (n p +1\right )}+\frac {2 \left (\tan ^{3}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}}{f \left (n p +3\right )}+\frac {\left (\tan ^{5}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}}{f \left (n p +5\right )} \]
command
int(sec(f*x+e)^6*(b*(c*tan(f*x+e))^n)^p,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(70270\) |
Maple 2021.1 output
\[ \int \left (\sec ^{6}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]________________________________________________________________________________________