\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))} \, dx \]
Optimal antiderivative \[ \frac {\csc \left (f x +e \right )}{a c f} \]
command
int(sec(f*x+e)/(a+a*sec(f*x+e))/(c-c*sec(f*x+e)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(\frac {1}{c a f \sin \left (f x +e \right )}\) | \(19\) |
| norman | \(\frac {\frac {1}{2 a c f}+\frac {\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a c f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}\) | \(47\) |
| risch | \(\frac {2 i {\mathrm e}^{i \left (f x +e \right )}}{f c a \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}\) | \(48\) |
| derivativedivides | error in RationalFunction: argument is not a rational function\ | N/A |
Maple 2021.1 output
\[ \int \frac {\sec \left (f x +e \right )}{\left (a +a \sec \left (f x +e \right )\right ) \left (c -c \sec \left (f x +e \right )\right )}\, dx \]______________________________________________________________________________________________________