\[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))} \, dx \]
Optimal antiderivative \[ \frac {\tan \left (f x +e \right )}{a c f} \]
command
int(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(\frac {\tan \left (f x +e \right )}{a c f}\) | \(17\) |
| risch | \(\frac {2 i}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a c f}\) | \(41\) |
| norman | \(-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a c f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\) | \(47\) |
| derivativedivides | error in RationalFunction: argument is not a rational function\ | N/A |
Maple 2021.1 output
\[ \int \frac {1}{\left (a +a \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )}\, dx \]______________________________________________________________________________________________________