\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {\csc \left (f x +e \right )}{a^{2} c^{2} f}-\frac {\csc ^{3}\left (f x +e \right )}{3 a^{2} c^{2} f} \]
command
int(sec(f*x+e)/(a+a*sec(f*x+e))^2/(c-c*sec(f*x+e))^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(\frac {-\frac {\cos ^{4}\left (f x +e \right )}{3 \sin \left (f x +e \right )^{3}}+\frac {\cos ^{4}\left (f x +e \right )}{3 \sin \left (f x +e \right )}+\frac {\left (2+\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )}{3}}{a^{2} c^{2} f}\) | \(66\) |
| risch | \(\frac {2 i \left (3 \,{\mathrm e}^{5 i \left (f x +e \right )}-2 \,{\mathrm e}^{3 i \left (f x +e \right )}+3 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 f \,a^{2} c^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{3}}\) | \(73\) |
| norman | \(\frac {-\frac {1}{24 a c f}+\frac {3 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 a c f}+\frac {3 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 a c f}-\frac {\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )}{24 a c f}}{a c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}\) | \(97\) |
| derivativedivides | error in RationalFunction: argument is not a rational function\ | N/A |
Maple 2021.1 output
\[ \int \frac {\sec \left (f x +e \right )}{\left (a +a \sec \left (f x +e \right )\right )^{2} \left (c -c \sec \left (f x +e \right )\right )^{2}}\, dx \]______________________________________________________________________________________________________