\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3} \, dx \]
Optimal antiderivative \[ \frac {\csc \left (f x +e \right )}{a^{3} c^{3} f}-\frac {2 \left (\csc ^{3}\left (f x +e \right )\right )}{3 a^{3} c^{3} f}+\frac {\csc ^{5}\left (f x +e \right )}{5 a^{3} c^{3} f} \]
command
int(sec(f*x+e)/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^3,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(-\frac {-\frac {\cos ^{6}\left (f x +e \right )}{5 \sin \left (f x +e \right )^{5}}+\frac {\cos ^{6}\left (f x +e \right )}{15 \sin \left (f x +e \right )^{3}}-\frac {\cos ^{6}\left (f x +e \right )}{5 \sin \left (f x +e \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (f x +e \right )+\frac {4 \left (\cos ^{2}\left (f x +e \right )\right )}{3}\right ) \sin \left (f x +e \right )}{5}}{c^{3} a^{3} f}\) | \(95\) |
| risch | \(\frac {2 i \left (15 \,{\mathrm e}^{9 i \left (f x +e \right )}-20 \,{\mathrm e}^{7 i \left (f x +e \right )}+58 \,{\mathrm e}^{5 i \left (f x +e \right )}-20 \,{\mathrm e}^{3 i \left (f x +e \right )}+15 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{15 f \,c^{3} a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5}}\) | \(95\) |
| norman | \(\frac {\frac {1}{160 a c f}-\frac {5 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{96 a c f}+\frac {5 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16 a c f}+\frac {5 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16 a c f}-\frac {5 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{96 a c f}+\frac {\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )}{160 a c f}}{a^{2} c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\) | \(141\) |
| derivativedivides | error in RationalFunction: argument is not a rational function\ | N/A |
Maple 2021.1 output
\[ \int \frac {\sec \left (f x +e \right )}{\left (a +a \sec \left (f x +e \right )\right )^{3} \left (c -c \sec \left (f x +e \right )\right )^{3}}\, dx \]______________________________________________________________________________________________________