\[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3} \, dx \]
Optimal antiderivative \[ \frac {x}{a^{3} c^{3}}+\frac {\cot \left (f x +e \right )}{a^{3} c^{3} f}-\frac {\cot ^{3}\left (f x +e \right )}{3 a^{3} c^{3} f}+\frac {\cot ^{5}\left (f x +e \right )}{5 a^{3} c^{3} f} \]
command
int(1/(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 | \(-\frac {-\frac {\left (\cot ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (\cot ^{3}\left (f x +e \right )\right )}{3}-\cot \left (f x +e \right )-f x -e}{c^{3} a^{3} f}\) | \(48\) |
| risch | \(\frac {x}{a^{3} c^{3}}+\frac {2 i \left (45 \,{\mathrm e}^{8 i \left (f x +e \right )}-90 \,{\mathrm e}^{6 i \left (f x +e \right )}+140 \,{\mathrm e}^{4 i \left (f x +e \right )}-70 \,{\mathrm e}^{2 i \left (f x +e \right )}+23\right )}{15 f \,c^{3} a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5}}\) | \(94\) |
| norman | \(\frac {\frac {x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c a}+\frac {1}{160 a c f}-\frac {7 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{96 a c f}+\frac {11 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16 a c f}-\frac {11 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16 a c f}+\frac {7 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{96 a c f}-\frac {\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )}{160 a c f}}{a^{2} c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\) | \(160\) |
| derivativedivides | error in RationalFunction: argument is not a rational function\ | N/A |
Maple 2021.1 output
\[ \int \frac {1}{\left (a +a \sec \left (f x +e \right )\right )^{3} \left (c -c \sec \left (f x +e \right )\right )^{3}}\, dx \]______________________________________________________________________________________________________