Integrand size = 141, antiderivative size = 28 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\frac {x^4}{\left (7-e^2+\frac {x}{4 \log \left (\frac {1}{3} x \log (3)\right )}\right )^2} \] Output:
x^4/(1/4*x/ln(1/3*x*ln(3))-exp(2)+7)^2
Time = 6.90 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\frac {16 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2} \] Input:
Integrate[(-32*x^4*Log[(x*Log[3])/3] - 32*x^4*Log[(x*Log[3])/3]^2 + (-1792 *x^3 + 256*E^2*x^3)*Log[(x*Log[3])/3]^3)/(-x^3 + (-84*x^2 + 12*E^2*x^2)*Lo g[(x*Log[3])/3] + (-2352*x + 672*E^2*x - 48*E^4*x)*Log[(x*Log[3])/3]^2 + ( -21952 + 9408*E^2 - 1344*E^4 + 64*E^6)*Log[(x*Log[3])/3]^3),x]
Output:
(16*x^4*Log[(x*Log[3])/3]^2)/(x - 4*(-7 + E^2)*Log[(x*Log[3])/3])^2
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )+\left (256 e^2 x^3-1792 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (12 e^2 x^2-84 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )+\left (-48 e^4 x+672 e^2 x-2352 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {32 x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (x-8 \left (e^2-7\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )\right )}{\left (x-4 \left (e^2-7\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (8 \left (7-e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x+4 \left (7-e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 32 \int \left (\frac {\left (-x+4 e^2-28\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (x-e^2+7\right ) x^4}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^3}{8 \left (-7+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+x \log \left (\frac {1}{3} x \log (3)\right )+x\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}dx\) |
Input:
Int[(-32*x^4*Log[(x*Log[3])/3] - 32*x^4*Log[(x*Log[3])/3]^2 + (-1792*x^3 + 256*E^2*x^3)*Log[(x*Log[3])/3]^3)/(-x^3 + (-84*x^2 + 12*E^2*x^2)*Log[(x*L og[3])/3] + (-2352*x + 672*E^2*x - 48*E^4*x)*Log[(x*Log[3])/3]^2 + (-21952 + 9408*E^2 - 1344*E^4 + 64*E^6)*Log[(x*Log[3])/3]^3),x]
Output:
$Aborted
Time = 2.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36
| method | result | size |
| norman | \(\frac {16 x^{4} \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2}}{\left (4 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) {\mathrm e}^{2}-28 \ln \left (\frac {x \ln \left (3\right )}{3}\right )-x \right )^{2}}\) | \(38\) |
| risch | \(\frac {16 x^{4} \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2}}{\left (4 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) {\mathrm e}^{2}-28 \ln \left (\frac {x \ln \left (3\right )}{3}\right )-x \right )^{2}}\) | \(38\) |
| derivativedivides | \(\frac {16 \ln \left (3\right )^{2} \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2} x^{4}}{\left (4 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) \ln \left (3\right ) {\mathrm e}^{2}-28 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) \ln \left (3\right )-x \ln \left (3\right )\right )^{2}}\) | \(48\) |
| default | \(\frac {16 \ln \left (3\right )^{2} \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2} x^{4}}{\left (4 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) \ln \left (3\right ) {\mathrm e}^{2}-28 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) \ln \left (3\right )-x \ln \left (3\right )\right )^{2}}\) | \(48\) |
| parallelrisch | \(\frac {16 x^{4} \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2}}{16 \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2} {\mathrm e}^{4}-8 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) {\mathrm e}^{2} x -224 \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2} {\mathrm e}^{2}+x^{2}+56 x \ln \left (\frac {x \ln \left (3\right )}{3}\right )+784 \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2}}\) | \(76\) |
Input:
int(((256*x^3*exp(2)-1792*x^3)*ln(1/3*x*ln(3))^3-32*x^4*ln(1/3*x*ln(3))^2- 32*x^4*ln(1/3*x*ln(3)))/((64*exp(2)^3-1344*exp(2)^2+9408*exp(2)-21952)*ln( 1/3*x*ln(3))^3+(-48*x*exp(2)^2+672*exp(2)*x-2352*x)*ln(1/3*x*ln(3))^2+(12* x^2*exp(2)-84*x^2)*ln(1/3*x*ln(3))-x^3),x,method=_RETURNVERBOSE)
Output:
16*x^4*ln(1/3*x*ln(3))^2/(4*ln(1/3*x*ln(3))*exp(2)-28*ln(1/3*x*ln(3))-x)^2
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\frac {16 \, x^{4} \log \left (\frac {1}{3} \, x \log \left (3\right )\right )^{2}}{16 \, {\left (e^{4} - 14 \, e^{2} + 49\right )} \log \left (\frac {1}{3} \, x \log \left (3\right )\right )^{2} + x^{2} - 8 \, {\left (x e^{2} - 7 \, x\right )} \log \left (\frac {1}{3} \, x \log \left (3\right )\right )} \] Input:
integrate(((256*x^3*exp(2)-1792*x^3)*log(1/3*x*log(3))^3-32*x^4*log(1/3*x* log(3))^2-32*x^4*log(1/3*x*log(3)))/((64*exp(2)^3-1344*exp(2)^2+9408*exp(2 )-21952)*log(1/3*x*log(3))^3+(-48*x*exp(2)^2+672*exp(2)*x-2352*x)*log(1/3* x*log(3))^2+(12*x^2*exp(2)-84*x^2)*log(1/3*x*log(3))-x^3),x, algorithm="fr icas")
Output:
16*x^4*log(1/3*x*log(3))^2/(16*(e^4 - 14*e^2 + 49)*log(1/3*x*log(3))^2 + x ^2 - 8*(x*e^2 - 7*x)*log(1/3*x*log(3)))
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.36 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\frac {x^{4}}{- 14 e^{2} + 49 + e^{4}} + \frac {- x^{6} + \left (- 56 x^{5} + 8 x^{5} e^{2}\right ) \log {\left (\frac {x \log {\left (3 \right )}}{3} \right )}}{- 14 x^{2} e^{2} + 49 x^{2} + x^{2} e^{4} + \left (- 1176 x e^{2} - 8 x e^{6} + 2744 x + 168 x e^{4}\right ) \log {\left (\frac {x \log {\left (3 \right )}}{3} \right )} + \left (- 448 e^{6} - 21952 e^{2} + 38416 + 16 e^{8} + 4704 e^{4}\right ) \log {\left (\frac {x \log {\left (3 \right )}}{3} \right )}^{2}} \] Input:
integrate(((256*x**3*exp(2)-1792*x**3)*ln(1/3*x*ln(3))**3-32*x**4*ln(1/3*x *ln(3))**2-32*x**4*ln(1/3*x*ln(3)))/((64*exp(2)**3-1344*exp(2)**2+9408*exp (2)-21952)*ln(1/3*x*ln(3))**3+(-48*x*exp(2)**2+672*exp(2)*x-2352*x)*ln(1/3 *x*ln(3))**2+(12*x**2*exp(2)-84*x**2)*ln(1/3*x*ln(3))-x**3),x)
Output:
x**4/(-14*exp(2) + 49 + exp(4)) + (-x**6 + (-56*x**5 + 8*x**5*exp(2))*log( x*log(3)/3))/(-14*x**2*exp(2) + 49*x**2 + x**2*exp(4) + (-1176*x*exp(2) - 8*x*exp(6) + 2744*x + 168*x*exp(4))*log(x*log(3)/3) + (-448*exp(6) - 21952 *exp(2) + 38416 + 16*exp(8) + 4704*exp(4))*log(x*log(3)/3)**2)
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (23) = 46\).
Time = 0.23 (sec) , antiderivative size = 199, normalized size of antiderivative = 7.11 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=-\frac {16 \, {\left (2 \, x^{4} {\left (\log \left (3\right ) - \log \left (\log \left (3\right )\right )\right )} \log \left (x\right ) - x^{4} \log \left (x\right )^{2} - {\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (\log \left (3\right )\right ) + \log \left (\log \left (3\right )\right )^{2}\right )} x^{4}\right )}}{16 \, {\left (e^{4} - 14 \, e^{2} + 49\right )} \log \left (x\right )^{2} + 8 \, {\left ({\left (\log \left (3\right ) - \log \left (\log \left (3\right )\right )\right )} e^{2} - 7 \, \log \left (3\right ) + 7 \, \log \left (\log \left (3\right )\right )\right )} x + x^{2} + 16 \, {\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (\log \left (3\right )\right ) + \log \left (\log \left (3\right )\right )^{2}\right )} e^{4} - 224 \, {\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (\log \left (3\right )\right ) + \log \left (\log \left (3\right )\right )^{2}\right )} e^{2} + 784 \, \log \left (3\right )^{2} - 8 \, {\left (x {\left (e^{2} - 7\right )} + 4 \, {\left (\log \left (3\right ) - \log \left (\log \left (3\right )\right )\right )} e^{4} - 56 \, {\left (\log \left (3\right ) - \log \left (\log \left (3\right )\right )\right )} e^{2} + 196 \, \log \left (3\right ) - 196 \, \log \left (\log \left (3\right )\right )\right )} \log \left (x\right ) - 1568 \, \log \left (3\right ) \log \left (\log \left (3\right )\right ) + 784 \, \log \left (\log \left (3\right )\right )^{2}} \] Input:
integrate(((256*x^3*exp(2)-1792*x^3)*log(1/3*x*log(3))^3-32*x^4*log(1/3*x* log(3))^2-32*x^4*log(1/3*x*log(3)))/((64*exp(2)^3-1344*exp(2)^2+9408*exp(2 )-21952)*log(1/3*x*log(3))^3+(-48*x*exp(2)^2+672*exp(2)*x-2352*x)*log(1/3* x*log(3))^2+(12*x^2*exp(2)-84*x^2)*log(1/3*x*log(3))-x^3),x, algorithm="ma xima")
Output:
-16*(2*x^4*(log(3) - log(log(3)))*log(x) - x^4*log(x)^2 - (log(3)^2 - 2*lo g(3)*log(log(3)) + log(log(3))^2)*x^4)/(16*(e^4 - 14*e^2 + 49)*log(x)^2 + 8*((log(3) - log(log(3)))*e^2 - 7*log(3) + 7*log(log(3)))*x + x^2 + 16*(lo g(3)^2 - 2*log(3)*log(log(3)) + log(log(3))^2)*e^4 - 224*(log(3)^2 - 2*log (3)*log(log(3)) + log(log(3))^2)*e^2 + 784*log(3)^2 - 8*(x*(e^2 - 7) + 4*( log(3) - log(log(3)))*e^4 - 56*(log(3) - log(log(3)))*e^2 + 196*log(3) - 1 96*log(log(3)))*log(x) - 1568*log(3)*log(log(3)) + 784*log(log(3))^2)
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (23) = 46\).
Time = 0.24 (sec) , antiderivative size = 154, normalized size of antiderivative = 5.50 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\frac {16 \, {\left (x^{4} \log \left (3\right )^{2} - 2 \, x^{4} \log \left (3\right ) \log \left (x \log \left (3\right )\right ) + x^{4} \log \left (x \log \left (3\right )\right )^{2}\right )}}{8 \, x e^{2} \log \left (3\right ) + 16 \, e^{4} \log \left (3\right )^{2} - 224 \, e^{2} \log \left (3\right )^{2} - 8 \, x e^{2} \log \left (x \log \left (3\right )\right ) - 32 \, e^{4} \log \left (3\right ) \log \left (x \log \left (3\right )\right ) + 448 \, e^{2} \log \left (3\right ) \log \left (x \log \left (3\right )\right ) + 16 \, e^{4} \log \left (x \log \left (3\right )\right )^{2} - 224 \, e^{2} \log \left (x \log \left (3\right )\right )^{2} + x^{2} - 56 \, x \log \left (3\right ) + 784 \, \log \left (3\right )^{2} + 56 \, x \log \left (x \log \left (3\right )\right ) - 1568 \, \log \left (3\right ) \log \left (x \log \left (3\right )\right ) + 784 \, \log \left (x \log \left (3\right )\right )^{2}} \] Input:
integrate(((256*x^3*exp(2)-1792*x^3)*log(1/3*x*log(3))^3-32*x^4*log(1/3*x* log(3))^2-32*x^4*log(1/3*x*log(3)))/((64*exp(2)^3-1344*exp(2)^2+9408*exp(2 )-21952)*log(1/3*x*log(3))^3+(-48*x*exp(2)^2+672*exp(2)*x-2352*x)*log(1/3* x*log(3))^2+(12*x^2*exp(2)-84*x^2)*log(1/3*x*log(3))-x^3),x, algorithm="gi ac")
Output:
16*(x^4*log(3)^2 - 2*x^4*log(3)*log(x*log(3)) + x^4*log(x*log(3))^2)/(8*x* e^2*log(3) + 16*e^4*log(3)^2 - 224*e^2*log(3)^2 - 8*x*e^2*log(x*log(3)) - 32*e^4*log(3)*log(x*log(3)) + 448*e^2*log(3)*log(x*log(3)) + 16*e^4*log(x* log(3))^2 - 224*e^2*log(x*log(3))^2 + x^2 - 56*x*log(3) + 784*log(3)^2 + 5 6*x*log(x*log(3)) - 1568*log(3)*log(x*log(3)) + 784*log(x*log(3))^2)
Timed out. \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=-\int \frac {32\,x^4\,{\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )}^2-{\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )}^3\,\left (256\,x^3\,{\mathrm {e}}^2-1792\,x^3\right )+32\,x^4\,\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )}{{\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )}^3\,\left (9408\,{\mathrm {e}}^2-1344\,{\mathrm {e}}^4+64\,{\mathrm {e}}^6-21952\right )-{\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )}^2\,\left (2352\,x-672\,x\,{\mathrm {e}}^2+48\,x\,{\mathrm {e}}^4\right )+\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )\,\left (12\,x^2\,{\mathrm {e}}^2-84\,x^2\right )-x^3} \,d x \] Input:
int(-(32*x^4*log((x*log(3))/3)^2 - log((x*log(3))/3)^3*(256*x^3*exp(2) - 1 792*x^3) + 32*x^4*log((x*log(3))/3))/(log((x*log(3))/3)^3*(9408*exp(2) - 1 344*exp(4) + 64*exp(6) - 21952) - log((x*log(3))/3)^2*(2352*x - 672*x*exp( 2) + 48*x*exp(4)) + log((x*log(3))/3)*(12*x^2*exp(2) - 84*x^2) - x^3),x)
Output:
-int((32*x^4*log((x*log(3))/3)^2 - log((x*log(3))/3)^3*(256*x^3*exp(2) - 1 792*x^3) + 32*x^4*log((x*log(3))/3))/(log((x*log(3))/3)^3*(9408*exp(2) - 1 344*exp(4) + 64*exp(6) - 21952) - log((x*log(3))/3)^2*(2352*x - 672*x*exp( 2) + 48*x*exp(4)) + log((x*log(3))/3)*(12*x^2*exp(2) - 84*x^2) - x^3), x)
Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.71 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\frac {16 \mathrm {log}\left (\frac {\mathrm {log}\left (3\right ) x}{3}\right )^{2} x^{4}}{16 \mathrm {log}\left (\frac {\mathrm {log}\left (3\right ) x}{3}\right )^{2} e^{4}-224 \mathrm {log}\left (\frac {\mathrm {log}\left (3\right ) x}{3}\right )^{2} e^{2}+784 \mathrm {log}\left (\frac {\mathrm {log}\left (3\right ) x}{3}\right )^{2}-8 \,\mathrm {log}\left (\frac {\mathrm {log}\left (3\right ) x}{3}\right ) e^{2} x +56 \,\mathrm {log}\left (\frac {\mathrm {log}\left (3\right ) x}{3}\right ) x +x^{2}} \] Input:
int(((256*x^3*exp(2)-1792*x^3)*log(1/3*x*log(3))^3-32*x^4*log(1/3*x*log(3) )^2-32*x^4*log(1/3*x*log(3)))/((64*exp(2)^3-1344*exp(2)^2+9408*exp(2)-2195 2)*log(1/3*x*log(3))^3+(-48*x*exp(2)^2+672*exp(2)*x-2352*x)*log(1/3*x*log( 3))^2+(12*x^2*exp(2)-84*x^2)*log(1/3*x*log(3))-x^3),x)
Output:
(16*log((log(3)*x)/3)**2*x**4)/(16*log((log(3)*x)/3)**2*e**4 - 224*log((lo g(3)*x)/3)**2*e**2 + 784*log((log(3)*x)/3)**2 - 8*log((log(3)*x)/3)*e**2*x + 56*log((log(3)*x)/3)*x + x**2)