Integrand size = 517, antiderivative size = 30 \[ \int \frac {-12-9 x+3 x^2+e^4 \left (3 x+3 x^2\right )+e^2 \left (-6 x+6 x^2\right )+\left (6 x-6 x^2+e^2 \left (-6 x-6 x^2\right )\right ) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)}{20 x^3-5 x^4-5 e^4 x^4+e^2 \left (20 x^3-10 x^4\right )+\left (-4 x^3+x^4+e^4 x^4+e^2 \left (-4 x^3+2 x^4\right )\right ) \log (3)+\left (-20 x^3+10 x^4+10 e^2 x^4+\left (4 x^3-2 x^4-2 e^2 x^4\right ) \log (3)\right ) \log (x)+\left (-5 x^4+x^4 \log (3)\right ) \log ^2(x)+\left (40 x^2-10 x^3-10 e^4 x^3+e^2 \left (40 x^2-20 x^3\right )+\left (-8 x^2+2 x^3+2 e^4 x^3+e^2 \left (-8 x^2+4 x^3\right )\right ) \log (3)+\left (-40 x^2+20 x^3+20 e^2 x^3+\left (8 x^2-4 x^3-4 e^2 x^3\right ) \log (3)\right ) \log (x)+\left (-10 x^3+2 x^3 \log (3)\right ) \log ^2(x)\right ) \log \left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )+\left (20 x-5 x^2-5 e^4 x^2+e^2 \left (20 x-10 x^2\right )+\left (-4 x+x^2+e^4 x^2+e^2 \left (-4 x+2 x^2\right )\right ) \log (3)+\left (-20 x+10 x^2+10 e^2 x^2+\left (4 x-2 x^2-2 e^2 x^2\right ) \log (3)\right ) \log (x)+\left (-5 x^2+x^2 \log (3)\right ) \log ^2(x)\right ) \log ^2\left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )} \, dx=\frac {3}{(5-\log (3)) \left (x+\log \left (x-\frac {4}{1+e^2-\log (x)}\right )\right )} \] Output:
3/(5-ln(3))/(x+ln(x-4/(exp(2)-ln(x)+1)))
Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {-12-9 x+3 x^2+e^4 \left (3 x+3 x^2\right )+e^2 \left (-6 x+6 x^2\right )+\left (6 x-6 x^2+e^2 \left (-6 x-6 x^2\right )\right ) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)}{20 x^3-5 x^4-5 e^4 x^4+e^2 \left (20 x^3-10 x^4\right )+\left (-4 x^3+x^4+e^4 x^4+e^2 \left (-4 x^3+2 x^4\right )\right ) \log (3)+\left (-20 x^3+10 x^4+10 e^2 x^4+\left (4 x^3-2 x^4-2 e^2 x^4\right ) \log (3)\right ) \log (x)+\left (-5 x^4+x^4 \log (3)\right ) \log ^2(x)+\left (40 x^2-10 x^3-10 e^4 x^3+e^2 \left (40 x^2-20 x^3\right )+\left (-8 x^2+2 x^3+2 e^4 x^3+e^2 \left (-8 x^2+4 x^3\right )\right ) \log (3)+\left (-40 x^2+20 x^3+20 e^2 x^3+\left (8 x^2-4 x^3-4 e^2 x^3\right ) \log (3)\right ) \log (x)+\left (-10 x^3+2 x^3 \log (3)\right ) \log ^2(x)\right ) \log \left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )+\left (20 x-5 x^2-5 e^4 x^2+e^2 \left (20 x-10 x^2\right )+\left (-4 x+x^2+e^4 x^2+e^2 \left (-4 x+2 x^2\right )\right ) \log (3)+\left (-20 x+10 x^2+10 e^2 x^2+\left (4 x-2 x^2-2 e^2 x^2\right ) \log (3)\right ) \log (x)+\left (-5 x^2+x^2 \log (3)\right ) \log ^2(x)\right ) \log ^2\left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )} \, dx=-\frac {3}{(-5+\log (3)) \left (x+\log \left (\frac {-4+x+e^2 x-x \log (x)}{1+e^2-\log (x)}\right )\right )} \] Input:
Integrate[(-12 - 9*x + 3*x^2 + E^4*(3*x + 3*x^2) + E^2*(-6*x + 6*x^2) + (6 *x - 6*x^2 + E^2*(-6*x - 6*x^2))*Log[x] + (3*x + 3*x^2)*Log[x]^2)/(20*x^3 - 5*x^4 - 5*E^4*x^4 + E^2*(20*x^3 - 10*x^4) + (-4*x^3 + x^4 + E^4*x^4 + E^ 2*(-4*x^3 + 2*x^4))*Log[3] + (-20*x^3 + 10*x^4 + 10*E^2*x^4 + (4*x^3 - 2*x ^4 - 2*E^2*x^4)*Log[3])*Log[x] + (-5*x^4 + x^4*Log[3])*Log[x]^2 + (40*x^2 - 10*x^3 - 10*E^4*x^3 + E^2*(40*x^2 - 20*x^3) + (-8*x^2 + 2*x^3 + 2*E^4*x^ 3 + E^2*(-8*x^2 + 4*x^3))*Log[3] + (-40*x^2 + 20*x^3 + 20*E^2*x^3 + (8*x^2 - 4*x^3 - 4*E^2*x^3)*Log[3])*Log[x] + (-10*x^3 + 2*x^3*Log[3])*Log[x]^2)* Log[(4 - x - E^2*x + x*Log[x])/(-1 - E^2 + Log[x])] + (20*x - 5*x^2 - 5*E^ 4*x^2 + E^2*(20*x - 10*x^2) + (-4*x + x^2 + E^4*x^2 + E^2*(-4*x + 2*x^2))* Log[3] + (-20*x + 10*x^2 + 10*E^2*x^2 + (4*x - 2*x^2 - 2*E^2*x^2)*Log[3])* Log[x] + (-5*x^2 + x^2*Log[3])*Log[x]^2)*Log[(4 - x - E^2*x + x*Log[x])/(- 1 - E^2 + Log[x])]^2),x]
Output:
-3/((-5 + Log[3])*(x + Log[(-4 + x + E^2*x - x*Log[x])/(1 + E^2 - Log[x])] ))
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.
| \(\displaystyle \int \frac {3 x^2+e^4 \left (3 x^2+3 x\right )+e^2 \left (6 x^2-6 x\right )+\left (3 x^2+3 x\right ) \log ^2(x)+\left (-6 x^2+e^2 \left (-6 x^2-6 x\right )+6 x\right ) \log (x)-9 x-12}{-5 e^4 x^4-5 x^4+\left (x^4 \log (3)-5 x^4\right ) \log ^2(x)+20 x^3+\left (-5 e^4 x^2-5 x^2+e^2 \left (20 x-10 x^2\right )+\left (x^2 \log (3)-5 x^2\right ) \log ^2(x)+\left (10 e^2 x^2+10 x^2+\left (-2 e^2 x^2-2 x^2+4 x\right ) \log (3)-20 x\right ) \log (x)+\left (e^4 x^2+x^2+e^2 \left (2 x^2-4 x\right )-4 x\right ) \log (3)+20 x\right ) \log ^2\left (\frac {-e^2 x-x+x \log (x)+4}{\log (x)-e^2-1}\right )+e^2 \left (20 x^3-10 x^4\right )+\left (10 e^2 x^4+10 x^4-20 x^3+\left (-2 e^2 x^4-2 x^4+4 x^3\right ) \log (3)\right ) \log (x)+\left (e^4 x^4+x^4-4 x^3+e^2 \left (2 x^4-4 x^3\right )\right ) \log (3)+\left (-10 e^4 x^3-10 x^3+\left (2 x^3 \log (3)-10 x^3\right ) \log ^2(x)+40 x^2+e^2 \left (40 x^2-20 x^3\right )+\left (20 e^2 x^3+20 x^3-40 x^2+\left (-4 e^2 x^3-4 x^3+8 x^2\right ) \log (3)\right ) \log (x)+\left (2 e^4 x^3+2 x^3-8 x^2+e^2 \left (4 x^3-8 x^2\right )\right ) \log (3)\right ) \log \left (\frac {-e^2 x-x+x \log (x)+4}{\log (x)-e^2-1}\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {3 x^2+e^4 \left (3 x^2+3 x\right )+e^2 \left (6 x^2-6 x\right )+\left (3 x^2+3 x\right ) \log ^2(x)+\left (-6 x^2+e^2 \left (-6 x^2-6 x\right )+6 x\right ) \log (x)-9 x-12}{\left (-5-5 e^4\right ) x^4+\left (x^4 \log (3)-5 x^4\right ) \log ^2(x)+20 x^3+\left (-5 e^4 x^2-5 x^2+e^2 \left (20 x-10 x^2\right )+\left (x^2 \log (3)-5 x^2\right ) \log ^2(x)+\left (10 e^2 x^2+10 x^2+\left (-2 e^2 x^2-2 x^2+4 x\right ) \log (3)-20 x\right ) \log (x)+\left (e^4 x^2+x^2+e^2 \left (2 x^2-4 x\right )-4 x\right ) \log (3)+20 x\right ) \log ^2\left (\frac {-e^2 x-x+x \log (x)+4}{\log (x)-e^2-1}\right )+e^2 \left (20 x^3-10 x^4\right )+\left (10 e^2 x^4+10 x^4-20 x^3+\left (-2 e^2 x^4-2 x^4+4 x^3\right ) \log (3)\right ) \log (x)+\left (e^4 x^4+x^4-4 x^3+e^2 \left (2 x^4-4 x^3\right )\right ) \log (3)+\left (-10 e^4 x^3-10 x^3+\left (2 x^3 \log (3)-10 x^3\right ) \log ^2(x)+40 x^2+e^2 \left (40 x^2-20 x^3\right )+\left (20 e^2 x^3+20 x^3-40 x^2+\left (-4 e^2 x^3-4 x^3+8 x^2\right ) \log (3)\right ) \log (x)+\left (2 e^4 x^3+2 x^3-8 x^2+e^2 \left (4 x^3-8 x^2\right )\right ) \log (3)\right ) \log \left (\frac {-e^2 x-x+x \log (x)+4}{\log (x)-e^2-1}\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {3 \left (\left (1+e^2\right )^2 x^2+\left (-3-2 e^2+e^4\right ) x+(x+1) x \log ^2(x)-2 \left (x+e^2 (x+1)-1\right ) x \log (x)-4\right )}{x (5-\log (3)) \left (-\log (x)+e^2+1\right ) \left (-\left (1+e^2\right ) x+x \log (x)+4\right ) \left (x+\log \left (\frac {e^2 x+x+x (-\log (x))-4}{-\log (x)+e^2+1}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int -\frac {-\left (1+e^2\right )^2 x^2-(x+1) \log ^2(x) x-2 \left (-x-e^2 (x+1)+1\right ) \log (x) x+\left (3+2 e^2-e^4\right ) x+4}{x \left (-\log (x)+e^2+1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (-\frac {\log (x) x-e^2 x-x+4}{-\log (x)+e^2+1}\right )\right )^2}dx}{5-\log (3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \int \frac {-\left (1+e^2\right )^2 x^2-(x+1) \log ^2(x) x-2 \left (-x-e^2 (x+1)+1\right ) \log (x) x+\left (3-e^2\right ) \left (1+e^2\right ) x+4}{x \left (-\log (x)+e^2+1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (-\frac {\log (x) x-e^2 x-x+4}{-\log (x)+e^2+1}\right )\right )^2}dx}{5-\log (3)}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \int \left (\frac {x \log ^2(x)}{\left (\log (x)-e^2-1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}+\frac {\log ^2(x)}{\left (\log (x)-e^2-1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}+\frac {2 \left (1+e^2\right ) x \log (x)}{\left (-\log (x)+e^2+1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}+\frac {2 \left (1-\frac {1}{e^2}\right ) e^2 \log (x)}{\left (-\log (x)+e^2+1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}+\frac {4}{x \left (-\log (x)+e^2+1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}+\frac {\left (3-e^2\right ) \left (1+e^2\right )}{\left (-\log (x)+e^2+1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}+\frac {\left (1+e^2\right )^2 x}{\left (\log (x)-e^2-1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}\right )dx}{5-\log (3)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \left (\int \frac {\log ^2(x)}{\left (\log (x)-e^2-1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}dx+\int \frac {x \log ^2(x)}{\left (\log (x)-e^2-1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}dx+\left (3-e^2\right ) \left (1+e^2\right ) \int \frac {1}{\left (-\log (x)+e^2+1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}dx+4 \int \frac {1}{x \left (-\log (x)+e^2+1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}dx-2 \left (1-e^2\right ) \int \frac {\log (x)}{\left (-\log (x)+e^2+1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}dx+2 \left (1+e^2\right ) \int \frac {x \log (x)}{\left (-\log (x)+e^2+1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}dx+\left (1+e^2\right )^2 \int \frac {x}{\left (\log (x)-e^2-1\right ) \left (\log (x) x-\left (1+e^2\right ) x+4\right ) \left (x+\log \left (\frac {-\log (x) x+e^2 x+x-4}{-\log (x)+e^2+1}\right )\right )^2}dx\right )}{5-\log (3)}\) |
Input:
Int[(-12 - 9*x + 3*x^2 + E^4*(3*x + 3*x^2) + E^2*(-6*x + 6*x^2) + (6*x - 6 *x^2 + E^2*(-6*x - 6*x^2))*Log[x] + (3*x + 3*x^2)*Log[x]^2)/(20*x^3 - 5*x^ 4 - 5*E^4*x^4 + E^2*(20*x^3 - 10*x^4) + (-4*x^3 + x^4 + E^4*x^4 + E^2*(-4* x^3 + 2*x^4))*Log[3] + (-20*x^3 + 10*x^4 + 10*E^2*x^4 + (4*x^3 - 2*x^4 - 2 *E^2*x^4)*Log[3])*Log[x] + (-5*x^4 + x^4*Log[3])*Log[x]^2 + (40*x^2 - 10*x ^3 - 10*E^4*x^3 + E^2*(40*x^2 - 20*x^3) + (-8*x^2 + 2*x^3 + 2*E^4*x^3 + E^ 2*(-8*x^2 + 4*x^3))*Log[3] + (-40*x^2 + 20*x^3 + 20*E^2*x^3 + (8*x^2 - 4*x ^3 - 4*E^2*x^3)*Log[3])*Log[x] + (-10*x^3 + 2*x^3*Log[3])*Log[x]^2)*Log[(4 - x - E^2*x + x*Log[x])/(-1 - E^2 + Log[x])] + (20*x - 5*x^2 - 5*E^4*x^2 + E^2*(20*x - 10*x^2) + (-4*x + x^2 + E^4*x^2 + E^2*(-4*x + 2*x^2))*Log[3] + (-20*x + 10*x^2 + 10*E^2*x^2 + (4*x - 2*x^2 - 2*E^2*x^2)*Log[3])*Log[x] + (-5*x^2 + x^2*Log[3])*Log[x]^2)*Log[(4 - x - E^2*x + x*Log[x])/(-1 - E^ 2 + Log[x])]^2),x]
Output:
$Aborted
Time = 15.58 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30
| method | result | size |
| parallelrisch | \(-\frac {3}{\left (\ln \left (\frac {x \ln \left (x \right )-{\mathrm e}^{2} x -x +4}{\ln \left (x \right )-1-{\mathrm e}^{2}}\right )+x \right ) \left (\ln \left (3\right )-5\right )}\) | \(39\) |
| risch | \(-\frac {6 i}{\left (\ln \left (3\right )-5\right ) \left (\pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2} x -4+x \left (1-\ln \left (x \right )\right )\right )\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2} x -4+x \left (1-\ln \left (x \right )\right )\right )}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right )-\pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2} x -4+x \left (1-\ln \left (x \right )\right )\right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2} x -4+x \left (1-\ln \left (x \right )\right )\right )}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right )}^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2} x -4+x \left (1-\ln \left (x \right )\right )\right )}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2} x -4+x \left (1-\ln \left (x \right )\right )\right )}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right )}^{3}+2 i x -2 i \ln \left ({\mathrm e}^{2}-\ln \left (x \right )+1\right )+2 i \ln \left ({\mathrm e}^{2} x -4+x \left (1-\ln \left (x \right )\right )\right )\right )}\) | \(239\) |
| default | \(-\frac {6 i}{\left (\ln \left (3\right )-5\right ) \left (\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right ) \operatorname {csgn}\left (i \left (x \ln \left (x \right )-x -{\mathrm e}^{\ln \left (x \right )+2}+4\right )\right ) \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-x -{\mathrm e}^{\ln \left (x \right )+2}+4\right )}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right ) {\operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-x -{\mathrm e}^{\ln \left (x \right )+2}+4\right )}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (x \ln \left (x \right )-x -{\mathrm e}^{\ln \left (x \right )+2}+4\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-x -{\mathrm e}^{\ln \left (x \right )+2}+4\right )}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right )}^{2}-\pi {\operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-x -{\mathrm e}^{\ln \left (x \right )+2}+4\right )}{{\mathrm e}^{2}-\ln \left (x \right )+1}\right )}^{3}+2 i x -2 i \ln \left ({\mathrm e}^{2}-\ln \left (x \right )+1\right )+2 i \ln \left (-x \ln \left (x \right )+x +{\mathrm e}^{\ln \left (x \right )+2}-4\right )\right )}\) | \(250\) |
Input:
int(((3*x^2+3*x)*ln(x)^2+((-6*x^2-6*x)*exp(2)-6*x^2+6*x)*ln(x)+(3*x^2+3*x) *exp(2)^2+(6*x^2-6*x)*exp(2)+3*x^2-9*x-12)/(((x^2*ln(3)-5*x^2)*ln(x)^2+((- 2*x^2*exp(2)-2*x^2+4*x)*ln(3)+10*x^2*exp(2)+10*x^2-20*x)*ln(x)+(x^2*exp(2) ^2+(2*x^2-4*x)*exp(2)+x^2-4*x)*ln(3)-5*x^2*exp(2)^2+(-10*x^2+20*x)*exp(2)- 5*x^2+20*x)*ln((x*ln(x)-exp(2)*x-x+4)/(ln(x)-1-exp(2)))^2+((2*x^3*ln(3)-10 *x^3)*ln(x)^2+((-4*x^3*exp(2)-4*x^3+8*x^2)*ln(3)+20*x^3*exp(2)+20*x^3-40*x ^2)*ln(x)+(2*x^3*exp(2)^2+(4*x^3-8*x^2)*exp(2)+2*x^3-8*x^2)*ln(3)-10*x^3*e xp(2)^2+(-20*x^3+40*x^2)*exp(2)-10*x^3+40*x^2)*ln((x*ln(x)-exp(2)*x-x+4)/( ln(x)-1-exp(2)))+(x^4*ln(3)-5*x^4)*ln(x)^2+((-2*x^4*exp(2)-2*x^4+4*x^3)*ln (3)+10*x^4*exp(2)+10*x^4-20*x^3)*ln(x)+(x^4*exp(2)^2+(2*x^4-4*x^3)*exp(2)+ x^4-4*x^3)*ln(3)-5*x^4*exp(2)^2+(-10*x^4+20*x^3)*exp(2)-5*x^4+20*x^3),x,me thod=_RETURNVERBOSE)
Output:
-3/(ln((x*ln(x)-exp(2)*x-x+4)/(ln(x)-1-exp(2)))+x)/(ln(3)-5)
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {-12-9 x+3 x^2+e^4 \left (3 x+3 x^2\right )+e^2 \left (-6 x+6 x^2\right )+\left (6 x-6 x^2+e^2 \left (-6 x-6 x^2\right )\right ) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)}{20 x^3-5 x^4-5 e^4 x^4+e^2 \left (20 x^3-10 x^4\right )+\left (-4 x^3+x^4+e^4 x^4+e^2 \left (-4 x^3+2 x^4\right )\right ) \log (3)+\left (-20 x^3+10 x^4+10 e^2 x^4+\left (4 x^3-2 x^4-2 e^2 x^4\right ) \log (3)\right ) \log (x)+\left (-5 x^4+x^4 \log (3)\right ) \log ^2(x)+\left (40 x^2-10 x^3-10 e^4 x^3+e^2 \left (40 x^2-20 x^3\right )+\left (-8 x^2+2 x^3+2 e^4 x^3+e^2 \left (-8 x^2+4 x^3\right )\right ) \log (3)+\left (-40 x^2+20 x^3+20 e^2 x^3+\left (8 x^2-4 x^3-4 e^2 x^3\right ) \log (3)\right ) \log (x)+\left (-10 x^3+2 x^3 \log (3)\right ) \log ^2(x)\right ) \log \left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )+\left (20 x-5 x^2-5 e^4 x^2+e^2 \left (20 x-10 x^2\right )+\left (-4 x+x^2+e^4 x^2+e^2 \left (-4 x+2 x^2\right )\right ) \log (3)+\left (-20 x+10 x^2+10 e^2 x^2+\left (4 x-2 x^2-2 e^2 x^2\right ) \log (3)\right ) \log (x)+\left (-5 x^2+x^2 \log (3)\right ) \log ^2(x)\right ) \log ^2\left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )} \, dx=-\frac {3}{x \log \left (3\right ) + {\left (\log \left (3\right ) - 5\right )} \log \left (\frac {x e^{2} - x \log \left (x\right ) + x - 4}{e^{2} - \log \left (x\right ) + 1}\right ) - 5 \, x} \] Input:
integrate(((3*x^2+3*x)*log(x)^2+((-6*x^2-6*x)*exp(2)-6*x^2+6*x)*log(x)+(3* x^2+3*x)*exp(2)^2+(6*x^2-6*x)*exp(2)+3*x^2-9*x-12)/(((x^2*log(3)-5*x^2)*lo g(x)^2+((-2*x^2*exp(2)-2*x^2+4*x)*log(3)+10*x^2*exp(2)+10*x^2-20*x)*log(x) +(x^2*exp(2)^2+(2*x^2-4*x)*exp(2)+x^2-4*x)*log(3)-5*x^2*exp(2)^2+(-10*x^2+ 20*x)*exp(2)-5*x^2+20*x)*log((x*log(x)-exp(2)*x-x+4)/(log(x)-1-exp(2)))^2+ ((2*x^3*log(3)-10*x^3)*log(x)^2+((-4*x^3*exp(2)-4*x^3+8*x^2)*log(3)+20*x^3 *exp(2)+20*x^3-40*x^2)*log(x)+(2*x^3*exp(2)^2+(4*x^3-8*x^2)*exp(2)+2*x^3-8 *x^2)*log(3)-10*x^3*exp(2)^2+(-20*x^3+40*x^2)*exp(2)-10*x^3+40*x^2)*log((x *log(x)-exp(2)*x-x+4)/(log(x)-1-exp(2)))+(x^4*log(3)-5*x^4)*log(x)^2+((-2* x^4*exp(2)-2*x^4+4*x^3)*log(3)+10*x^4*exp(2)+10*x^4-20*x^3)*log(x)+(x^4*ex p(2)^2+(2*x^4-4*x^3)*exp(2)+x^4-4*x^3)*log(3)-5*x^4*exp(2)^2+(-10*x^4+20*x ^3)*exp(2)-5*x^4+20*x^3),x, algorithm="fricas")
Output:
-3/(x*log(3) + (log(3) - 5)*log((x*e^2 - x*log(x) + x - 4)/(e^2 - log(x) + 1)) - 5*x)
Time = 0.48 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-12-9 x+3 x^2+e^4 \left (3 x+3 x^2\right )+e^2 \left (-6 x+6 x^2\right )+\left (6 x-6 x^2+e^2 \left (-6 x-6 x^2\right )\right ) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)}{20 x^3-5 x^4-5 e^4 x^4+e^2 \left (20 x^3-10 x^4\right )+\left (-4 x^3+x^4+e^4 x^4+e^2 \left (-4 x^3+2 x^4\right )\right ) \log (3)+\left (-20 x^3+10 x^4+10 e^2 x^4+\left (4 x^3-2 x^4-2 e^2 x^4\right ) \log (3)\right ) \log (x)+\left (-5 x^4+x^4 \log (3)\right ) \log ^2(x)+\left (40 x^2-10 x^3-10 e^4 x^3+e^2 \left (40 x^2-20 x^3\right )+\left (-8 x^2+2 x^3+2 e^4 x^3+e^2 \left (-8 x^2+4 x^3\right )\right ) \log (3)+\left (-40 x^2+20 x^3+20 e^2 x^3+\left (8 x^2-4 x^3-4 e^2 x^3\right ) \log (3)\right ) \log (x)+\left (-10 x^3+2 x^3 \log (3)\right ) \log ^2(x)\right ) \log \left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )+\left (20 x-5 x^2-5 e^4 x^2+e^2 \left (20 x-10 x^2\right )+\left (-4 x+x^2+e^4 x^2+e^2 \left (-4 x+2 x^2\right )\right ) \log (3)+\left (-20 x+10 x^2+10 e^2 x^2+\left (4 x-2 x^2-2 e^2 x^2\right ) \log (3)\right ) \log (x)+\left (-5 x^2+x^2 \log (3)\right ) \log ^2(x)\right ) \log ^2\left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )} \, dx=- \frac {3}{- 5 x + x \log {\left (3 \right )} + \left (-5 + \log {\left (3 \right )}\right ) \log {\left (\frac {x \log {\left (x \right )} - x e^{2} - x + 4}{\log {\left (x \right )} - e^{2} - 1} \right )}} \] Input:
integrate(((3*x**2+3*x)*ln(x)**2+((-6*x**2-6*x)*exp(2)-6*x**2+6*x)*ln(x)+( 3*x**2+3*x)*exp(2)**2+(6*x**2-6*x)*exp(2)+3*x**2-9*x-12)/(((x**2*ln(3)-5*x **2)*ln(x)**2+((-2*x**2*exp(2)-2*x**2+4*x)*ln(3)+10*x**2*exp(2)+10*x**2-20 *x)*ln(x)+(x**2*exp(2)**2+(2*x**2-4*x)*exp(2)+x**2-4*x)*ln(3)-5*x**2*exp(2 )**2+(-10*x**2+20*x)*exp(2)-5*x**2+20*x)*ln((x*ln(x)-exp(2)*x-x+4)/(ln(x)- 1-exp(2)))**2+((2*x**3*ln(3)-10*x**3)*ln(x)**2+((-4*x**3*exp(2)-4*x**3+8*x **2)*ln(3)+20*x**3*exp(2)+20*x**3-40*x**2)*ln(x)+(2*x**3*exp(2)**2+(4*x**3 -8*x**2)*exp(2)+2*x**3-8*x**2)*ln(3)-10*x**3*exp(2)**2+(-20*x**3+40*x**2)* exp(2)-10*x**3+40*x**2)*ln((x*ln(x)-exp(2)*x-x+4)/(ln(x)-1-exp(2)))+(x**4* ln(3)-5*x**4)*ln(x)**2+((-2*x**4*exp(2)-2*x**4+4*x**3)*ln(3)+10*x**4*exp(2 )+10*x**4-20*x**3)*ln(x)+(x**4*exp(2)**2+(2*x**4-4*x**3)*exp(2)+x**4-4*x** 3)*ln(3)-5*x**4*exp(2)**2+(-10*x**4+20*x**3)*exp(2)-5*x**4+20*x**3),x)
Output:
-3/(-5*x + x*log(3) + (-5 + log(3))*log((x*log(x) - x*exp(2) - x + 4)/(log (x) - exp(2) - 1)))
Time = 0.41 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {-12-9 x+3 x^2+e^4 \left (3 x+3 x^2\right )+e^2 \left (-6 x+6 x^2\right )+\left (6 x-6 x^2+e^2 \left (-6 x-6 x^2\right )\right ) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)}{20 x^3-5 x^4-5 e^4 x^4+e^2 \left (20 x^3-10 x^4\right )+\left (-4 x^3+x^4+e^4 x^4+e^2 \left (-4 x^3+2 x^4\right )\right ) \log (3)+\left (-20 x^3+10 x^4+10 e^2 x^4+\left (4 x^3-2 x^4-2 e^2 x^4\right ) \log (3)\right ) \log (x)+\left (-5 x^4+x^4 \log (3)\right ) \log ^2(x)+\left (40 x^2-10 x^3-10 e^4 x^3+e^2 \left (40 x^2-20 x^3\right )+\left (-8 x^2+2 x^3+2 e^4 x^3+e^2 \left (-8 x^2+4 x^3\right )\right ) \log (3)+\left (-40 x^2+20 x^3+20 e^2 x^3+\left (8 x^2-4 x^3-4 e^2 x^3\right ) \log (3)\right ) \log (x)+\left (-10 x^3+2 x^3 \log (3)\right ) \log ^2(x)\right ) \log \left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )+\left (20 x-5 x^2-5 e^4 x^2+e^2 \left (20 x-10 x^2\right )+\left (-4 x+x^2+e^4 x^2+e^2 \left (-4 x+2 x^2\right )\right ) \log (3)+\left (-20 x+10 x^2+10 e^2 x^2+\left (4 x-2 x^2-2 e^2 x^2\right ) \log (3)\right ) \log (x)+\left (-5 x^2+x^2 \log (3)\right ) \log ^2(x)\right ) \log ^2\left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )} \, dx=-\frac {3}{x {\left (\log \left (3\right ) - 5\right )} + {\left (\log \left (3\right ) - 5\right )} \log \left (-x {\left (e^{2} + 1\right )} + x \log \left (x\right ) + 4\right ) - {\left (\log \left (3\right ) - 5\right )} \log \left (-e^{2} + \log \left (x\right ) - 1\right )} \] Input:
integrate(((3*x^2+3*x)*log(x)^2+((-6*x^2-6*x)*exp(2)-6*x^2+6*x)*log(x)+(3* x^2+3*x)*exp(2)^2+(6*x^2-6*x)*exp(2)+3*x^2-9*x-12)/(((x^2*log(3)-5*x^2)*lo g(x)^2+((-2*x^2*exp(2)-2*x^2+4*x)*log(3)+10*x^2*exp(2)+10*x^2-20*x)*log(x) +(x^2*exp(2)^2+(2*x^2-4*x)*exp(2)+x^2-4*x)*log(3)-5*x^2*exp(2)^2+(-10*x^2+ 20*x)*exp(2)-5*x^2+20*x)*log((x*log(x)-exp(2)*x-x+4)/(log(x)-1-exp(2)))^2+ ((2*x^3*log(3)-10*x^3)*log(x)^2+((-4*x^3*exp(2)-4*x^3+8*x^2)*log(3)+20*x^3 *exp(2)+20*x^3-40*x^2)*log(x)+(2*x^3*exp(2)^2+(4*x^3-8*x^2)*exp(2)+2*x^3-8 *x^2)*log(3)-10*x^3*exp(2)^2+(-20*x^3+40*x^2)*exp(2)-10*x^3+40*x^2)*log((x *log(x)-exp(2)*x-x+4)/(log(x)-1-exp(2)))+(x^4*log(3)-5*x^4)*log(x)^2+((-2* x^4*exp(2)-2*x^4+4*x^3)*log(3)+10*x^4*exp(2)+10*x^4-20*x^3)*log(x)+(x^4*ex p(2)^2+(2*x^4-4*x^3)*exp(2)+x^4-4*x^3)*log(3)-5*x^4*exp(2)^2+(-10*x^4+20*x ^3)*exp(2)-5*x^4+20*x^3),x, algorithm="maxima")
Output:
-3/(x*(log(3) - 5) + (log(3) - 5)*log(-x*(e^2 + 1) + x*log(x) + 4) - (log( 3) - 5)*log(-e^2 + log(x) - 1))
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (27) = 54\).
Time = 1.78 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.23 \[ \int \frac {-12-9 x+3 x^2+e^4 \left (3 x+3 x^2\right )+e^2 \left (-6 x+6 x^2\right )+\left (6 x-6 x^2+e^2 \left (-6 x-6 x^2\right )\right ) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)}{20 x^3-5 x^4-5 e^4 x^4+e^2 \left (20 x^3-10 x^4\right )+\left (-4 x^3+x^4+e^4 x^4+e^2 \left (-4 x^3+2 x^4\right )\right ) \log (3)+\left (-20 x^3+10 x^4+10 e^2 x^4+\left (4 x^3-2 x^4-2 e^2 x^4\right ) \log (3)\right ) \log (x)+\left (-5 x^4+x^4 \log (3)\right ) \log ^2(x)+\left (40 x^2-10 x^3-10 e^4 x^3+e^2 \left (40 x^2-20 x^3\right )+\left (-8 x^2+2 x^3+2 e^4 x^3+e^2 \left (-8 x^2+4 x^3\right )\right ) \log (3)+\left (-40 x^2+20 x^3+20 e^2 x^3+\left (8 x^2-4 x^3-4 e^2 x^3\right ) \log (3)\right ) \log (x)+\left (-10 x^3+2 x^3 \log (3)\right ) \log ^2(x)\right ) \log \left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )+\left (20 x-5 x^2-5 e^4 x^2+e^2 \left (20 x-10 x^2\right )+\left (-4 x+x^2+e^4 x^2+e^2 \left (-4 x+2 x^2\right )\right ) \log (3)+\left (-20 x+10 x^2+10 e^2 x^2+\left (4 x-2 x^2-2 e^2 x^2\right ) \log (3)\right ) \log (x)+\left (-5 x^2+x^2 \log (3)\right ) \log ^2(x)\right ) \log ^2\left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )} \, dx=-\frac {3}{x \log \left (3\right ) + \log \left (3\right ) \log \left (x e^{2} - x \log \left (x\right ) + x - 4\right ) - \log \left (3\right ) \log \left (e^{2} - \log \left (x\right ) + 1\right ) - 5 \, x - 5 \, \log \left (x e^{2} - x \log \left (x\right ) + x - 4\right ) + 5 \, \log \left (e^{2} - \log \left (x\right ) + 1\right )} \] Input:
integrate(((3*x^2+3*x)*log(x)^2+((-6*x^2-6*x)*exp(2)-6*x^2+6*x)*log(x)+(3* x^2+3*x)*exp(2)^2+(6*x^2-6*x)*exp(2)+3*x^2-9*x-12)/(((x^2*log(3)-5*x^2)*lo g(x)^2+((-2*x^2*exp(2)-2*x^2+4*x)*log(3)+10*x^2*exp(2)+10*x^2-20*x)*log(x) +(x^2*exp(2)^2+(2*x^2-4*x)*exp(2)+x^2-4*x)*log(3)-5*x^2*exp(2)^2+(-10*x^2+ 20*x)*exp(2)-5*x^2+20*x)*log((x*log(x)-exp(2)*x-x+4)/(log(x)-1-exp(2)))^2+ ((2*x^3*log(3)-10*x^3)*log(x)^2+((-4*x^3*exp(2)-4*x^3+8*x^2)*log(3)+20*x^3 *exp(2)+20*x^3-40*x^2)*log(x)+(2*x^3*exp(2)^2+(4*x^3-8*x^2)*exp(2)+2*x^3-8 *x^2)*log(3)-10*x^3*exp(2)^2+(-20*x^3+40*x^2)*exp(2)-10*x^3+40*x^2)*log((x *log(x)-exp(2)*x-x+4)/(log(x)-1-exp(2)))+(x^4*log(3)-5*x^4)*log(x)^2+((-2* x^4*exp(2)-2*x^4+4*x^3)*log(3)+10*x^4*exp(2)+10*x^4-20*x^3)*log(x)+(x^4*ex p(2)^2+(2*x^4-4*x^3)*exp(2)+x^4-4*x^3)*log(3)-5*x^4*exp(2)^2+(-10*x^4+20*x ^3)*exp(2)-5*x^4+20*x^3),x, algorithm="giac")
Output:
-3/(x*log(3) + log(3)*log(x*e^2 - x*log(x) + x - 4) - log(3)*log(e^2 - log (x) + 1) - 5*x - 5*log(x*e^2 - x*log(x) + x - 4) + 5*log(e^2 - log(x) + 1) )
Time = 3.75 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {-12-9 x+3 x^2+e^4 \left (3 x+3 x^2\right )+e^2 \left (-6 x+6 x^2\right )+\left (6 x-6 x^2+e^2 \left (-6 x-6 x^2\right )\right ) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)}{20 x^3-5 x^4-5 e^4 x^4+e^2 \left (20 x^3-10 x^4\right )+\left (-4 x^3+x^4+e^4 x^4+e^2 \left (-4 x^3+2 x^4\right )\right ) \log (3)+\left (-20 x^3+10 x^4+10 e^2 x^4+\left (4 x^3-2 x^4-2 e^2 x^4\right ) \log (3)\right ) \log (x)+\left (-5 x^4+x^4 \log (3)\right ) \log ^2(x)+\left (40 x^2-10 x^3-10 e^4 x^3+e^2 \left (40 x^2-20 x^3\right )+\left (-8 x^2+2 x^3+2 e^4 x^3+e^2 \left (-8 x^2+4 x^3\right )\right ) \log (3)+\left (-40 x^2+20 x^3+20 e^2 x^3+\left (8 x^2-4 x^3-4 e^2 x^3\right ) \log (3)\right ) \log (x)+\left (-10 x^3+2 x^3 \log (3)\right ) \log ^2(x)\right ) \log \left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )+\left (20 x-5 x^2-5 e^4 x^2+e^2 \left (20 x-10 x^2\right )+\left (-4 x+x^2+e^4 x^2+e^2 \left (-4 x+2 x^2\right )\right ) \log (3)+\left (-20 x+10 x^2+10 e^2 x^2+\left (4 x-2 x^2-2 e^2 x^2\right ) \log (3)\right ) \log (x)+\left (-5 x^2+x^2 \log (3)\right ) \log ^2(x)\right ) \log ^2\left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )} \, dx=-\frac {3}{\left (\ln \left (3\right )-5\right )\,\left (x+\ln \left (\frac {x+x\,{\mathrm {e}}^2-x\,\ln \left (x\right )-4}{{\mathrm {e}}^2-\ln \left (x\right )+1}\right )\right )} \] Input:
int((9*x - log(x)^2*(3*x + 3*x^2) - exp(4)*(3*x + 3*x^2) + exp(2)*(6*x - 6 *x^2) - 3*x^2 + log(x)*(exp(2)*(6*x + 6*x^2) - 6*x + 6*x^2) + 12)/(log((x + x*exp(2) - x*log(x) - 4)/(exp(2) - log(x) + 1))^2*(log(3)*(4*x + exp(2)* (4*x - 2*x^2) - x^2*exp(4) - x^2) - 20*x + log(x)*(20*x - 10*x^2*exp(2) + log(3)*(2*x^2*exp(2) - 4*x + 2*x^2) - 10*x^2) - exp(2)*(20*x - 10*x^2) + 5 *x^2*exp(4) + 5*x^2 - log(x)^2*(x^2*log(3) - 5*x^2)) - exp(2)*(20*x^3 - 10 *x^4) + 5*x^4*exp(4) + log(3)*(exp(2)*(4*x^3 - 2*x^4) - x^4*exp(4) + 4*x^3 - x^4) - 20*x^3 + 5*x^4 - log(x)^2*(x^4*log(3) - 5*x^4) + log(x)*(log(3)* (2*x^4*exp(2) - 4*x^3 + 2*x^4) - 10*x^4*exp(2) + 20*x^3 - 10*x^4) + log((x + x*exp(2) - x*log(x) - 4)/(exp(2) - log(x) + 1))*(10*x^3*exp(4) - exp(2) *(40*x^2 - 20*x^3) + log(3)*(exp(2)*(8*x^2 - 4*x^3) - 2*x^3*exp(4) + 8*x^2 - 2*x^3) - 40*x^2 + 10*x^3 - log(x)^2*(2*x^3*log(3) - 10*x^3) + log(x)*(l og(3)*(4*x^3*exp(2) - 8*x^2 + 4*x^3) - 20*x^3*exp(2) + 40*x^2 - 20*x^3))), x)
Output:
-3/((log(3) - 5)*(x + log((x + x*exp(2) - x*log(x) - 4)/(exp(2) - log(x) + 1))))
Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.43 \[ \int \frac {-12-9 x+3 x^2+e^4 \left (3 x+3 x^2\right )+e^2 \left (-6 x+6 x^2\right )+\left (6 x-6 x^2+e^2 \left (-6 x-6 x^2\right )\right ) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)}{20 x^3-5 x^4-5 e^4 x^4+e^2 \left (20 x^3-10 x^4\right )+\left (-4 x^3+x^4+e^4 x^4+e^2 \left (-4 x^3+2 x^4\right )\right ) \log (3)+\left (-20 x^3+10 x^4+10 e^2 x^4+\left (4 x^3-2 x^4-2 e^2 x^4\right ) \log (3)\right ) \log (x)+\left (-5 x^4+x^4 \log (3)\right ) \log ^2(x)+\left (40 x^2-10 x^3-10 e^4 x^3+e^2 \left (40 x^2-20 x^3\right )+\left (-8 x^2+2 x^3+2 e^4 x^3+e^2 \left (-8 x^2+4 x^3\right )\right ) \log (3)+\left (-40 x^2+20 x^3+20 e^2 x^3+\left (8 x^2-4 x^3-4 e^2 x^3\right ) \log (3)\right ) \log (x)+\left (-10 x^3+2 x^3 \log (3)\right ) \log ^2(x)\right ) \log \left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )+\left (20 x-5 x^2-5 e^4 x^2+e^2 \left (20 x-10 x^2\right )+\left (-4 x+x^2+e^4 x^2+e^2 \left (-4 x+2 x^2\right )\right ) \log (3)+\left (-20 x+10 x^2+10 e^2 x^2+\left (4 x-2 x^2-2 e^2 x^2\right ) \log (3)\right ) \log (x)+\left (-5 x^2+x^2 \log (3)\right ) \log ^2(x)\right ) \log ^2\left (\frac {4-x-e^2 x+x \log (x)}{-1-e^2+\log (x)}\right )} \, dx=-\frac {3}{\mathrm {log}\left (\frac {\mathrm {log}\left (x \right ) x -e^{2} x -x +4}{\mathrm {log}\left (x \right )-e^{2}-1}\right ) \mathrm {log}\left (3\right )-5 \,\mathrm {log}\left (\frac {\mathrm {log}\left (x \right ) x -e^{2} x -x +4}{\mathrm {log}\left (x \right )-e^{2}-1}\right )+\mathrm {log}\left (3\right ) x -5 x} \] Input:
int(((3*x^2+3*x)*log(x)^2+((-6*x^2-6*x)*exp(2)-6*x^2+6*x)*log(x)+(3*x^2+3* x)*exp(2)^2+(6*x^2-6*x)*exp(2)+3*x^2-9*x-12)/(((x^2*log(3)-5*x^2)*log(x)^2 +((-2*x^2*exp(2)-2*x^2+4*x)*log(3)+10*x^2*exp(2)+10*x^2-20*x)*log(x)+(x^2* exp(2)^2+(2*x^2-4*x)*exp(2)+x^2-4*x)*log(3)-5*x^2*exp(2)^2+(-10*x^2+20*x)* exp(2)-5*x^2+20*x)*log((x*log(x)-exp(2)*x-x+4)/(log(x)-1-exp(2)))^2+((2*x^ 3*log(3)-10*x^3)*log(x)^2+((-4*x^3*exp(2)-4*x^3+8*x^2)*log(3)+20*x^3*exp(2 )+20*x^3-40*x^2)*log(x)+(2*x^3*exp(2)^2+(4*x^3-8*x^2)*exp(2)+2*x^3-8*x^2)* log(3)-10*x^3*exp(2)^2+(-20*x^3+40*x^2)*exp(2)-10*x^3+40*x^2)*log((x*log(x )-exp(2)*x-x+4)/(log(x)-1-exp(2)))+(x^4*log(3)-5*x^4)*log(x)^2+((-2*x^4*ex p(2)-2*x^4+4*x^3)*log(3)+10*x^4*exp(2)+10*x^4-20*x^3)*log(x)+(x^4*exp(2)^2 +(2*x^4-4*x^3)*exp(2)+x^4-4*x^3)*log(3)-5*x^4*exp(2)^2+(-10*x^4+20*x^3)*ex p(2)-5*x^4+20*x^3),x)
Output:
( - 3)/(log((log(x)*x - e**2*x - x + 4)/(log(x) - e**2 - 1))*log(3) - 5*lo g((log(x)*x - e**2*x - x + 4)/(log(x) - e**2 - 1)) + log(3)*x - 5*x)