Integrand size = 152, antiderivative size = 17 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\log \left (\log \left (\log \left (x+\frac {x^2}{(-x+\log (x))^2}\right )\right )\right ) \] Output:
ln(ln(ln(x^2/(ln(x)-x)^2+x)))
Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\log \left (\log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )\right ) \] Input:
Integrate[(-2*x - x^3 + (2*x + 3*x^2)*Log[x] - 3*x*Log[x]^2 + Log[x]^3)/(( -x^3 - x^4 + (x^2 + 3*x^3)*Log[x] - 3*x^2*Log[x]^2 + x*Log[x]^3)*Log[(x^2 + x^3 - 2*x^2*Log[x] + x*Log[x]^2)/(x^2 - 2*x*Log[x] + Log[x]^2)]*Log[Log[ (x^2 + x^3 - 2*x^2*Log[x] + x*Log[x]^2)/(x^2 - 2*x*Log[x] + Log[x]^2)]]),x ]
Output:
Log[Log[Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/(x - Log[x])^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.
| \(\displaystyle \int \frac {-x^3+\left (3 x^2+2 x\right ) \log (x)-2 x+\log ^3(x)-3 x \log ^2(x)}{\left (-x^4-x^3-3 x^2 \log ^2(x)+\left (3 x^3+x^2\right ) \log (x)+x \log ^3(x)\right ) \log \left (\frac {x^3+x^2-2 x^2 \log (x)+x \log ^2(x)}{x^2+\log ^2(x)-2 x \log (x)}\right ) \log \left (\log \left (\frac {x^3+x^2-2 x^2 \log (x)+x \log ^2(x)}{x^2+\log ^2(x)-2 x \log (x)}\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {x^3-\left (3 x^2+2 x\right ) \log (x)+2 x-\log ^3(x)+3 x \log ^2(x)}{\left (x^4+x^3+3 x^2 \log ^2(x)-\left (3 x^3+x^2\right ) \log (x)-x \log ^3(x)\right ) \log \left (\frac {x^3+x^2-2 x^2 \log (x)+x \log ^2(x)}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x^3+x^2-2 x^2 \log (x)+x \log ^2(x)}{(x-\log (x))^2}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\log ^3(x)}{x \left (x^3+x^2-3 x^2 \log (x)-\log ^3(x)+3 x \log ^2(x)-x \log (x)\right ) \log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right )\right )}-\frac {3 \log ^2(x)}{\left (-x^3-x^2+3 x^2 \log (x)+\log ^3(x)-3 x \log ^2(x)+x \log (x)\right ) \log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right )\right )}-\frac {3 x \log (x)}{\left (x^3+x^2-3 x^2 \log (x)-\log ^3(x)+3 x \log ^2(x)-x \log (x)\right ) \log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right )\right )}+\frac {2 \log (x)}{\left (-x^3-x^2+3 x^2 \log (x)+\log ^3(x)-3 x \log ^2(x)+x \log (x)\right ) \log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right )\right )}+\frac {x^2}{\left (x^3+x^2-3 x^2 \log (x)-\log ^3(x)+3 x \log ^2(x)-x \log (x)\right ) \log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right )\right )}+\frac {2}{\left (x^3+x^2-3 x^2 \log (x)-\log ^3(x)+3 x \log ^2(x)-x \log (x)\right ) \log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2+x+\log ^2(x)-2 x \log (x)\right )}{(x-\log (x))^2}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {1}{\left (x^3-3 \log (x) x^2+x^2+3 \log ^2(x) x-\log (x) x-\log ^3(x)\right ) \log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}dx+\int \frac {x^2}{\left (x^3-3 \log (x) x^2+x^2+3 \log ^2(x) x-\log (x) x-\log ^3(x)\right ) \log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}dx-3 \int \frac {x \log (x)}{\left (x^3-3 \log (x) x^2+x^2+3 \log ^2(x) x-\log (x) x-\log ^3(x)\right ) \log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}dx-\int \frac {\log ^3(x)}{x \left (x^3-3 \log (x) x^2+x^2+3 \log ^2(x) x-\log (x) x-\log ^3(x)\right ) \log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}dx+2 \int \frac {\log (x)}{\left (-x^3+3 \log (x) x^2-x^2-3 \log ^2(x) x+\log (x) x+\log ^3(x)\right ) \log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}dx-3 \int \frac {\log ^2(x)}{\left (-x^3+3 \log (x) x^2-x^2-3 \log ^2(x) x+\log (x) x+\log ^3(x)\right ) \log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x^2-2 \log (x) x+x+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}dx\) |
Input:
Int[(-2*x - x^3 + (2*x + 3*x^2)*Log[x] - 3*x*Log[x]^2 + Log[x]^3)/((-x^3 - x^4 + (x^2 + 3*x^3)*Log[x] - 3*x^2*Log[x]^2 + x*Log[x]^3)*Log[(x^2 + x^3 - 2*x^2*Log[x] + x*Log[x]^2)/(x^2 - 2*x*Log[x] + Log[x]^2)]*Log[Log[(x^2 + x^3 - 2*x^2*Log[x] + x*Log[x]^2)/(x^2 - 2*x*Log[x] + Log[x]^2)]]),x]
Output:
$Aborted
Time = 61.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.06
| method | result | size |
| parallelrisch | \(\ln \left (\ln \left (\ln \left (\frac {x \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}}\right )\right )\right )\) | \(35\) |
| default | \(\ln \left (\ln \left (\ln \left (x \right )-2 \ln \left (\ln \left (x \right )-x \right )+\ln \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )^{2}\right )+\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (\frac {i}{\left (\ln \left (x \right )-x \right )^{2}}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (i \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )\right )}{2}\right )\right )\) | \(313\) |
| risch | \(\ln \left (\ln \left (\ln \left (x \right )-2 \ln \left (x -\ln \left (x \right )\right )+\ln \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )^{2}\right )-\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (\frac {i}{\left (\ln \left (x \right )-x \right )^{2}}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (i \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )\right )}{2}\right )\right )\) | \(333\) |
Input:
int((ln(x)^3-3*x*ln(x)^2+(3*x^2+2*x)*ln(x)-x^3-2*x)/(x*ln(x)^3-3*x^2*ln(x) ^2+(3*x^3+x^2)*ln(x)-x^4-x^3)/ln((x*ln(x)^2-2*x^2*ln(x)+x^3+x^2)/(ln(x)^2- 2*x*ln(x)+x^2))/ln(ln((x*ln(x)^2-2*x^2*ln(x)+x^3+x^2)/(ln(x)^2-2*x*ln(x)+x ^2))),x,method=_RETURNVERBOSE)
Output:
ln(ln(ln(x*(ln(x)^2-2*x*ln(x)+x^2+x)/(ln(x)^2-2*x*ln(x)+x^2))))
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.29 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\log \left (\log \left (\log \left (\frac {x^{3} - 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + x^{2}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}}\right )\right )\right ) \] Input:
integrate((log(x)^3-3*x*log(x)^2+(3*x^2+2*x)*log(x)-x^3-2*x)/(x*log(x)^3-3 *x^2*log(x)^2+(3*x^3+x^2)*log(x)-x^4-x^3)/log((x*log(x)^2-2*x^2*log(x)+x^3 +x^2)/(log(x)^2-2*x*log(x)+x^2))/log(log((x*log(x)^2-2*x^2*log(x)+x^3+x^2) /(log(x)^2-2*x*log(x)+x^2))),x, algorithm="fricas")
Output:
log(log(log((x^3 - 2*x^2*log(x) + x*log(x)^2 + x^2)/(x^2 - 2*x*log(x) + lo g(x)^2))))
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (15) = 30\).
Time = 2.40 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\log {\left (\log {\left (\log {\left (\frac {x^{3} - 2 x^{2} \log {\left (x \right )} + x^{2} + x \log {\left (x \right )}^{2}}{x^{2} - 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} \right )} \right )} \right )} \] Input:
integrate((ln(x)**3-3*x*ln(x)**2+(3*x**2+2*x)*ln(x)-x**3-2*x)/(x*ln(x)**3- 3*x**2*ln(x)**2+(3*x**3+x**2)*ln(x)-x**4-x**3)/ln((x*ln(x)**2-2*x**2*ln(x) +x**3+x**2)/(ln(x)**2-2*x*ln(x)+x**2))/ln(ln((x*ln(x)**2-2*x**2*ln(x)+x**3 +x**2)/(ln(x)**2-2*x*ln(x)+x**2))),x)
Output:
log(log(log((x**3 - 2*x**2*log(x) + x**2 + x*log(x)**2)/(x**2 - 2*x*log(x) + log(x)**2))))
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\log \left (\log \left (\log \left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} + x\right ) + \log \left (x\right ) - 2 \, \log \left (-x + \log \left (x\right )\right )\right )\right ) \] Input:
integrate((log(x)^3-3*x*log(x)^2+(3*x^2+2*x)*log(x)-x^3-2*x)/(x*log(x)^3-3 *x^2*log(x)^2+(3*x^3+x^2)*log(x)-x^4-x^3)/log((x*log(x)^2-2*x^2*log(x)+x^3 +x^2)/(log(x)^2-2*x*log(x)+x^2))/log(log((x*log(x)^2-2*x^2*log(x)+x^3+x^2) /(log(x)^2-2*x*log(x)+x^2))),x, algorithm="maxima")
Output:
log(log(log(x^2 - 2*x*log(x) + log(x)^2 + x) + log(x) - 2*log(-x + log(x)) ))
\[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\int { \frac {x^{3} + 3 \, x \log \left (x\right )^{2} - \log \left (x\right )^{3} - {\left (3 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2 \, x}{{\left (x^{4} + 3 \, x^{2} \log \left (x\right )^{2} - x \log \left (x\right )^{3} + x^{3} - {\left (3 \, x^{3} + x^{2}\right )} \log \left (x\right )\right )} \log \left (\frac {x^{3} - 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + x^{2}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{3} - 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + x^{2}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}}\right )\right )} \,d x } \] Input:
integrate((log(x)^3-3*x*log(x)^2+(3*x^2+2*x)*log(x)-x^3-2*x)/(x*log(x)^3-3 *x^2*log(x)^2+(3*x^3+x^2)*log(x)-x^4-x^3)/log((x*log(x)^2-2*x^2*log(x)+x^3 +x^2)/(log(x)^2-2*x*log(x)+x^2))/log(log((x*log(x)^2-2*x^2*log(x)+x^3+x^2) /(log(x)^2-2*x*log(x)+x^2))),x, algorithm="giac")
Output:
integrate((x^3 + 3*x*log(x)^2 - log(x)^3 - (3*x^2 + 2*x)*log(x) + 2*x)/((x ^4 + 3*x^2*log(x)^2 - x*log(x)^3 + x^3 - (3*x^3 + x^2)*log(x))*log((x^3 - 2*x^2*log(x) + x*log(x)^2 + x^2)/(x^2 - 2*x*log(x) + log(x)^2))*log(log((x ^3 - 2*x^2*log(x) + x*log(x)^2 + x^2)/(x^2 - 2*x*log(x) + log(x)^2)))), x)
Time = 3.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\ln \left (\ln \left (\ln \left (x+\frac {x^2}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}\right )\right )\right ) \] Input:
int((2*x + 3*x*log(x)^2 - log(x)^3 - log(x)*(2*x + 3*x^2) + x^3)/(log(log( (x*log(x)^2 - 2*x^2*log(x) + x^2 + x^3)/(log(x)^2 - 2*x*log(x) + x^2)))*lo g((x*log(x)^2 - 2*x^2*log(x) + x^2 + x^3)/(log(x)^2 - 2*x*log(x) + x^2))*( 3*x^2*log(x)^2 - x*log(x)^3 - log(x)*(x^2 + 3*x^3) + x^3 + x^4)),x)
Output:
log(log(log(x + x^2/(log(x)^2 - 2*x*log(x) + x^2))))
Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.29 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )^{2} x -2 \,\mathrm {log}\left (x \right ) x^{2}+x^{3}+x^{2}}{\mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) x +x^{2}}\right )\right )\right ) \] Input:
int((log(x)^3-3*x*log(x)^2+(3*x^2+2*x)*log(x)-x^3-2*x)/(x*log(x)^3-3*x^2*l og(x)^2+(3*x^3+x^2)*log(x)-x^4-x^3)/log((x*log(x)^2-2*x^2*log(x)+x^3+x^2)/ (log(x)^2-2*x*log(x)+x^2))/log(log((x*log(x)^2-2*x^2*log(x)+x^3+x^2)/(log( x)^2-2*x*log(x)+x^2))),x)
Output:
log(log(log((log(x)**2*x - 2*log(x)*x**2 + x**3 + x**2)/(log(x)**2 - 2*log (x)*x + x**2))))