Integrand size = 65, antiderivative size = 21 \[ \int \frac {-1-x \log (x)+\left (2-x \log ^2(x)\right ) \log (\log (x))+\left (-1+x \log ^2(x)\right ) \log ^2(\log (x))}{x \log ^2(x)-2 x \log ^2(x) \log (\log (x))+x \log ^2(x) \log ^2(\log (x))} \, dx=x+\frac {1}{\log (x)}-\frac {x^2}{x-x \log (\log (x))} \] Output:
1/ln(x)+x-x^2/(x-x*ln(ln(x)))
Time = 0.11 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {-1-x \log (x)+\left (2-x \log ^2(x)\right ) \log (\log (x))+\left (-1+x \log ^2(x)\right ) \log ^2(\log (x))}{x \log ^2(x)-2 x \log ^2(x) \log (\log (x))+x \log ^2(x) \log ^2(\log (x))} \, dx=x+\frac {1}{\log (x)}+\frac {x}{-1+\log (\log (x))} \] Input:
Integrate[(-1 - x*Log[x] + (2 - x*Log[x]^2)*Log[Log[x]] + (-1 + x*Log[x]^2 )*Log[Log[x]]^2)/(x*Log[x]^2 - 2*x*Log[x]^2*Log[Log[x]] + x*Log[x]^2*Log[L og[x]]^2),x]
Output:
x + Log[x]^(-1) + x/(-1 + Log[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.
\(\displaystyle \int \frac {\left (x \log ^2(x)-1\right ) \log ^2(\log (x))+\left (2-x \log ^2(x)\right ) \log (\log (x))-x \log (x)-1}{x \log ^2(\log (x)) \log ^2(x)+x \log ^2(x)-2 x \log (\log (x)) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (-\frac {1}{x \log ^2(x)}+\frac {\log (\log (x))}{\log (\log (x))-1}-\frac {1}{\log (x) (\log (\log (x))-1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {1}{\log (x) (\log (\log (x))-1)^2}dx+\int \frac {1}{\log (\log (x))-1}dx+x+\frac {1}{\log (x)}\) |
Input:
Int[(-1 - x*Log[x] + (2 - x*Log[x]^2)*Log[Log[x]] + (-1 + x*Log[x]^2)*Log[ Log[x]]^2)/(x*Log[x]^2 - 2*x*Log[x]^2*Log[Log[x]] + x*Log[x]^2*Log[Log[x]] ^2),x]
Output:
$Aborted
Time = 0.48 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76
method | result | size |
default | \(x +\frac {1}{\ln \left (x \right )}+\frac {x}{\ln \left (\ln \left (x \right )\right )-1}\) | \(16\) |
parts | \(x +\frac {1}{\ln \left (x \right )}+\frac {x}{\ln \left (\ln \left (x \right )\right )-1}\) | \(16\) |
risch | \(\frac {x \ln \left (x \right )+1}{\ln \left (x \right )}+\frac {x}{\ln \left (\ln \left (x \right )\right )-1}\) | \(22\) |
parallelrisch | \(-\frac {1-x \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x \right )\right )}{\ln \left (x \right ) \left (\ln \left (\ln \left (x \right )\right )-1\right )}\) | \(29\) |
Input:
int(((x*ln(x)^2-1)*ln(ln(x))^2+(-x*ln(x)^2+2)*ln(ln(x))-x*ln(x)-1)/(x*ln(x )^2*ln(ln(x))^2-2*x*ln(x)^2*ln(ln(x))+x*ln(x)^2),x,method=_RETURNVERBOSE)
Output:
x+1/ln(x)+x/(ln(ln(x))-1)
Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {-1-x \log (x)+\left (2-x \log ^2(x)\right ) \log (\log (x))+\left (-1+x \log ^2(x)\right ) \log ^2(\log (x))}{x \log ^2(x)-2 x \log ^2(x) \log (\log (x))+x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {{\left (x \log \left (x\right ) + 1\right )} \log \left (\log \left (x\right )\right ) - 1}{\log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right )} \] Input:
integrate(((x*log(x)^2-1)*log(log(x))^2+(-x*log(x)^2+2)*log(log(x))-x*log( x)-1)/(x*log(x)^2*log(log(x))^2-2*x*log(x)^2*log(log(x))+x*log(x)^2),x, al gorithm="fricas")
Output:
((x*log(x) + 1)*log(log(x)) - 1)/(log(x)*log(log(x)) - log(x))
Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {-1-x \log (x)+\left (2-x \log ^2(x)\right ) \log (\log (x))+\left (-1+x \log ^2(x)\right ) \log ^2(\log (x))}{x \log ^2(x)-2 x \log ^2(x) \log (\log (x))+x \log ^2(x) \log ^2(\log (x))} \, dx=x + \frac {x}{\log {\left (\log {\left (x \right )} \right )} - 1} + \frac {1}{\log {\left (x \right )}} \] Input:
integrate(((x*ln(x)**2-1)*ln(ln(x))**2+(-x*ln(x)**2+2)*ln(ln(x))-x*ln(x)-1 )/(x*ln(x)**2*ln(ln(x))**2-2*x*ln(x)**2*ln(ln(x))+x*ln(x)**2),x)
Output:
x + x/(log(log(x)) - 1) + 1/log(x)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {-1-x \log (x)+\left (2-x \log ^2(x)\right ) \log (\log (x))+\left (-1+x \log ^2(x)\right ) \log ^2(\log (x))}{x \log ^2(x)-2 x \log ^2(x) \log (\log (x))+x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {{\left (x \log \left (x\right ) + 1\right )} \log \left (\log \left (x\right )\right ) - 1}{\log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right )} \] Input:
integrate(((x*log(x)^2-1)*log(log(x))^2+(-x*log(x)^2+2)*log(log(x))-x*log( x)-1)/(x*log(x)^2*log(log(x))^2-2*x*log(x)^2*log(log(x))+x*log(x)^2),x, al gorithm="maxima")
Output:
((x*log(x) + 1)*log(log(x)) - 1)/(log(x)*log(log(x)) - log(x))
Time = 0.14 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {-1-x \log (x)+\left (2-x \log ^2(x)\right ) \log (\log (x))+\left (-1+x \log ^2(x)\right ) \log ^2(\log (x))}{x \log ^2(x)-2 x \log ^2(x) \log (\log (x))+x \log ^2(x) \log ^2(\log (x))} \, dx=x + \frac {x}{\log \left (\log \left (x\right )\right ) - 1} + \frac {1}{\log \left (x\right )} \] Input:
integrate(((x*log(x)^2-1)*log(log(x))^2+(-x*log(x)^2+2)*log(log(x))-x*log( x)-1)/(x*log(x)^2*log(log(x))^2-2*x*log(x)^2*log(log(x))+x*log(x)^2),x, al gorithm="giac")
Output:
x + x/(log(log(x)) - 1) + 1/log(x)
Time = 2.66 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {-1-x \log (x)+\left (2-x \log ^2(x)\right ) \log (\log (x))+\left (-1+x \log ^2(x)\right ) \log ^2(\log (x))}{x \log ^2(x)-2 x \log ^2(x) \log (\log (x))+x \log ^2(x) \log ^2(\log (x))} \, dx=x+\frac {1}{\ln \left (x\right )}+x\,\ln \left (x\right )+\frac {x\,\left (\ln \left (x\right )+1\right )-x\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )}{\ln \left (\ln \left (x\right )\right )-1} \] Input:
int(-(log(log(x))*(x*log(x)^2 - 2) - log(log(x))^2*(x*log(x)^2 - 1) + x*lo g(x) + 1)/(x*log(x)^2 - 2*x*log(log(x))*log(x)^2 + x*log(log(x))^2*log(x)^ 2),x)
Output:
x + 1/log(x) + x*log(x) + (x*(log(x) + 1) - x*log(log(x))*log(x))/(log(log (x)) - 1)
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {-1-x \log (x)+\left (2-x \log ^2(x)\right ) \log (\log (x))+\left (-1+x \log ^2(x)\right ) \log ^2(\log (x))}{x \log ^2(x)-2 x \log ^2(x) \log (\log (x))+x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (x \right ) x +\mathrm {log}\left (\mathrm {log}\left (x \right )\right )-1}{\mathrm {log}\left (x \right ) \left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right )-1\right )} \] Input:
int(((x*log(x)^2-1)*log(log(x))^2+(-x*log(x)^2+2)*log(log(x))-x*log(x)-1)/ (x*log(x)^2*log(log(x))^2-2*x*log(x)^2*log(log(x))+x*log(x)^2),x)
Output:
(log(log(x))*log(x)*x + log(log(x)) - 1)/(log(x)*(log(log(x)) - 1))