Integrand size = 53, antiderivative size = 19 \[ \int \frac {12 \log \left (\frac {\log (2)}{3}\right )+e^x x \log (x) \log ^2(\log (x))}{-12 x \log (x) \log \left (\frac {\log (2)}{3}\right ) \log (\log (x))+e^x x \log (x) \log ^2(\log (x))} \, dx=\log \left (e^x-\frac {12 \log \left (\frac {\log (2)}{3}\right )}{\log (\log (x))}\right ) \]
Time = 0.52 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32 \[ \int \frac {12 \log \left (\frac {\log (2)}{3}\right )+e^x x \log (x) \log ^2(\log (x))}{-12 x \log (x) \log \left (\frac {\log (2)}{3}\right ) \log (\log (x))+e^x x \log (x) \log ^2(\log (x))} \, dx=-\log (\log (\log (x)))+\log \left (-12 \log \left (\frac {\log (2)}{3}\right )+e^x \log (\log (x))\right ) \]
Integrate[(12*Log[Log[2]/3] + E^x*x*Log[x]*Log[Log[x]]^2)/(-12*x*Log[x]*Lo g[Log[2]/3]*Log[Log[x]] + E^x*x*Log[x]*Log[Log[x]]^2),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.
\(\displaystyle \int \frac {e^x x \log (x) \log ^2(\log (x))+12 \log \left (\frac {\log (2)}{3}\right )}{e^x x \log (x) \log ^2(\log (x))-12 x \log \left (\frac {\log (2)}{3}\right ) \log (x) \log (\log (x))} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-e^x x \log (x) \log ^2(\log (x))-12 \log \left (\frac {\log (2)}{3}\right )}{x \log (x) \log (\log (x)) \left (12 \log \left (\frac {\log (2)}{3}\right )-e^x \log (\log (x))\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (1-\frac {12 \log \left (\frac {\log (2)}{3}\right ) (x \log (x) \log (\log (x))+1)}{x \log (x) \log (\log (x)) \left (12 \log \left (\frac {\log (2)}{3}\right )-e^x \log (\log (x))\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -12 \log \left (\frac {\log (2)}{3}\right ) \int \frac {1}{12 \log \left (\frac {\log (2)}{3}\right )-e^x \log (\log (x))}dx+12 \log \left (\frac {\log (2)}{3}\right ) \int \frac {1}{x \log (x) \log (\log (x)) \left (e^x \log (\log (x))-12 \log \left (\frac {\log (2)}{3}\right )\right )}dx+x\) |
Int[(12*Log[Log[2]/3] + E^x*x*Log[x]*Log[Log[x]]^2)/(-12*x*Log[x]*Log[Log[ 2]/3]*Log[Log[x]] + E^x*x*Log[x]*Log[Log[x]]^2),x]
3.26.48.3.1 Defintions of rubi rules used
Time = 3.82 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21
method | result | size |
parallelrisch | \(-\ln \left (\ln \left (\ln \left (x \right )\right )\right )+\ln \left ({\mathrm e}^{x} \ln \left (\ln \left (x \right )\right )-12 \ln \left (\frac {\ln \left (2\right )}{3}\right )\right )\) | \(23\) |
risch | \(x -\ln \left (\ln \left (\ln \left (x \right )\right )\right )+\ln \left (\ln \left (\ln \left (x \right )\right )-12 \left (-\ln \left (3\right )+\ln \left (\ln \left (2\right )\right )\right ) {\mathrm e}^{-x}\right )\) | \(28\) |
int((x*exp(x)*ln(x)*ln(ln(x))^2+12*ln(1/3*ln(2)))/(x*exp(x)*ln(x)*ln(ln(x) )^2-12*x*ln(x)*ln(1/3*ln(2))*ln(ln(x))),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {12 \log \left (\frac {\log (2)}{3}\right )+e^x x \log (x) \log ^2(\log (x))}{-12 x \log (x) \log \left (\frac {\log (2)}{3}\right ) \log (\log (x))+e^x x \log (x) \log ^2(\log (x))} \, dx=x + \log \left ({\left (e^{x} \log \left (\log \left (x\right )\right ) - 12 \, \log \left (\frac {1}{3} \, \log \left (2\right )\right )\right )} e^{\left (-x\right )}\right ) - \log \left (\log \left (\log \left (x\right )\right )\right ) \]
integrate((x*exp(x)*log(x)*log(log(x))^2+12*log(1/3*log(2)))/(x*exp(x)*log (x)*log(log(x))^2-12*x*log(x)*log(1/3*log(2))*log(log(x))),x, algorithm=\
Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {12 \log \left (\frac {\log (2)}{3}\right )+e^x x \log (x) \log ^2(\log (x))}{-12 x \log (x) \log \left (\frac {\log (2)}{3}\right ) \log (\log (x))+e^x x \log (x) \log ^2(\log (x))} \, dx=\log {\left (e^{x} + \frac {- 12 \log {\left (\log {\left (2 \right )} \right )} + 12 \log {\left (3 \right )}}{\log {\left (\log {\left (x \right )} \right )}} \right )} \]
integrate((x*exp(x)*ln(x)*ln(ln(x))**2+12*ln(1/3*ln(2)))/(x*exp(x)*ln(x)*l n(ln(x))**2-12*x*ln(x)*ln(1/3*ln(2))*ln(ln(x))),x)
Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {12 \log \left (\frac {\log (2)}{3}\right )+e^x x \log (x) \log ^2(\log (x))}{-12 x \log (x) \log \left (\frac {\log (2)}{3}\right ) \log (\log (x))+e^x x \log (x) \log ^2(\log (x))} \, dx=x + \log \left ({\left (e^{x} \log \left (\log \left (x\right )\right ) + 12 \, \log \left (3\right ) - 12 \, \log \left (\log \left (2\right )\right )\right )} e^{\left (-x\right )}\right ) - \log \left (\log \left (\log \left (x\right )\right )\right ) \]
integrate((x*exp(x)*log(x)*log(log(x))^2+12*log(1/3*log(2)))/(x*exp(x)*log (x)*log(log(x))^2-12*x*log(x)*log(1/3*log(2))*log(log(x))),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {12 \log \left (\frac {\log (2)}{3}\right )+e^x x \log (x) \log ^2(\log (x))}{-12 x \log (x) \log \left (\frac {\log (2)}{3}\right ) \log (\log (x))+e^x x \log (x) \log ^2(\log (x))} \, dx=\log \left (e^{x} \log \left (\log \left (x\right )\right ) + 12 \, \log \left (3\right ) - 12 \, \log \left (\log \left (2\right )\right )\right ) - \log \left (\log \left (\log \left (x\right )\right )\right ) \]
integrate((x*exp(x)*log(x)*log(log(x))^2+12*log(1/3*log(2)))/(x*exp(x)*log (x)*log(log(x))^2-12*x*log(x)*log(1/3*log(2))*log(log(x))),x, algorithm=\
Timed out. \[ \int \frac {12 \log \left (\frac {\log (2)}{3}\right )+e^x x \log (x) \log ^2(\log (x))}{-12 x \log (x) \log \left (\frac {\log (2)}{3}\right ) \log (\log (x))+e^x x \log (x) \log ^2(\log (x))} \, dx=\int -\frac {x\,{\mathrm {e}}^x\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^2+12\,\ln \left (\frac {\ln \left (2\right )}{3}\right )}{12\,x\,\ln \left (\frac {\ln \left (2\right )}{3}\right )\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )-x\,{\ln \left (\ln \left (x\right )\right )}^2\,{\mathrm {e}}^x\,\ln \left (x\right )} \,d x \]
int(-(12*log(log(2)/3) + x*log(log(x))^2*exp(x)*log(x))/(12*x*log(log(2)/3 )*log(log(x))*log(x) - x*log(log(x))^2*exp(x)*log(x)),x)