Integrand size = 121, antiderivative size = 27 \[ \int \frac {\left (-2+(4+2 x) \log (4 x)+2 x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right ) \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{\left (2 x-x^2\right ) \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )+x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )} \, dx=5-\log ^2\left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right ) \]
Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-2+(4+2 x) \log (4 x)+2 x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right ) \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{\left (2 x-x^2\right ) \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )+x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )} \, dx=-\log ^2\left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right ) \]
Integrate[((-2 + (4 + 2*x)*Log[4*x] + 2*x*Log[4*x]*Log[Log[4*x]/(E^x*x^2)] )*Log[-2 + x - Log[Log[Log[4*x]/(E^x*x^2)]]])/((2*x - x^2)*Log[4*x]*Log[Lo g[4*x]/(E^x*x^2)] + x*Log[4*x]*Log[Log[4*x]/(E^x*x^2)]*Log[Log[Log[4*x]/(E ^x*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.
| \(\displaystyle \int \frac {\left (2 x \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \log (4 x)+(2 x+4) \log (4 x)-2\right ) \log \left (-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )+x-2\right )}{\left (2 x-x^2\right ) \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )+x \log (4 x) \log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right ) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (2 x \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \log (4 x)+(2 x+4) \log (4 x)-2\right ) \log \left (-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )+x-2\right )}{x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )-x+2\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 \log \left (-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )+x-2\right )}{x \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )+x-2\right )}+\frac {2 \log \left (-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )+x-2\right )}{x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )+x-2\right )}-\frac {2 \log \left (-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )+x-2\right )}{\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )+x-2\right )}-\frac {2 \log \left (-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )+x-2\right )}{-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )+x-2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \frac {\log \left (x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )-2\right )}{x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )-2}dx-2 \int \frac {\log \left (x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )-2\right )}{\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )-2\right )}dx-4 \int \frac {\log \left (x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )-2\right )}{x \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )-2\right )}dx+2 \int \frac {\log \left (x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )-2\right )}{x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )-2\right )}dx\) |
Int[((-2 + (4 + 2*x)*Log[4*x] + 2*x*Log[4*x]*Log[Log[4*x]/(E^x*x^2)])*Log[ -2 + x - Log[Log[Log[4*x]/(E^x*x^2)]]])/((2*x - x^2)*Log[4*x]*Log[Log[4*x] /(E^x*x^2)] + x*Log[4*x]*Log[Log[4*x]/(E^x*x^2)]*Log[Log[Log[4*x]/(E^x*x^2 )]]),x]
3.2.68.3.1 Defintions of rubi rules used
\[\int \frac {\left (2 x \ln \left (4 x \right ) \ln \left (\frac {\ln \left (4 x \right ) {\mathrm e}^{-x}}{x^{2}}\right )+\left (4+2 x \right ) \ln \left (4 x \right )-2\right ) \ln \left (-\ln \left (\ln \left (\frac {\ln \left (4 x \right ) {\mathrm e}^{-x}}{x^{2}}\right )\right )+x -2\right )}{x \ln \left (4 x \right ) \ln \left (\frac {\ln \left (4 x \right ) {\mathrm e}^{-x}}{x^{2}}\right ) \ln \left (\ln \left (\frac {\ln \left (4 x \right ) {\mathrm e}^{-x}}{x^{2}}\right )\right )+\left (-x^{2}+2 x \right ) \ln \left (4 x \right ) \ln \left (\frac {\ln \left (4 x \right ) {\mathrm e}^{-x}}{x^{2}}\right )}d x\]
int((2*x*ln(4*x)*ln(ln(4*x)/exp(x)/x^2)+(4+2*x)*ln(4*x)-2)*ln(-ln(ln(ln(4* x)/exp(x)/x^2))+x-2)/(x*ln(4*x)*ln(ln(4*x)/exp(x)/x^2)*ln(ln(ln(4*x)/exp(x )/x^2))+(-x^2+2*x)*ln(4*x)*ln(ln(4*x)/exp(x)/x^2)),x)
int((2*x*ln(4*x)*ln(ln(4*x)/exp(x)/x^2)+(4+2*x)*ln(4*x)-2)*ln(-ln(ln(ln(4* x)/exp(x)/x^2))+x-2)/(x*ln(4*x)*ln(ln(4*x)/exp(x)/x^2)*ln(ln(ln(4*x)/exp(x )/x^2))+(-x^2+2*x)*ln(4*x)*ln(ln(4*x)/exp(x)/x^2)),x)
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-2+(4+2 x) \log (4 x)+2 x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right ) \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{\left (2 x-x^2\right ) \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )+x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )} \, dx=-\log \left (x - \log \left (\log \left (\frac {e^{\left (-x\right )} \log \left (4 \, x\right )}{x^{2}}\right )\right ) - 2\right )^{2} \]
integrate((2*x*log(4*x)*log(log(4*x)/exp(x)/x^2)+(4+2*x)*log(4*x)-2)*log(- log(log(log(4*x)/exp(x)/x^2))+x-2)/(x*log(4*x)*log(log(4*x)/exp(x)/x^2)*lo g(log(log(4*x)/exp(x)/x^2))+(-x^2+2*x)*log(4*x)*log(log(4*x)/exp(x)/x^2)), x, algorithm=\
Timed out. \[ \int \frac {\left (-2+(4+2 x) \log (4 x)+2 x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right ) \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{\left (2 x-x^2\right ) \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )+x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )} \, dx=\text {Timed out} \]
integrate((2*x*ln(4*x)*ln(ln(4*x)/exp(x)/x**2)+(4+2*x)*ln(4*x)-2)*ln(-ln(l n(ln(4*x)/exp(x)/x**2))+x-2)/(x*ln(4*x)*ln(ln(4*x)/exp(x)/x**2)*ln(ln(ln(4 *x)/exp(x)/x**2))+(-x**2+2*x)*ln(4*x)*ln(ln(4*x)/exp(x)/x**2)),x)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (26) = 52\).
Time = 0.37 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.63 \[ \int \frac {\left (-2+(4+2 x) \log (4 x)+2 x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right ) \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{\left (2 x-x^2\right ) \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )+x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )} \, dx=-2 \, \log \left (x - \log \left (\log \left (\frac {e^{\left (-x\right )} \log \left (4 \, x\right )}{x^{2}}\right )\right ) - 2\right ) \log \left (-x + \log \left (-x - 2 \, \log \left (x\right ) + \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right )\right ) + 2\right ) + \log \left (-x + \log \left (-x - 2 \, \log \left (x\right ) + \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right )\right ) + 2\right )^{2} \]
integrate((2*x*log(4*x)*log(log(4*x)/exp(x)/x^2)+(4+2*x)*log(4*x)-2)*log(- log(log(log(4*x)/exp(x)/x^2))+x-2)/(x*log(4*x)*log(log(4*x)/exp(x)/x^2)*lo g(log(log(4*x)/exp(x)/x^2))+(-x^2+2*x)*log(4*x)*log(log(4*x)/exp(x)/x^2)), x, algorithm=\
-2*log(x - log(log(e^(-x)*log(4*x)/x^2)) - 2)*log(-x + log(-x - 2*log(x) + log(2*log(2) + log(x))) + 2) + log(-x + log(-x - 2*log(x) + log(2*log(2) + log(x))) + 2)^2
\[ \int \frac {\left (-2+(4+2 x) \log (4 x)+2 x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right ) \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{\left (2 x-x^2\right ) \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )+x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )} \, dx=\int { \frac {2 \, {\left (x \log \left (4 \, x\right ) \log \left (\frac {e^{\left (-x\right )} \log \left (4 \, x\right )}{x^{2}}\right ) + {\left (x + 2\right )} \log \left (4 \, x\right ) - 1\right )} \log \left (x - \log \left (\log \left (\frac {e^{\left (-x\right )} \log \left (4 \, x\right )}{x^{2}}\right )\right ) - 2\right )}{x \log \left (4 \, x\right ) \log \left (\frac {e^{\left (-x\right )} \log \left (4 \, x\right )}{x^{2}}\right ) \log \left (\log \left (\frac {e^{\left (-x\right )} \log \left (4 \, x\right )}{x^{2}}\right )\right ) - {\left (x^{2} - 2 \, x\right )} \log \left (4 \, x\right ) \log \left (\frac {e^{\left (-x\right )} \log \left (4 \, x\right )}{x^{2}}\right )} \,d x } \]
integrate((2*x*log(4*x)*log(log(4*x)/exp(x)/x^2)+(4+2*x)*log(4*x)-2)*log(- log(log(log(4*x)/exp(x)/x^2))+x-2)/(x*log(4*x)*log(log(4*x)/exp(x)/x^2)*lo g(log(log(4*x)/exp(x)/x^2))+(-x^2+2*x)*log(4*x)*log(log(4*x)/exp(x)/x^2)), x, algorithm=\
integrate(2*(x*log(4*x)*log(e^(-x)*log(4*x)/x^2) + (x + 2)*log(4*x) - 1)*l og(x - log(log(e^(-x)*log(4*x)/x^2)) - 2)/(x*log(4*x)*log(e^(-x)*log(4*x)/ x^2)*log(log(e^(-x)*log(4*x)/x^2)) - (x^2 - 2*x)*log(4*x)*log(e^(-x)*log(4 *x)/x^2)), x)
Time = 10.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-2+(4+2 x) \log (4 x)+2 x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right ) \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{\left (2 x-x^2\right ) \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )+x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )} \, dx=-{\ln \left (x-\ln \left (\ln \left (\frac {\ln \left (4\,x\right )\,{\mathrm {e}}^{-x}}{x^2}\right )\right )-2\right )}^2 \]