Integrand size = 195, antiderivative size = 30 \[ \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )+\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )} \, dx=4 \left (4+\left (1+\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )^2\right ) \] Output:
4*(1+exp(2*ln(ln(1/9*x*ln(x/exp(x))^2))/x))^2+16
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )+\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )} \, dx=4 \log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (2+\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right ) \] Input:
Integrate[(Log[(x*Log[x/E^x]^2)/9]^(2/x)*(32 - 32*x + 16*Log[x/E^x] - 16*L og[x/E^x]*Log[(x*Log[x/E^x]^2)/9]*Log[Log[(x*Log[x/E^x]^2)/9]]) + Log[(x*L og[x/E^x]^2)/9]^(4/x)*(32 - 32*x + 16*Log[x/E^x] - 16*Log[x/E^x]*Log[(x*Lo g[x/E^x]^2)/9]*Log[Log[(x*Log[x/E^x]^2)/9]]))/(x^2*Log[x/E^x]*Log[(x*Log[x /E^x]^2)/9]),x]
Output:
4*Log[(x*Log[x/E^x]^2)/9]^(2/x)*(2 + Log[(x*Log[x/E^x]^2)/9]^(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.
| \(\displaystyle \int \frac {\left (-32 x-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )+16 \log \left (e^{-x} x\right )+32\right ) \log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )+\left (-32 x-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )+16 \log \left (e^{-x} x\right )+32\right ) \log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {16 \log ^{\frac {2}{x}-1}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )+1\right ) \left (-2 (x-1)-\log \left (e^{-x} x\right ) \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )-1\right )\right )}{x^2 \log \left (e^{-x} x\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 16 \int \frac {\log ^{\frac {2}{x}-1}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )+1\right ) \left (2 (1-x)+\log \left (e^{-x} x\right ) \left (1-\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 16 \int \left (\frac {\log ^{\frac {2}{x}-1}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (-2 x+\log \left (e^{-x} x\right )-\log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )+2\right )}{x^2 \log \left (e^{-x} x\right )}-\frac {\log ^{\frac {4}{x}-1}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (2 x-\log \left (e^{-x} x\right )+\log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )-2\right )}{x^2 \log \left (e^{-x} x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 \left (\int \frac {\log ^{\frac {2}{x}-1}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2}dx+2 \int \frac {\log ^{\frac {2}{x}-1}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )}dx+\int \frac {\log ^{\frac {4}{x}-1}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2}dx+2 \int \frac {\log ^{\frac {4}{x}-1}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )}dx-\int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2}dx-\int \frac {\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2}dx-2 \int \frac {\log ^{\frac {2}{x}-1}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x \log \left (e^{-x} x\right )}dx-2 \int \frac {\log ^{\frac {4}{x}-1}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x \log \left (e^{-x} x\right )}dx\right )\) |
Input:
Int[(Log[(x*Log[x/E^x]^2)/9]^(2/x)*(32 - 32*x + 16*Log[x/E^x] - 16*Log[x/E ^x]*Log[(x*Log[x/E^x]^2)/9]*Log[Log[(x*Log[x/E^x]^2)/9]]) + Log[(x*Log[x/E ^x]^2)/9]^(4/x)*(32 - 32*x + 16*Log[x/E^x] - 16*Log[x/E^x]*Log[(x*Log[x/E^ x]^2)/9]*Log[Log[(x*Log[x/E^x]^2)/9]]))/(x^2*Log[x/E^x]*Log[(x*Log[x/E^x]^ 2)/9]),x]
Output:
$Aborted
Time = 24.81 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60
| method | result | size |
| parallelrisch | \(4 \,{\mathrm e}^{\frac {4 \ln \left (\ln \left (\frac {x \ln \left (x \,{\mathrm e}^{-x}\right )^{2}}{9}\right )\right )}{x}}+8 \,{\mathrm e}^{\frac {2 \ln \left (\ln \left (\frac {x \ln \left (x \,{\mathrm e}^{-x}\right )^{2}}{9}\right )\right )}{x}}\) | \(48\) |
Input:
int(((-16*ln(x/exp(x))*ln(1/9*x*ln(x/exp(x))^2)*ln(ln(1/9*x*ln(x/exp(x))^2 ))+16*ln(x/exp(x))-32*x+32)*exp(2*ln(ln(1/9*x*ln(x/exp(x))^2))/x)^2+(-16*l n(x/exp(x))*ln(1/9*x*ln(x/exp(x))^2)*ln(ln(1/9*x*ln(x/exp(x))^2))+16*ln(x/ exp(x))-32*x+32)*exp(2*ln(ln(1/9*x*ln(x/exp(x))^2))/x))/x^2/ln(x/exp(x))/l n(1/9*x*ln(x/exp(x))^2),x,method=_RETURNVERBOSE)
Output:
4*exp(2*ln(ln(1/9*x*ln(x/exp(x))^2))/x)^2+8*exp(2*ln(ln(1/9*x*ln(x/exp(x)) ^2))/x)
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )+\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )} \, dx=4 \, \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )^{\frac {4}{x}} + 8 \, \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )^{\frac {2}{x}} \] Input:
integrate(((-16*log(x/exp(x))*log(1/9*x*log(x/exp(x))^2)*log(log(1/9*x*log (x/exp(x))^2))+16*log(x/exp(x))-32*x+32)*exp(2*log(log(1/9*x*log(x/exp(x)) ^2))/x)^2+(-16*log(x/exp(x))*log(1/9*x*log(x/exp(x))^2)*log(log(1/9*x*log( x/exp(x))^2))+16*log(x/exp(x))-32*x+32)*exp(2*log(log(1/9*x*log(x/exp(x))^ 2))/x))/x^2/log(x/exp(x))/log(1/9*x*log(x/exp(x))^2),x, algorithm="fricas" )
Output:
4*log(1/9*x*log(x*e^(-x))^2)^(4/x) + 8*log(1/9*x*log(x*e^(-x))^2)^(2/x)
Timed out. \[ \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )+\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )} \, dx=\text {Timed out} \] Input:
integrate(((-16*ln(x/exp(x))*ln(1/9*x*ln(x/exp(x))**2)*ln(ln(1/9*x*ln(x/ex p(x))**2))+16*ln(x/exp(x))-32*x+32)*exp(2*ln(ln(1/9*x*ln(x/exp(x))**2))/x) **2+(-16*ln(x/exp(x))*ln(1/9*x*ln(x/exp(x))**2)*ln(ln(1/9*x*ln(x/exp(x))** 2))+16*ln(x/exp(x))-32*x+32)*exp(2*ln(ln(1/9*x*ln(x/exp(x))**2))/x))/x**2/ ln(x/exp(x))/ln(1/9*x*ln(x/exp(x))**2),x)
Output:
Timed out
\[ \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )+\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )} \, dx=\int { -\frac {16 \, {\left ({\left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right ) \log \left (x e^{\left (-x\right )}\right ) \log \left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )\right ) + 2 \, x - \log \left (x e^{\left (-x\right )}\right ) - 2\right )} \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )^{\frac {4}{x}} + {\left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right ) \log \left (x e^{\left (-x\right )}\right ) \log \left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )\right ) + 2 \, x - \log \left (x e^{\left (-x\right )}\right ) - 2\right )} \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )^{\frac {2}{x}}\right )}}{x^{2} \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right ) \log \left (x e^{\left (-x\right )}\right )} \,d x } \] Input:
integrate(((-16*log(x/exp(x))*log(1/9*x*log(x/exp(x))^2)*log(log(1/9*x*log (x/exp(x))^2))+16*log(x/exp(x))-32*x+32)*exp(2*log(log(1/9*x*log(x/exp(x)) ^2))/x)^2+(-16*log(x/exp(x))*log(1/9*x*log(x/exp(x))^2)*log(log(1/9*x*log( x/exp(x))^2))+16*log(x/exp(x))-32*x+32)*exp(2*log(log(1/9*x*log(x/exp(x))^ 2))/x))/x^2/log(x/exp(x))/log(1/9*x*log(x/exp(x))^2),x, algorithm="maxima" )
Output:
4*(-2*log(3) + log(x) + 2*log(-x + log(x)))^(4/x) + 8*(-2*log(3) + log(x) + 2*log(-x + log(x)))^(2/x) + 16*integrate(-2*((3*x - log(x) - 2)*log(x - log(x)) - (3*x - log(x) - 2)*log(-x + log(x)))*(-2*log(3) + log(x) + 2*log (-x + log(x)))^(4/x)/(4*x^3*log(3)^2 - x^2*log(x)^3 + (x^3 + 4*x^2*log(3)) *log(x)^2 - 2*(2*x^3*log(3) + x^2*log(x)^2 - (x^3 + 2*x^2*log(3))*log(x))* log(x - log(x)) - 4*(x^3*log(3) + x^2*log(3)^2)*log(x) - 2*(2*x^3*log(3) + x^2*log(x)^2 - 2*(x^3 - x^2*log(x))*log(x - log(x)) - (x^3 + 2*x^2*log(3) )*log(x))*log(-x + log(x))), x) + 16*integrate(-2*((3*x - log(x) - 2)*log( x - log(x)) - (3*x - log(x) - 2)*log(-x + log(x)))*(-2*log(3) + log(x) + 2 *log(-x + log(x)))^(2/x)/(4*x^3*log(3)^2 - x^2*log(x)^3 + (x^3 + 4*x^2*log (3))*log(x)^2 - 2*(2*x^3*log(3) + x^2*log(x)^2 - (x^3 + 2*x^2*log(3))*log( x))*log(x - log(x)) - 4*(x^3*log(3) + x^2*log(3)^2)*log(x) - 2*(2*x^3*log( 3) + x^2*log(x)^2 - 2*(x^3 - x^2*log(x))*log(x - log(x)) - (x^3 + 2*x^2*lo g(3))*log(x))*log(-x + log(x))), x)
\[ \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )+\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )} \, dx=\int { -\frac {16 \, {\left ({\left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right ) \log \left (x e^{\left (-x\right )}\right ) \log \left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )\right ) + 2 \, x - \log \left (x e^{\left (-x\right )}\right ) - 2\right )} \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )^{\frac {4}{x}} + {\left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right ) \log \left (x e^{\left (-x\right )}\right ) \log \left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )\right ) + 2 \, x - \log \left (x e^{\left (-x\right )}\right ) - 2\right )} \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )^{\frac {2}{x}}\right )}}{x^{2} \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right ) \log \left (x e^{\left (-x\right )}\right )} \,d x } \] Input:
integrate(((-16*log(x/exp(x))*log(1/9*x*log(x/exp(x))^2)*log(log(1/9*x*log (x/exp(x))^2))+16*log(x/exp(x))-32*x+32)*exp(2*log(log(1/9*x*log(x/exp(x)) ^2))/x)^2+(-16*log(x/exp(x))*log(1/9*x*log(x/exp(x))^2)*log(log(1/9*x*log( x/exp(x))^2))+16*log(x/exp(x))-32*x+32)*exp(2*log(log(1/9*x*log(x/exp(x))^ 2))/x))/x^2/log(x/exp(x))/log(1/9*x*log(x/exp(x))^2),x, algorithm="giac")
Output:
integrate(-16*((log(1/9*x*log(x*e^(-x))^2)*log(x*e^(-x))*log(log(1/9*x*log (x*e^(-x))^2)) + 2*x - log(x*e^(-x)) - 2)*log(1/9*x*log(x*e^(-x))^2)^(4/x) + (log(1/9*x*log(x*e^(-x))^2)*log(x*e^(-x))*log(log(1/9*x*log(x*e^(-x))^2 )) + 2*x - log(x*e^(-x)) - 2)*log(1/9*x*log(x*e^(-x))^2)^(2/x))/(x^2*log(1 /9*x*log(x*e^(-x))^2)*log(x*e^(-x))), x)
Time = 2.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )+\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )} \, dx=4\,{\ln \left (\frac {x^3}{9}-\frac {2\,x^2\,\ln \left (x\right )}{9}+\frac {x\,{\ln \left (x\right )}^2}{9}\right )}^{2/x}\,\left ({\ln \left (\frac {x^3}{9}-\frac {2\,x^2\,\ln \left (x\right )}{9}+\frac {x\,{\ln \left (x\right )}^2}{9}\right )}^{2/x}+2\right ) \] Input:
int(-(exp((2*log(log((x*log(x*exp(-x))^2)/9)))/x)*(32*x - 16*log(x*exp(-x) ) + 16*log((x*log(x*exp(-x))^2)/9)*log(x*exp(-x))*log(log((x*log(x*exp(-x) )^2)/9)) - 32) + exp((4*log(log((x*log(x*exp(-x))^2)/9)))/x)*(32*x - 16*lo g(x*exp(-x)) + 16*log((x*log(x*exp(-x))^2)/9)*log(x*exp(-x))*log(log((x*lo g(x*exp(-x))^2)/9)) - 32))/(x^2*log((x*log(x*exp(-x))^2)/9)*log(x*exp(-x)) ),x)
Output:
4*log((x*log(x)^2)/9 - (2*x^2*log(x))/9 + x^3/9)^(2/x)*(log((x*log(x)^2)/9 - (2*x^2*log(x))/9 + x^3/9)^(2/x) + 2)
Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )+\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )} \, dx=4 e^{\frac {2 \,\mathrm {log}\left (\mathrm {log}\left (\frac {\mathrm {log}\left (\frac {x}{e^{x}}\right )^{2} x}{9}\right )\right )}{x}} \left (e^{\frac {2 \,\mathrm {log}\left (\mathrm {log}\left (\frac {\mathrm {log}\left (\frac {x}{e^{x}}\right )^{2} x}{9}\right )\right )}{x}}+2\right ) \] Input:
int(((-16*log(x/exp(x))*log(1/9*x*log(x/exp(x))^2)*log(log(1/9*x*log(x/exp (x))^2))+16*log(x/exp(x))-32*x+32)*exp(2*log(log(1/9*x*log(x/exp(x))^2))/x )^2+(-16*log(x/exp(x))*log(1/9*x*log(x/exp(x))^2)*log(log(1/9*x*log(x/exp( x))^2))+16*log(x/exp(x))-32*x+32)*exp(2*log(log(1/9*x*log(x/exp(x))^2))/x) )/x^2/log(x/exp(x))/log(1/9*x*log(x/exp(x))^2),x)
Output:
4*e**((2*log(log((log(x/e**x)**2*x)/9)))/x)*(e**((2*log(log((log(x/e**x)** 2*x)/9)))/x) + 2)