\(\int \frac {e^{-e-2 x+x^2+2 x \log ^2(\frac {x-\log (x)}{\log (x)})} ((2 x-2 x^2) \log (x)+(-2+2 x) \log ^2(x)+(4 x-4 x \log (x)) \log (\frac {x-\log (x)}{\log (x)})+(-2 x \log (x)+2 \log ^2(x)) \log ^2(\frac {x-\log (x)}{\log (x)}))}{-x \log (x)+\log ^2(x)} \, dx\) [1806]

Optimal result

Integrand size = 114, antiderivative size = 26 \[ \int \frac {e^{-e-2 x+x^2+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )} \left (\left (2 x-2 x^2\right ) \log (x)+(-2+2 x) \log ^2(x)+(4 x-4 x \log (x)) \log \left (\frac {x-\log (x)}{\log (x)}\right )+\left (-2 x \log (x)+2 \log ^2(x)\right ) \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )\right )}{-x \log (x)+\log ^2(x)} \, dx=e^{-e-2 x+x^2+2 x \log ^2\left (-1+\frac {x}{\log (x)}\right )} \] Output:

exp(x^2-2*x+2*x*ln(x/ln(x)-1)^2-exp(1))
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-e-2 x+x^2+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )} \left (\left (2 x-2 x^2\right ) \log (x)+(-2+2 x) \log ^2(x)+(4 x-4 x \log (x)) \log \left (\frac {x-\log (x)}{\log (x)}\right )+\left (-2 x \log (x)+2 \log ^2(x)\right ) \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )\right )}{-x \log (x)+\log ^2(x)} \, dx=e^{-e-2 x+x^2+2 x \log ^2\left (-1+\frac {x}{\log (x)}\right )} \] Input:

Integrate[(E^(-E - 2*x + x^2 + 2*x*Log[(x - Log[x])/Log[x]]^2)*((2*x - 2*x 
^2)*Log[x] + (-2 + 2*x)*Log[x]^2 + (4*x - 4*x*Log[x])*Log[(x - Log[x])/Log 
[x]] + (-2*x*Log[x] + 2*Log[x]^2)*Log[(x - Log[x])/Log[x]]^2))/(-(x*Log[x] 
) + Log[x]^2),x]
 

Output:

E^(-E - 2*x + x^2 + 2*x*Log[-1 + x/Log[x]]^2)
 

Rubi [F]

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.

\(\displaystyle \int \frac {e^{x^2-2 x+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )-e} \left (\left (2 x-2 x^2\right ) \log (x)+(2 x-2) \log ^2(x)+\left (2 \log ^2(x)-2 x \log (x)\right ) \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )+(4 x-4 x \log (x)) \log \left (\frac {x-\log (x)}{\log (x)}\right )\right )}{\log ^2(x)-x \log (x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{x^2-2 x+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )-e} \left (-\left (2 x-2 x^2\right ) \log (x)-\left ((2 x-2) \log ^2(x)\right )-\left (2 \log ^2(x)-2 x \log (x)\right ) \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )-(4 x-4 x \log (x)) \log \left (\frac {x-\log (x)}{\log (x)}\right )\right )}{(x-\log (x)) \log (x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (2 e^{x^2-2 x+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )-e} \log ^2\left (\frac {x}{\log (x)}-1\right )+\frac {4 x e^{x^2-2 x+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )-e} (\log (x)-1) \log \left (\frac {x}{\log (x)}-1\right )}{(x-\log (x)) \log (x)}+2 (x-1) e^{x^2-2 x+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )-e}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \int e^{x^2+2 \log ^2\left (\frac {x-\log (x)}{\log (x)}\right ) x-2 x-e}dx+2 \int e^{x^2+2 \log ^2\left (\frac {x-\log (x)}{\log (x)}\right ) x-2 x-e} xdx+4 \int \frac {e^{x^2+2 \log ^2\left (\frac {x-\log (x)}{\log (x)}\right ) x-2 x-e} x \log \left (\frac {x}{\log (x)}-1\right )}{x-\log (x)}dx-4 \int \frac {e^{x^2+2 \log ^2\left (\frac {x-\log (x)}{\log (x)}\right ) x-2 x-e} x \log \left (\frac {x}{\log (x)}-1\right )}{(x-\log (x)) \log (x)}dx+2 \int e^{x^2+2 \log ^2\left (\frac {x-\log (x)}{\log (x)}\right ) x-2 x-e} \log ^2\left (\frac {x}{\log (x)}-1\right )dx\)

Input:

Int[(E^(-E - 2*x + x^2 + 2*x*Log[(x - Log[x])/Log[x]]^2)*((2*x - 2*x^2)*Lo 
g[x] + (-2 + 2*x)*Log[x]^2 + (4*x - 4*x*Log[x])*Log[(x - Log[x])/Log[x]] + 
 (-2*x*Log[x] + 2*Log[x]^2)*Log[(x - Log[x])/Log[x]]^2))/(-(x*Log[x]) + Lo 
g[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.69 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19

method	result	size
parallelrisch	\({\mathrm e}^{2 x \ln \left (-\frac {\ln \left (x \right )-x}{\ln \left (x \right )}\right )^{2}-{\mathrm e}+x^{2}-2 x}\)	\(31\)
risch	\(\text {Expression too large to display}\)	\(571\)

Input:

int(((2*ln(x)^2-2*x*ln(x))*ln((x-ln(x))/ln(x))^2+(-4*x*ln(x)+4*x)*ln((x-ln 
(x))/ln(x))+(-2+2*x)*ln(x)^2+(-2*x^2+2*x)*ln(x))*exp(2*x*ln((x-ln(x))/ln(x 
))^2-exp(1)+x^2-2*x)/(ln(x)^2-x*ln(x)),x,method=_RETURNVERBOSE)
 

Output:

exp(2*x*ln(-(ln(x)-x)/ln(x))^2-exp(1)+x^2-2*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-e-2 x+x^2+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )} \left (\left (2 x-2 x^2\right ) \log (x)+(-2+2 x) \log ^2(x)+(4 x-4 x \log (x)) \log \left (\frac {x-\log (x)}{\log (x)}\right )+\left (-2 x \log (x)+2 \log ^2(x)\right ) \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )\right )}{-x \log (x)+\log ^2(x)} \, dx=e^{\left (2 \, x \log \left (\frac {x - \log \left (x\right )}{\log \left (x\right )}\right )^{2} + x^{2} - 2 \, x - e\right )} \] Input:

integrate(((2*log(x)^2-2*x*log(x))*log((x-log(x))/log(x))^2+(-4*x*log(x)+4 
*x)*log((x-log(x))/log(x))+(2*x-2)*log(x)^2+(-2*x^2+2*x)*log(x))*exp(2*x*l 
og((x-log(x))/log(x))^2-exp(1)+x^2-2*x)/(log(x)^2-x*log(x)),x, algorithm=" 
fricas")
 

Output:

e^(2*x*log((x - log(x))/log(x))^2 + x^2 - 2*x - e)
 

Sympy [A] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-e-2 x+x^2+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )} \left (\left (2 x-2 x^2\right ) \log (x)+(-2+2 x) \log ^2(x)+(4 x-4 x \log (x)) \log \left (\frac {x-\log (x)}{\log (x)}\right )+\left (-2 x \log (x)+2 \log ^2(x)\right ) \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )\right )}{-x \log (x)+\log ^2(x)} \, dx=e^{x^{2} + 2 x \log {\left (\frac {x - \log {\left (x \right )}}{\log {\left (x \right )}} \right )}^{2} - 2 x - e} \] Input:

integrate(((2*ln(x)**2-2*x*ln(x))*ln((x-ln(x))/ln(x))**2+(-4*x*ln(x)+4*x)* 
ln((x-ln(x))/ln(x))+(2*x-2)*ln(x)**2+(-2*x**2+2*x)*ln(x))*exp(2*x*ln((x-ln 
(x))/ln(x))**2-exp(1)+x**2-2*x)/(ln(x)**2-x*ln(x)),x)
 

Output:

exp(x**2 + 2*x*log((x - log(x))/log(x))**2 - 2*x - E)
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \frac {e^{-e-2 x+x^2+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )} \left (\left (2 x-2 x^2\right ) \log (x)+(-2+2 x) \log ^2(x)+(4 x-4 x \log (x)) \log \left (\frac {x-\log (x)}{\log (x)}\right )+\left (-2 x \log (x)+2 \log ^2(x)\right ) \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )\right )}{-x \log (x)+\log ^2(x)} \, dx=e^{\left (2 \, x \log \left (x - \log \left (x\right )\right )^{2} - 4 \, x \log \left (x - \log \left (x\right )\right ) \log \left (\log \left (x\right )\right ) + 2 \, x \log \left (\log \left (x\right )\right )^{2} + x^{2} - 2 \, x - e\right )} \] Input:

integrate(((2*log(x)^2-2*x*log(x))*log((x-log(x))/log(x))^2+(-4*x*log(x)+4 
*x)*log((x-log(x))/log(x))+(2*x-2)*log(x)^2+(-2*x^2+2*x)*log(x))*exp(2*x*l 
og((x-log(x))/log(x))^2-exp(1)+x^2-2*x)/(log(x)^2-x*log(x)),x, algorithm=" 
maxima")
 

Output:

e^(2*x*log(x - log(x))^2 - 4*x*log(x - log(x))*log(log(x)) + 2*x*log(log(x 
))^2 + x^2 - 2*x - e)
 

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-e-2 x+x^2+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )} \left (\left (2 x-2 x^2\right ) \log (x)+(-2+2 x) \log ^2(x)+(4 x-4 x \log (x)) \log \left (\frac {x-\log (x)}{\log (x)}\right )+\left (-2 x \log (x)+2 \log ^2(x)\right ) \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )\right )}{-x \log (x)+\log ^2(x)} \, dx=e^{\left (2 \, x \log \left (\frac {x}{\log \left (x\right )} - 1\right )^{2} + x^{2} - 2 \, x - e\right )} \] Input:

integrate(((2*log(x)^2-2*x*log(x))*log((x-log(x))/log(x))^2+(-4*x*log(x)+4 
*x)*log((x-log(x))/log(x))+(2*x-2)*log(x)^2+(-2*x^2+2*x)*log(x))*exp(2*x*l 
og((x-log(x))/log(x))^2-exp(1)+x^2-2*x)/(log(x)^2-x*log(x)),x, algorithm=" 
giac")
 

Output:

e^(2*x*log(x/log(x) - 1)^2 + x^2 - 2*x - e)
 

Mupad [B] (verification not implemented)

Time = 1.88 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-e-2 x+x^2+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )} \left (\left (2 x-2 x^2\right ) \log (x)+(-2+2 x) \log ^2(x)+(4 x-4 x \log (x)) \log \left (\frac {x-\log (x)}{\log (x)}\right )+\left (-2 x \log (x)+2 \log ^2(x)\right ) \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )\right )}{-x \log (x)+\log ^2(x)} \, dx={\mathrm {e}}^{-\mathrm {e}}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{2\,x\,{\ln \left (\frac {x-\ln \left (x\right )}{\ln \left (x\right )}\right )}^2} \] Input:

int((exp(2*x*log((x - log(x))/log(x))^2 - exp(1) - 2*x + x^2)*(log((x - lo 
g(x))/log(x))*(4*x - 4*x*log(x)) + log((x - log(x))/log(x))^2*(2*log(x)^2 
- 2*x*log(x)) + log(x)*(2*x - 2*x^2) + log(x)^2*(2*x - 2)))/(log(x)^2 - x* 
log(x)),x)
 

Output:

exp(-exp(1))*exp(-2*x)*exp(x^2)*exp(2*x*log((x - log(x))/log(x))^2)
 

Reduce [F]

\[ \int \frac {e^{-e-2 x+x^2+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )} \left (\left (2 x-2 x^2\right ) \log (x)+(-2+2 x) \log ^2(x)+(4 x-4 x \log (x)) \log \left (\frac {x-\log (x)}{\log (x)}\right )+\left (-2 x \log (x)+2 \log ^2(x)\right ) \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )\right )}{-x \log (x)+\log ^2(x)} \, dx=\int \frac {\left (\left (2 \mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) x \right ) \mathrm {log}\left (\frac {x -\mathrm {log}\left (x \right )}{\mathrm {log}\left (x \right )}\right )^{2}+\left (-4 \,\mathrm {log}\left (x \right ) x +4 x \right ) \mathrm {log}\left (\frac {x -\mathrm {log}\left (x \right )}{\mathrm {log}\left (x \right )}\right )+\left (2 x -2\right ) \mathrm {log}\left (x \right )^{2}+\left (-2 x^{2}+2 x \right ) \mathrm {log}\left (x \right )\right ) {\mathrm e}^{2 x \mathrm {log}\left (\frac {x -\mathrm {log}\left (x \right )}{\mathrm {log}\left (x \right )}\right )^{2}-{\mathrm e}+x^{2}-2 x}}{\mathrm {log}\left (x \right )^{2}-\mathrm {log}\left (x \right ) x}d x \] Input:

int(((2*log(x)^2-2*x*log(x))*log((x-log(x))/log(x))^2+(-4*x*log(x)+4*x)*lo 
g((x-log(x))/log(x))+(2*x-2)*log(x)^2+(-2*x^2+2*x)*log(x))*exp(2*x*log((x- 
log(x))/log(x))^2-exp(1)+x^2-2*x)/(log(x)^2-x*log(x)),x)
 

Output:

int(((2*log(x)^2-2*x*log(x))*log((x-log(x))/log(x))^2+(-4*x*log(x)+4*x)*lo 
g((x-log(x))/log(x))+(2*x-2)*log(x)^2+(-2*x^2+2*x)*log(x))*exp(2*x*log((x- 
log(x))/log(x))^2-exp(1)+x^2-2*x)/(log(x)^2-x*log(x)),x)