\(\int \frac {-4 x-4 x^2+8 x^3 \log (x)+(8 x^2-4 x^3) \log ^2(x)-4 x^2 \log ^3(x)+(8 x^2+8 x \log (x)) \log (x+\log (x))}{x^3 \log ^4(x)+x^2 \log ^5(x)+(-2 x^2 \log ^2(x)-2 x \log ^3(x)) \log (x+\log (x))+(x+\log (x)) \log ^2(x+\log (x))} \, dx\) [2453]

Optimal result

Integrand size = 113, antiderivative size = 20 \[ \int \frac {-4 x-4 x^2+8 x^3 \log (x)+\left (8 x^2-4 x^3\right ) \log ^2(x)-4 x^2 \log ^3(x)+\left (8 x^2+8 x \log (x)\right ) \log (x+\log (x))}{x^3 \log ^4(x)+x^2 \log ^5(x)+\left (-2 x^2 \log ^2(x)-2 x \log ^3(x)\right ) \log (x+\log (x))+(x+\log (x)) \log ^2(x+\log (x))} \, dx=\frac {4 x^2}{-x \log ^2(x)+\log (x+\log (x))} \] Output:

4*x^2/(ln(x+ln(x))-x*ln(x)^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x-4 x^2+8 x^3 \log (x)+\left (8 x^2-4 x^3\right ) \log ^2(x)-4 x^2 \log ^3(x)+\left (8 x^2+8 x \log (x)\right ) \log (x+\log (x))}{x^3 \log ^4(x)+x^2 \log ^5(x)+\left (-2 x^2 \log ^2(x)-2 x \log ^3(x)\right ) \log (x+\log (x))+(x+\log (x)) \log ^2(x+\log (x))} \, dx=\frac {4 x^2}{-x \log ^2(x)+\log (x+\log (x))} \] Input:

Integrate[(-4*x - 4*x^2 + 8*x^3*Log[x] + (8*x^2 - 4*x^3)*Log[x]^2 - 4*x^2* 
Log[x]^3 + (8*x^2 + 8*x*Log[x])*Log[x + Log[x]])/(x^3*Log[x]^4 + x^2*Log[x 
]^5 + (-2*x^2*Log[x]^2 - 2*x*Log[x]^3)*Log[x + Log[x]] + (x + Log[x])*Log[ 
x + Log[x]]^2),x]
 

Output:

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

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.

\(\displaystyle \int \frac {8 x^3 \log (x)-4 x^2-4 x^2 \log ^3(x)+\left (8 x^2+8 x \log (x)\right ) \log (x+\log (x))+\left (8 x^2-4 x^3\right ) \log ^2(x)-4 x}{x^3 \log ^4(x)+x^2 \log ^5(x)+\left (-2 x^2 \log ^2(x)-2 x \log ^3(x)\right ) \log (x+\log (x))+(x+\log (x)) \log ^2(x+\log (x))} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {4 x \left (-x^2 \log ^2(x)+2 x^2 \log (x)-x-x \log ^3(x)+2 x \log ^2(x)+2 \log (x+\log (x)) \log (x)+2 x \log (x+\log (x))-1\right )}{(x+\log (x)) \left (x \log ^2(x)-\log (x+\log (x))\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 4 \int -\frac {x \left (x \log ^3(x)+x^2 \log ^2(x)-2 x \log ^2(x)-2 x^2 \log (x)-2 \log (x+\log (x)) \log (x)+x-2 x \log (x+\log (x))+1\right )}{(x+\log (x)) \left (x \log ^2(x)-\log (x+\log (x))\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -4 \int \frac {x \left (x \log ^3(x)+x^2 \log ^2(x)-2 x \log ^2(x)-2 x^2 \log (x)-2 \log (x+\log (x)) \log (x)+x-2 x \log (x+\log (x))+1\right )}{(x+\log (x)) \left (x \log ^2(x)-\log (x+\log (x))\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {2 x}{x \log ^2(x)-\log (x+\log (x))}-\frac {x \left (x \log ^3(x)+x^2 \log ^2(x)+2 x \log ^2(x)+2 x^2 \log (x)-x-1\right )}{(x+\log (x)) \left (x \log ^2(x)-\log (x+\log (x))\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -4 \left (-2 \int \frac {x^3 \log (x)}{(x+\log (x)) \left (x \log ^2(x)-\log (x+\log (x))\right )^2}dx-\int \frac {x^3 \log ^2(x)}{(x+\log (x)) \left (x \log ^2(x)-\log (x+\log (x))\right )^2}dx+\int \frac {x^2}{(x+\log (x)) \left (x \log ^2(x)-\log (x+\log (x))\right )^2}dx-2 \int \frac {x^2 \log ^2(x)}{(x+\log (x)) \left (x \log ^2(x)-\log (x+\log (x))\right )^2}dx-\int \frac {x^2 \log ^3(x)}{(x+\log (x)) \left (x \log ^2(x)-\log (x+\log (x))\right )^2}dx+\int \frac {x}{(x+\log (x)) \left (x \log ^2(x)-\log (x+\log (x))\right )^2}dx+2 \int \frac {x}{x \log ^2(x)-\log (x+\log (x))}dx\right )\)

Input:

Int[(-4*x - 4*x^2 + 8*x^3*Log[x] + (8*x^2 - 4*x^3)*Log[x]^2 - 4*x^2*Log[x] 
^3 + (8*x^2 + 8*x*Log[x])*Log[x + Log[x]])/(x^3*Log[x]^4 + x^2*Log[x]^5 + 
(-2*x^2*Log[x]^2 - 2*x*Log[x]^3)*Log[x + Log[x]] + (x + Log[x])*Log[x + Lo 
g[x]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10

method	result	size
default	\(-\frac {4 x^{2}}{x \ln \left (x \right )^{2}-\ln \left (x +\ln \left (x \right )\right )}\)	\(22\)
risch	\(-\frac {4 x^{2}}{x \ln \left (x \right )^{2}-\ln \left (x +\ln \left (x \right )\right )}\)	\(22\)
parallelrisch	\(-\frac {4 x^{2}}{x \ln \left (x \right )^{2}-\ln \left (x +\ln \left (x \right )\right )}\)	\(22\)

Input:

int(((8*x*ln(x)+8*x^2)*ln(x+ln(x))-4*x^2*ln(x)^3+(-4*x^3+8*x^2)*ln(x)^2+8* 
x^3*ln(x)-4*x^2-4*x)/((x+ln(x))*ln(x+ln(x))^2+(-2*x*ln(x)^3-2*x^2*ln(x)^2) 
*ln(x+ln(x))+x^2*ln(x)^5+x^3*ln(x)^4),x,method=_RETURNVERBOSE)
 

Output:

-4*x^2/(x*ln(x)^2-ln(x+ln(x)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {-4 x-4 x^2+8 x^3 \log (x)+\left (8 x^2-4 x^3\right ) \log ^2(x)-4 x^2 \log ^3(x)+\left (8 x^2+8 x \log (x)\right ) \log (x+\log (x))}{x^3 \log ^4(x)+x^2 \log ^5(x)+\left (-2 x^2 \log ^2(x)-2 x \log ^3(x)\right ) \log (x+\log (x))+(x+\log (x)) \log ^2(x+\log (x))} \, dx=-\frac {4 \, x^{2}}{x \log \left (x\right )^{2} - \log \left (x + \log \left (x\right )\right )} \] Input:

integrate(((8*x*log(x)+8*x^2)*log(x+log(x))-4*x^2*log(x)^3+(-4*x^3+8*x^2)* 
log(x)^2+8*x^3*log(x)-4*x^2-4*x)/((x+log(x))*log(x+log(x))^2+(-2*x*log(x)^ 
3-2*x^2*log(x)^2)*log(x+log(x))+x^2*log(x)^5+x^3*log(x)^4),x, algorithm="f 
ricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-4 x-4 x^2+8 x^3 \log (x)+\left (8 x^2-4 x^3\right ) \log ^2(x)-4 x^2 \log ^3(x)+\left (8 x^2+8 x \log (x)\right ) \log (x+\log (x))}{x^3 \log ^4(x)+x^2 \log ^5(x)+\left (-2 x^2 \log ^2(x)-2 x \log ^3(x)\right ) \log (x+\log (x))+(x+\log (x)) \log ^2(x+\log (x))} \, dx=\frac {4 x^{2}}{- x \log {\left (x \right )}^{2} + \log {\left (x + \log {\left (x \right )} \right )}} \] Input:

integrate(((8*x*ln(x)+8*x**2)*ln(x+ln(x))-4*x**2*ln(x)**3+(-4*x**3+8*x**2) 
*ln(x)**2+8*x**3*ln(x)-4*x**2-4*x)/((x+ln(x))*ln(x+ln(x))**2+(-2*x*ln(x)** 
3-2*x**2*ln(x)**2)*ln(x+ln(x))+x**2*ln(x)**5+x**3*ln(x)**4),x)
 

Output:

4*x**2/(-x*log(x)**2 + log(x + log(x)))
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {-4 x-4 x^2+8 x^3 \log (x)+\left (8 x^2-4 x^3\right ) \log ^2(x)-4 x^2 \log ^3(x)+\left (8 x^2+8 x \log (x)\right ) \log (x+\log (x))}{x^3 \log ^4(x)+x^2 \log ^5(x)+\left (-2 x^2 \log ^2(x)-2 x \log ^3(x)\right ) \log (x+\log (x))+(x+\log (x)) \log ^2(x+\log (x))} \, dx=-\frac {4 \, x^{2}}{x \log \left (x\right )^{2} - \log \left (x + \log \left (x\right )\right )} \] Input:

integrate(((8*x*log(x)+8*x^2)*log(x+log(x))-4*x^2*log(x)^3+(-4*x^3+8*x^2)* 
log(x)^2+8*x^3*log(x)-4*x^2-4*x)/((x+log(x))*log(x+log(x))^2+(-2*x*log(x)^ 
3-2*x^2*log(x)^2)*log(x+log(x))+x^2*log(x)^5+x^3*log(x)^4),x, algorithm="m 
axima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {-4 x-4 x^2+8 x^3 \log (x)+\left (8 x^2-4 x^3\right ) \log ^2(x)-4 x^2 \log ^3(x)+\left (8 x^2+8 x \log (x)\right ) \log (x+\log (x))}{x^3 \log ^4(x)+x^2 \log ^5(x)+\left (-2 x^2 \log ^2(x)-2 x \log ^3(x)\right ) \log (x+\log (x))+(x+\log (x)) \log ^2(x+\log (x))} \, dx=-\frac {4 \, x^{2}}{x \log \left (x\right )^{2} - \log \left (x + \log \left (x\right )\right )} \] Input:

integrate(((8*x*log(x)+8*x^2)*log(x+log(x))-4*x^2*log(x)^3+(-4*x^3+8*x^2)* 
log(x)^2+8*x^3*log(x)-4*x^2-4*x)/((x+log(x))*log(x+log(x))^2+(-2*x*log(x)^ 
3-2*x^2*log(x)^2)*log(x+log(x))+x^2*log(x)^5+x^3*log(x)^4),x, algorithm="g 
iac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-4 x-4 x^2+8 x^3 \log (x)+\left (8 x^2-4 x^3\right ) \log ^2(x)-4 x^2 \log ^3(x)+\left (8 x^2+8 x \log (x)\right ) \log (x+\log (x))}{x^3 \log ^4(x)+x^2 \log ^5(x)+\left (-2 x^2 \log ^2(x)-2 x \log ^3(x)\right ) \log (x+\log (x))+(x+\log (x)) \log ^2(x+\log (x))} \, dx=\int -\frac {4\,x-8\,x^3\,\ln \left (x\right )-{\ln \left (x\right )}^2\,\left (8\,x^2-4\,x^3\right )+4\,x^2\,{\ln \left (x\right )}^3+4\,x^2-\ln \left (x+\ln \left (x\right )\right )\,\left (8\,x\,\ln \left (x\right )+8\,x^2\right )}{x^2\,{\ln \left (x\right )}^5+x^3\,{\ln \left (x\right )}^4-\ln \left (x+\ln \left (x\right )\right )\,\left (2\,x^2\,{\ln \left (x\right )}^2+2\,x\,{\ln \left (x\right )}^3\right )+{\ln \left (x+\ln \left (x\right )\right )}^2\,\left (x+\ln \left (x\right )\right )} \,d x \] Input:

int(-(4*x - 8*x^3*log(x) - log(x)^2*(8*x^2 - 4*x^3) + 4*x^2*log(x)^3 + 4*x 
^2 - log(x + log(x))*(8*x*log(x) + 8*x^2))/(x^2*log(x)^5 + x^3*log(x)^4 - 
log(x + log(x))*(2*x*log(x)^3 + 2*x^2*log(x)^2) + log(x + log(x))^2*(x + l 
og(x))),x)
 

Output:

int(-(4*x - 8*x^3*log(x) - log(x)^2*(8*x^2 - 4*x^3) + 4*x^2*log(x)^3 + 4*x 
^2 - log(x + log(x))*(8*x*log(x) + 8*x^2))/(x^2*log(x)^5 + x^3*log(x)^4 - 
log(x + log(x))*(2*x*log(x)^3 + 2*x^2*log(x)^2) + log(x + log(x))^2*(x + l 
og(x))), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x-4 x^2+8 x^3 \log (x)+\left (8 x^2-4 x^3\right ) \log ^2(x)-4 x^2 \log ^3(x)+\left (8 x^2+8 x \log (x)\right ) \log (x+\log (x))}{x^3 \log ^4(x)+x^2 \log ^5(x)+\left (-2 x^2 \log ^2(x)-2 x \log ^3(x)\right ) \log (x+\log (x))+(x+\log (x)) \log ^2(x+\log (x))} \, dx=\frac {4 x^{2}}{\mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )-\mathrm {log}\left (x \right )^{2} x} \] Input:

int(((8*x*log(x)+8*x^2)*log(x+log(x))-4*x^2*log(x)^3+(-4*x^3+8*x^2)*log(x) 
^2+8*x^3*log(x)-4*x^2-4*x)/((x+log(x))*log(x+log(x))^2+(-2*x*log(x)^3-2*x^ 
2*log(x)^2)*log(x+log(x))+x^2*log(x)^5+x^3*log(x)^4),x)
 

Output:

(4*x**2)/(log(log(x) + x) - log(x)**2*x)