\(\int \frac {(-8+2 x) \log (x)+x \log (x) \log (x^2) \log (\log (x^2))+((-2+2 x \log (x)) \log (x^2)+(4-x+(-4 x+x^2) \log (x)) \log (x^2) \log (\log (x^2))) \log (2+(-4+x) \log (\log (x^2)))}{(2 x \log (x) \log (x^2)+(-4 x+x^2) \log (x) \log (x^2) \log (\log (x^2))) \log (2+(-4+x) \log (\log (x^2)))} \, dx\) [2372]

Optimal result

Integrand size = 119, antiderivative size = 23 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x-\log \left (\frac {4 \log (x)}{\log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x-\log (\log (x))+\log \left (\log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )\right ) \] Input:

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

Output:

x - Log[Log[x]] + Log[Log[2 + (-4 + x)*Log[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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+2 x-8}{x \log \left (x^2\right ) \left (x \log \left (\log \left (x^2\right )\right )-4 \log \left (\log \left (x^2\right )\right )+2\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )}+\frac {x \log (x)-1}{x \log (x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \int \frac {1}{\log \left (x^2\right ) \left (x \log \left (\log \left (x^2\right )\right )-4 \log \left (\log \left (x^2\right )\right )+2\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )}dx-8 \int \frac {1}{x \log \left (x^2\right ) \left (x \log \left (\log \left (x^2\right )\right )-4 \log \left (\log \left (x^2\right )\right )+2\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )}dx+\int \frac {\log \left (\log \left (x^2\right )\right )}{\left (x \log \left (\log \left (x^2\right )\right )-4 \log \left (\log \left (x^2\right )\right )+2\right ) \log \left ((x-4) \log \left (\log \left (x^2\right )\right )+2\right )}dx+x-\log (\log (x))\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 85.91 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
parallelrisch	\(2-\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (\left (x -4\right ) \ln \left (\ln \left (x^{2}\right )\right )+2\right )\right )+x\)	\(22\)
risch	\(x -\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (\left (x -4\right ) \ln \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )+2\right )\right )\)	\(50\)

Input:

int(((((x^2-4*x)*ln(x)-x+4)*ln(x^2)*ln(ln(x^2))+(2*x*ln(x)-2)*ln(x^2))*ln( 
(x-4)*ln(ln(x^2))+2)+x*ln(x)*ln(x^2)*ln(ln(x^2))+(2*x-8)*ln(x))/((x^2-4*x) 
*ln(x)*ln(x^2)*ln(ln(x^2))+2*x*ln(x)*ln(x^2))/ln((x-4)*ln(ln(x^2))+2),x,me 
thod=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x - \log \left (2 \, \log \left (x\right )\right ) + \log \left (\log \left ({\left (x - 4\right )} \log \left (2 \, \log \left (x\right )\right ) + 2\right )\right ) \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x - \log {\left (\log {\left (x \right )} \right )} + \log {\left (\log {\left (\left (x - 4\right ) \log {\left (2 \log {\left (x \right )} \right )} + 2 \right )} \right )} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x + \log \left (\log \left (x \log \left (2\right ) + {\left (x - 4\right )} \log \left (\log \left (x\right )\right ) - 4 \, \log \left (2\right ) + 2\right )\right ) - \log \left (\log \left (x\right )\right ) \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=\int { \frac {x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right )\right ) + {\left ({\left ({\left (x^{2} - 4 \, x\right )} \log \left (x\right ) - x + 4\right )} \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right ) + 2 \, {\left (x \log \left (x\right ) - 1\right )} \log \left (x^{2}\right )\right )} \log \left ({\left (x - 4\right )} \log \left (\log \left (x^{2}\right )\right ) + 2\right ) + 2 \, {\left (x - 4\right )} \log \left (x\right )}{{\left ({\left (x^{2} - 4 \, x\right )} \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right )\right ) + 2 \, x \log \left (x^{2}\right ) \log \left (x\right )\right )} \log \left ({\left (x - 4\right )} \log \left (\log \left (x^{2}\right )\right ) + 2\right )} \,d x } \] Input:

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

Output:

integrate((x*log(x^2)*log(x)*log(log(x^2)) + (((x^2 - 4*x)*log(x) - x + 4) 
*log(x^2)*log(log(x^2)) + 2*(x*log(x) - 1)*log(x^2))*log((x - 4)*log(log(x 
^2)) + 2) + 2*(x - 4)*log(x))/(((x^2 - 4*x)*log(x^2)*log(x)*log(log(x^2)) 
+ 2*x*log(x^2)*log(x))*log((x - 4)*log(log(x^2)) + 2)), x)
 

Mupad [B] (verification not implemented)

Time = 3.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=x+\ln \left (\ln \left (\ln \left (\ln \left (x^2\right )\right )\,\left (x-4\right )+2\right )\right )-\ln \left (\ln \left (x\right )\right ) \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right ) x -4 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )+2\right )\right )-\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+x \] Input:

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

Output:

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