\(\int \frac {4+(4 x-2 x^3) \log (x)-4 x^2 \log (x) \log (\log (x))-2 x \log (x) \log ^2(\log (x))}{((4 x^2+x^3+2 x^4) \log (x)+(4 x+2 x^2+4 x^3) \log (x) \log (\log (x))+(x+2 x^2) \log (x) \log ^2(\log (x))) \log ^2(\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))})} \, dx\) [2411]

Optimal result

Integrand size = 120, antiderivative size = 17 \[ \int \frac {4+\left (4 x-2 x^3\right ) \log (x)-4 x^2 \log (x) \log (\log (x))-2 x \log (x) \log ^2(\log (x))}{\left (\left (4 x^2+x^3+2 x^4\right ) \log (x)+\left (4 x+2 x^2+4 x^3\right ) \log (x) \log (\log (x))+\left (x+2 x^2\right ) \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))}\right )} \, dx=\frac {1}{\log \left (-1-2 x-\frac {4}{x+\log (\log (x))}\right )} \] Output:

1/ln(-2*x-4/(ln(ln(x))+x)-1)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {4+\left (4 x-2 x^3\right ) \log (x)-4 x^2 \log (x) \log (\log (x))-2 x \log (x) \log ^2(\log (x))}{\left (\left (4 x^2+x^3+2 x^4\right ) \log (x)+\left (4 x+2 x^2+4 x^3\right ) \log (x) \log (\log (x))+\left (x+2 x^2\right ) \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))}\right )} \, dx=\frac {1}{\log \left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )} \] Input:

Integrate[(4 + (4*x - 2*x^3)*Log[x] - 4*x^2*Log[x]*Log[Log[x]] - 2*x*Log[x 
]*Log[Log[x]]^2)/(((4*x^2 + x^3 + 2*x^4)*Log[x] + (4*x + 2*x^2 + 4*x^3)*Lo 
g[x]*Log[Log[x]] + (x + 2*x^2)*Log[x]*Log[Log[x]]^2)*Log[(-4 - x - 2*x^2 + 
 (-1 - 2*x)*Log[Log[x]])/(x + Log[Log[x]])]^2),x]
 

Output:

Log[-((4 + x + 2*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])/(x + Log[Log[x]]))]^ 
(-1)
 

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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(4 + (4*x - 2*x^3)*Log[x] - 4*x^2*Log[x]*Log[Log[x]] - 2*x*Log[x]*Log[ 
Log[x]]^2)/(((4*x^2 + x^3 + 2*x^4)*Log[x] + (4*x + 2*x^2 + 4*x^3)*Log[x]*L 
og[Log[x]] + (x + 2*x^2)*Log[x]*Log[Log[x]]^2)*Log[(-4 - x - 2*x^2 + (-1 - 
 2*x)*Log[Log[x]])/(x + Log[Log[x]])]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 95.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82

method	result	size
parallelrisch	\(\frac {1}{\ln \left (\frac {\left (-1-2 x \right ) \ln \left (\ln \left (x \right )\right )-2 x^{2}-x -4}{\ln \left (\ln \left (x \right )\right )+x}\right )}\)	\(31\)
risch	\(\frac {2}{2 \ln \left (2\right )+2 i \pi -2 \ln \left (\ln \left (\ln \left (x \right )\right )+x \right )+2 \ln \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\ln \left (x \right )\right )+x}\right ) \operatorname {csgn}\left (i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )}{\ln \left (\ln \left (x \right )\right )+x}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\ln \left (x \right )\right )+x}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )}{\ln \left (\ln \left (x \right )\right )+x}\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )}{\ln \left (\ln \left (x \right )\right )+x}\right )}^{2}-2 i \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )}{\ln \left (\ln \left (x \right )\right )+x}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )}{\ln \left (\ln \left (x \right )\right )+x}\right )}^{3}}\)	\(274\)

Input:

int((-2*x*ln(x)*ln(ln(x))^2-4*x^2*ln(x)*ln(ln(x))+(-2*x^3+4*x)*ln(x)+4)/(( 
2*x^2+x)*ln(x)*ln(ln(x))^2+(4*x^3+2*x^2+4*x)*ln(x)*ln(ln(x))+(2*x^4+x^3+4* 
x^2)*ln(x))/ln(((-1-2*x)*ln(ln(x))-2*x^2-x-4)/(ln(ln(x))+x))^2,x,method=_R 
ETURNVERBOSE)
 

Output:

1/ln(((-1-2*x)*ln(ln(x))-2*x^2-x-4)/(ln(ln(x))+x))
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {4+\left (4 x-2 x^3\right ) \log (x)-4 x^2 \log (x) \log (\log (x))-2 x \log (x) \log ^2(\log (x))}{\left (\left (4 x^2+x^3+2 x^4\right ) \log (x)+\left (4 x+2 x^2+4 x^3\right ) \log (x) \log (\log (x))+\left (x+2 x^2\right ) \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))}\right )} \, dx=\frac {1}{\log \left (-\frac {2 \, x^{2} + {\left (2 \, x + 1\right )} \log \left (\log \left (x\right )\right ) + x + 4}{x + \log \left (\log \left (x\right )\right )}\right )} \] Input:

integrate((-2*x*log(x)*log(log(x))^2-4*x^2*log(x)*log(log(x))+(-2*x^3+4*x) 
*log(x)+4)/((2*x^2+x)*log(x)*log(log(x))^2+(4*x^3+2*x^2+4*x)*log(x)*log(lo 
g(x))+(2*x^4+x^3+4*x^2)*log(x))/log(((-1-2*x)*log(log(x))-2*x^2-x-4)/(log( 
log(x))+x))^2,x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 1.53 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {4+\left (4 x-2 x^3\right ) \log (x)-4 x^2 \log (x) \log (\log (x))-2 x \log (x) \log ^2(\log (x))}{\left (\left (4 x^2+x^3+2 x^4\right ) \log (x)+\left (4 x+2 x^2+4 x^3\right ) \log (x) \log (\log (x))+\left (x+2 x^2\right ) \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))}\right )} \, dx=\frac {1}{\log {\left (\frac {- 2 x^{2} - x + \left (- 2 x - 1\right ) \log {\left (\log {\left (x \right )} \right )} - 4}{x + \log {\left (\log {\left (x \right )} \right )}} \right )}} \] Input:

integrate((-2*x*ln(x)*ln(ln(x))**2-4*x**2*ln(x)*ln(ln(x))+(-2*x**3+4*x)*ln 
(x)+4)/((2*x**2+x)*ln(x)*ln(ln(x))**2+(4*x**3+2*x**2+4*x)*ln(x)*ln(ln(x))+ 
(2*x**4+x**3+4*x**2)*ln(x))/ln(((-2*x-1)*ln(ln(x))-2*x**2-x-4)/(ln(ln(x))+ 
x))**2,x)
 

Output:

1/log((-2*x**2 - x + (-2*x - 1)*log(log(x)) - 4)/(x + log(log(x))))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88 \[ \int \frac {4+\left (4 x-2 x^3\right ) \log (x)-4 x^2 \log (x) \log (\log (x))-2 x \log (x) \log ^2(\log (x))}{\left (\left (4 x^2+x^3+2 x^4\right ) \log (x)+\left (4 x+2 x^2+4 x^3\right ) \log (x) \log (\log (x))+\left (x+2 x^2\right ) \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))}\right )} \, dx=\frac {1}{\log \left (-2 \, x^{2} - {\left (2 \, x + 1\right )} \log \left (\log \left (x\right )\right ) - x - 4\right ) - \log \left (x + \log \left (\log \left (x\right )\right )\right )} \] Input:

integrate((-2*x*log(x)*log(log(x))^2-4*x^2*log(x)*log(log(x))+(-2*x^3+4*x) 
*log(x)+4)/((2*x^2+x)*log(x)*log(log(x))^2+(4*x^3+2*x^2+4*x)*log(x)*log(lo 
g(x))+(2*x^4+x^3+4*x^2)*log(x))/log(((-1-2*x)*log(log(x))-2*x^2-x-4)/(log( 
log(x))+x))^2,x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {4+\left (4 x-2 x^3\right ) \log (x)-4 x^2 \log (x) \log (\log (x))-2 x \log (x) \log ^2(\log (x))}{\left (\left (4 x^2+x^3+2 x^4\right ) \log (x)+\left (4 x+2 x^2+4 x^3\right ) \log (x) \log (\log (x))+\left (x+2 x^2\right ) \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))}\right )} \, dx=\frac {1}{\log \left (-2 \, x^{2} - 2 \, x \log \left (\log \left (x\right )\right ) - x - \log \left (\log \left (x\right )\right ) - 4\right ) - \log \left (x + \log \left (\log \left (x\right )\right )\right )} \] Input:

integrate((-2*x*log(x)*log(log(x))^2-4*x^2*log(x)*log(log(x))+(-2*x^3+4*x) 
*log(x)+4)/((2*x^2+x)*log(x)*log(log(x))^2+(4*x^3+2*x^2+4*x)*log(x)*log(lo 
g(x))+(2*x^4+x^3+4*x^2)*log(x))/log(((-1-2*x)*log(log(x))-2*x^2-x-4)/(log( 
log(x))+x))^2,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {4+\left (4 x-2 x^3\right ) \log (x)-4 x^2 \log (x) \log (\log (x))-2 x \log (x) \log ^2(\log (x))}{\left (\left (4 x^2+x^3+2 x^4\right ) \log (x)+\left (4 x+2 x^2+4 x^3\right ) \log (x) \log (\log (x))+\left (x+2 x^2\right ) \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))}\right )} \, dx=\frac {1}{\ln \left (-\frac {x+2\,x^2+\ln \left (\ln \left (x\right )\right )\,\left (2\,x+1\right )+4}{x+\ln \left (\ln \left (x\right )\right )}\right )} \] Input:

int((log(x)*(4*x - 2*x^3) - 2*x*log(log(x))^2*log(x) - 4*x^2*log(log(x))*l 
og(x) + 4)/(log(-(x + 2*x^2 + log(log(x))*(2*x + 1) + 4)/(x + log(log(x))) 
)^2*(log(x)*(4*x^2 + x^3 + 2*x^4) + log(log(x))^2*log(x)*(x + 2*x^2) + log 
(log(x))*log(x)*(4*x + 2*x^2 + 4*x^3))),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88 \[ \int \frac {4+\left (4 x-2 x^3\right ) \log (x)-4 x^2 \log (x) \log (\log (x))-2 x \log (x) \log ^2(\log (x))}{\left (\left (4 x^2+x^3+2 x^4\right ) \log (x)+\left (4 x+2 x^2+4 x^3\right ) \log (x) \log (\log (x))+\left (x+2 x^2\right ) \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))}\right )} \, dx=\frac {1}{\mathrm {log}\left (\frac {-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x -\mathrm {log}\left (\mathrm {log}\left (x \right )\right )-2 x^{2}-x -4}{\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+x}\right )} \] Input:

int((-2*x*log(x)*log(log(x))^2-4*x^2*log(x)*log(log(x))+(-2*x^3+4*x)*log(x 
)+4)/((2*x^2+x)*log(x)*log(log(x))^2+(4*x^3+2*x^2+4*x)*log(x)*log(log(x))+ 
(2*x^4+x^3+4*x^2)*log(x))/log(((-2*x-1)*log(log(x))-2*x^2-x-4)/(log(log(x) 
)+x))^2,x)
 

Output:

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