\(\int \frac {-x+(-x+x^2) \log (x)+(-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx\) [1209]

Optimal result

Integrand size = 76, antiderivative size = 18 \[ \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^2}{\log (8 (x-\log (x \log (x))))} \] Output:

x^2/ln(-8*ln(x*ln(x))+8*x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^2}{\log (8 (x-\log (x \log (x))))} \] Input:

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

Output:

x^2/Log[8*(x - 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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(-x + (-x + x^2)*Log[x] + (-2*x^2*Log[x] + 2*x*Log[x]*Log[x*Log[x]])*L 
og[8*x - 8*Log[x*Log[x]]])/((-(x*Log[x]) + Log[x]*Log[x*Log[x]])*Log[8*x - 
 8*Log[x*Log[x]]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 9.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06

method	result	size
parallelrisch	\(\frac {x^{2}}{\ln \left (-8 \ln \left (x \ln \left (x \right )\right )+8 x \right )}\)	\(19\)
risch	\(\frac {x^{2}}{\ln \left (-8 \ln \left (x \right )-8 \ln \left (\ln \left (x \right )\right )+4 i \pi \,\operatorname {csgn}\left (i x \ln \left (x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )+8 x \right )}\)	\(63\)
default	\(\frac {\ln \left (x \right ) x^{2} \left (-1+x \right )}{\left (x \ln \left (x \right )-\ln \left (x \right )-1\right ) \left (3 \ln \left (2\right )+\ln \left (-\ln \left (\ln \left (x \right )\right )-\ln \left (x \right )+\frac {i \pi \,\operatorname {csgn}\left (i x \ln \left (x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )}{2}+x \right )\right )}-\frac {x^{2}}{\left (x \ln \left (x \right )-\ln \left (x \right )-1\right ) \left (3 \ln \left (2\right )+\ln \left (-\ln \left (\ln \left (x \right )\right )-\ln \left (x \right )+\frac {i \pi \,\operatorname {csgn}\left (i x \ln \left (x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )}{2}+x \right )\right )}\)	\(162\)
parts	\(\frac {\ln \left (x \right ) x^{2} \left (-1+x \right )}{\left (x \ln \left (x \right )-\ln \left (x \right )-1\right ) \left (3 \ln \left (2\right )+\ln \left (-\ln \left (\ln \left (x \right )\right )-\ln \left (x \right )+\frac {i \pi \,\operatorname {csgn}\left (i x \ln \left (x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )}{2}+x \right )\right )}-\frac {x^{2}}{\left (x \ln \left (x \right )-\ln \left (x \right )-1\right ) \left (3 \ln \left (2\right )+\ln \left (-\ln \left (\ln \left (x \right )\right )-\ln \left (x \right )+\frac {i \pi \,\operatorname {csgn}\left (i x \ln \left (x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )}{2}+x \right )\right )}\)	\(162\)

Input:

int(((2*x*ln(x)*ln(x*ln(x))-2*x^2*ln(x))*ln(-8*ln(x*ln(x))+8*x)+ln(x)*(x^2 
-x)-x)/(ln(x)*ln(x*ln(x))-x*ln(x))/ln(-8*ln(x*ln(x))+8*x)^2,x,method=_RETU 
RNVERBOSE)
 

Output:

x^2/ln(-8*ln(x*ln(x))+8*x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^{2}}{\log \left (8 \, x - 8 \, \log \left (x \log \left (x\right )\right )\right )} \] Input:

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

Output:

x^2/log(8*x - 8*log(x*log(x)))
 

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^{2}}{\log {\left (8 x - 8 \log {\left (x \log {\left (x \right )} \right )} \right )}} \] Input:

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

Output:

x**2/log(8*x - 8*log(x*log(x)))
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^{2}}{i \, \pi + 3 \, \log \left (2\right ) + \log \left (-x + \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right )} \] Input:

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

Output:

x^2/(I*pi + 3*log(2) + log(-x + log(x) + log(log(x))))
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^{2}}{\log \left (8 \, x - 8 \, \log \left (x\right ) - 8 \, \log \left (\log \left (x\right )\right )\right )} \] Input:

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

Output:

x^2/log(8*x - 8*log(x) - 8*log(log(x)))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx=\frac {x^{2}}{\mathrm {log}\left (-8 \,\mathrm {log}\left (\mathrm {log}\left (x \right ) x \right )+8 x \right )} \] Input:

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

Output:

x**2/log( - 8*log(log(x)*x) + 8*x)