\(\int \frac {-6+3 x+(-3 x+(12-6 x) \log (4)) \log (x)+(12-6 x) \log (x) \log (\frac {-2+x}{\log (x)})}{(-2 x^3+x^4) \log ^2(4) \log (x)+(-4 x^3+2 x^4) \log (4) \log (x) \log (\frac {-2+x}{\log (x)})+(-2 x^3+x^4) \log (x) \log ^2(\frac {-2+x}{\log (x)})} \, dx\) [2645]

Optimal result

Integrand size = 105, antiderivative size = 19 \[ \int \frac {-6+3 x+(-3 x+(12-6 x) \log (4)) \log (x)+(12-6 x) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )}{\left (-2 x^3+x^4\right ) \log ^2(4) \log (x)+\left (-4 x^3+2 x^4\right ) \log (4) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )+\left (-2 x^3+x^4\right ) \log (x) \log ^2\left (\frac {-2+x}{\log (x)}\right )} \, dx=\frac {3}{x^2 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-6+3 x+(-3 x+(12-6 x) \log (4)) \log (x)+(12-6 x) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )}{\left (-2 x^3+x^4\right ) \log ^2(4) \log (x)+\left (-4 x^3+2 x^4\right ) \log (4) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )+\left (-2 x^3+x^4\right ) \log (x) \log ^2\left (\frac {-2+x}{\log (x)}\right )} \, dx=\frac {3}{x^2 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )} \] Input:

Integrate[(-6 + 3*x + (-3*x + (12 - 6*x)*Log[4])*Log[x] + (12 - 6*x)*Log[x 
]*Log[(-2 + x)/Log[x]])/((-2*x^3 + x^4)*Log[4]^2*Log[x] + (-4*x^3 + 2*x^4) 
*Log[4]*Log[x]*Log[(-2 + x)/Log[x]] + (-2*x^3 + x^4)*Log[x]*Log[(-2 + x)/L 
og[x]]^2),x]
 

Output:

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

\(\displaystyle \int \frac {3 x+((12-6 x) \log (4)-3 x) \log (x)+(12-6 x) \log (x) \log \left (\frac {x-2}{\log (x)}\right )-6}{\left (x^4-2 x^3\right ) \log (x) \log ^2\left (\frac {x-2}{\log (x)}\right )+\left (x^4-2 x^3\right ) \log ^2(4) \log (x)+\left (2 x^4-4 x^3\right ) \log (4) \log (x) \log \left (\frac {x-2}{\log (x)}\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-3 x+3 \log (x) \left (x+x \log (16)+2 (x-2) \log \left (\frac {x-2}{\log (x)}\right )-4 \log (4)\right )+6}{(2-x) x^3 \log (x) \left (\log \left (\frac {x-2}{\log (x)}\right )+\log (4)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {3 (-x+x \log (x)+2)}{(x-2) x^3 \log (x) \left (\log \left (\frac {x-2}{\log (x)}\right )+\log (4)\right )^2}-\frac {6}{x^3 \left (\log \left (\frac {x-2}{\log (x)}\right )+\log (4)\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \int \frac {1}{x^3 \log (x) \left (\log \left (\frac {x-2}{\log (x)}\right )+\log (4)\right )^2}dx-6 \int \frac {1}{x^3 \left (\log \left (\frac {x-2}{\log (x)}\right )+\log (4)\right )}dx+\frac {3}{2} \int \frac {1}{x^2 \left (\log \left (\frac {x-2}{\log (x)}\right )+\log (4)\right )^2}dx-\frac {3}{4} \int \frac {1}{(x-2) \left (\log \left (\frac {x-2}{\log (x)}\right )+\log (4)\right )^2}dx+\frac {3}{4} \int \frac {1}{x \left (\log \left (\frac {x-2}{\log (x)}\right )+\log (4)\right )^2}dx\)

Input:

Int[(-6 + 3*x + (-3*x + (12 - 6*x)*Log[4])*Log[x] + (12 - 6*x)*Log[x]*Log[ 
(-2 + x)/Log[x]])/((-2*x^3 + x^4)*Log[4]^2*Log[x] + (-4*x^3 + 2*x^4)*Log[4 
]*Log[x]*Log[(-2 + x)/Log[x]] + (-2*x^3 + x^4)*Log[x]*Log[(-2 + x)/Log[x]] 
^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16

method	result	size
parallelrisch	\(\frac {3}{x^{2} \left (2 \ln \left (2\right )+\ln \left (\frac {-2+x}{\ln \left (x \right )}\right )\right )}\)	\(22\)
risch	\(-\frac {6 i}{x^{2} \left (\pi \,\operatorname {csgn}\left (i \left (-2+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (-2+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )-\pi \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{3}+\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{2}-4 i \ln \left (2\right )+2 i \ln \left (\ln \left (x \right )\right )-2 i \ln \left (-2+x \right )\right )}\)	\(118\)
default	\(-\frac {6 i \left (-2+x \right )}{\left (x \ln \left (x \right )-x +2\right ) x^{2} \left (\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (-2+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (-2+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{3}+4 i \ln \left (2\right )-2 i \ln \left (\ln \left (x \right )\right )+2 i \ln \left (-2+x \right )\right )}+\frac {6 i \ln \left (x \right )}{\left (x \ln \left (x \right )-x +2\right ) x \left (\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (-2+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (-2+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{3}+4 i \ln \left (2\right )-2 i \ln \left (\ln \left (x \right )\right )+2 i \ln \left (-2+x \right )\right )}\)	\(263\)
parts	\(-\frac {6 i \left (-2+x \right )}{\left (x \ln \left (x \right )-x +2\right ) x^{2} \left (\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (-2+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (-2+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{3}+4 i \ln \left (2\right )-2 i \ln \left (\ln \left (x \right )\right )+2 i \ln \left (-2+x \right )\right )}+\frac {6 i \ln \left (x \right )}{\left (x \ln \left (x \right )-x +2\right ) x \left (\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (-2+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (-2+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (-2+x \right )}{\ln \left (x \right )}\right )^{3}+4 i \ln \left (2\right )-2 i \ln \left (\ln \left (x \right )\right )+2 i \ln \left (-2+x \right )\right )}\)	\(263\)

Input:

int(((12-6*x)*ln(x)*ln((-2+x)/ln(x))+(2*(12-6*x)*ln(2)-3*x)*ln(x)+3*x-6)/( 
(x^4-2*x^3)*ln(x)*ln((-2+x)/ln(x))^2+2*(2*x^4-4*x^3)*ln(2)*ln(x)*ln((-2+x) 
/ln(x))+4*(x^4-2*x^3)*ln(2)^2*ln(x)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32 \[ \int \frac {-6+3 x+(-3 x+(12-6 x) \log (4)) \log (x)+(12-6 x) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )}{\left (-2 x^3+x^4\right ) \log ^2(4) \log (x)+\left (-4 x^3+2 x^4\right ) \log (4) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )+\left (-2 x^3+x^4\right ) \log (x) \log ^2\left (\frac {-2+x}{\log (x)}\right )} \, dx=\frac {3}{2 \, x^{2} \log \left (2\right ) + x^{2} \log \left (\frac {x - 2}{\log \left (x\right )}\right )} \] Input:

integrate(((12-6*x)*log(x)*log((-2+x)/log(x))+(2*(12-6*x)*log(2)-3*x)*log( 
x)+3*x-6)/((x^4-2*x^3)*log(x)*log((-2+x)/log(x))^2+2*(2*x^4-4*x^3)*log(2)* 
log(x)*log((-2+x)/log(x))+4*(x^4-2*x^3)*log(2)^2*log(x)),x, algorithm="fri 
cas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {-6+3 x+(-3 x+(12-6 x) \log (4)) \log (x)+(12-6 x) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )}{\left (-2 x^3+x^4\right ) \log ^2(4) \log (x)+\left (-4 x^3+2 x^4\right ) \log (4) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )+\left (-2 x^3+x^4\right ) \log (x) \log ^2\left (\frac {-2+x}{\log (x)}\right )} \, dx=\frac {3}{x^{2} \log {\left (\frac {x - 2}{\log {\left (x \right )}} \right )} + 2 x^{2} \log {\left (2 \right )}} \] Input:

integrate(((12-6*x)*ln(x)*ln((-2+x)/ln(x))+(2*(12-6*x)*ln(2)-3*x)*ln(x)+3* 
x-6)/((x**4-2*x**3)*ln(x)*ln((-2+x)/ln(x))**2+2*(2*x**4-4*x**3)*ln(2)*ln(x 
)*ln((-2+x)/ln(x))+4*(x**4-2*x**3)*ln(2)**2*ln(x)),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {-6+3 x+(-3 x+(12-6 x) \log (4)) \log (x)+(12-6 x) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )}{\left (-2 x^3+x^4\right ) \log ^2(4) \log (x)+\left (-4 x^3+2 x^4\right ) \log (4) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )+\left (-2 x^3+x^4\right ) \log (x) \log ^2\left (\frac {-2+x}{\log (x)}\right )} \, dx=\frac {3}{2 \, x^{2} \log \left (2\right ) + x^{2} \log \left (x - 2\right ) - x^{2} \log \left (\log \left (x\right )\right )} \] Input:

integrate(((12-6*x)*log(x)*log((-2+x)/log(x))+(2*(12-6*x)*log(2)-3*x)*log( 
x)+3*x-6)/((x^4-2*x^3)*log(x)*log((-2+x)/log(x))^2+2*(2*x^4-4*x^3)*log(2)* 
log(x)*log((-2+x)/log(x))+4*(x^4-2*x^3)*log(2)^2*log(x)),x, algorithm="max 
ima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {-6+3 x+(-3 x+(12-6 x) \log (4)) \log (x)+(12-6 x) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )}{\left (-2 x^3+x^4\right ) \log ^2(4) \log (x)+\left (-4 x^3+2 x^4\right ) \log (4) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )+\left (-2 x^3+x^4\right ) \log (x) \log ^2\left (\frac {-2+x}{\log (x)}\right )} \, dx=\frac {3}{2 \, x^{2} \log \left (2\right ) + x^{2} \log \left (x - 2\right ) - x^{2} \log \left (\log \left (x\right )\right )} \] Input:

integrate(((12-6*x)*log(x)*log((-2+x)/log(x))+(2*(12-6*x)*log(2)-3*x)*log( 
x)+3*x-6)/((x^4-2*x^3)*log(x)*log((-2+x)/log(x))^2+2*(2*x^4-4*x^3)*log(2)* 
log(x)*log((-2+x)/log(x))+4*(x^4-2*x^3)*log(2)^2*log(x)),x, algorithm="gia 
c")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-6+3 x+(-3 x+(12-6 x) \log (4)) \log (x)+(12-6 x) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )}{\left (-2 x^3+x^4\right ) \log ^2(4) \log (x)+\left (-4 x^3+2 x^4\right ) \log (4) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )+\left (-2 x^3+x^4\right ) \log (x) \log ^2\left (\frac {-2+x}{\log (x)}\right )} \, dx=\int \frac {\ln \left (x\right )\,\left (3\,x+2\,\ln \left (2\right )\,\left (6\,x-12\right )\right )-3\,x+\ln \left (x\right )\,\ln \left (\frac {x-2}{\ln \left (x\right )}\right )\,\left (6\,x-12\right )+6}{\ln \left (x\right )\,\left (2\,x^3-x^4\right )\,{\ln \left (\frac {x-2}{\ln \left (x\right )}\right )}^2+2\,\ln \left (2\right )\,\ln \left (x\right )\,\left (4\,x^3-2\,x^4\right )\,\ln \left (\frac {x-2}{\ln \left (x\right )}\right )+4\,{\ln \left (2\right )}^2\,\ln \left (x\right )\,\left (2\,x^3-x^4\right )} \,d x \] Input:

int((log(x)*(3*x + 2*log(2)*(6*x - 12)) - 3*x + log(x)*log((x - 2)/log(x)) 
*(6*x - 12) + 6)/(log(x)*log((x - 2)/log(x))^2*(2*x^3 - x^4) + 4*log(2)^2* 
log(x)*(2*x^3 - x^4) + 2*log(2)*log(x)*log((x - 2)/log(x))*(4*x^3 - 2*x^4) 
),x)
 

Output:

int((log(x)*(3*x + 2*log(2)*(6*x - 12)) - 3*x + log(x)*log((x - 2)/log(x)) 
*(6*x - 12) + 6)/(log(x)*log((x - 2)/log(x))^2*(2*x^3 - x^4) + 4*log(2)^2* 
log(x)*(2*x^3 - x^4) + 2*log(2)*log(x)*log((x - 2)/log(x))*(4*x^3 - 2*x^4) 
), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {-6+3 x+(-3 x+(12-6 x) \log (4)) \log (x)+(12-6 x) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )}{\left (-2 x^3+x^4\right ) \log ^2(4) \log (x)+\left (-4 x^3+2 x^4\right ) \log (4) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )+\left (-2 x^3+x^4\right ) \log (x) \log ^2\left (\frac {-2+x}{\log (x)}\right )} \, dx=\frac {3}{x^{2} \left (\mathrm {log}\left (\frac {x -2}{\mathrm {log}\left (x \right )}\right )+2 \,\mathrm {log}\left (2\right )\right )} \] Input:

int(((12-6*x)*log(x)*log((-2+x)/log(x))+(2*(12-6*x)*log(2)-3*x)*log(x)+3*x 
-6)/((x^4-2*x^3)*log(x)*log((-2+x)/log(x))^2+2*(2*x^4-4*x^3)*log(2)*log(x) 
*log((-2+x)/log(x))+4*(x^4-2*x^3)*log(2)^2*log(x)),x)
 

Output:

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