\(\int \frac {-4 x^2+6 x^3+(-2 x^2+9 x^3) \log (x)+(2 x^2-6 x^3) \log ^2(x)+((-14 x^2+21 x^3) \log (x)+(6 x^2-9 x^3) \log ^2(x)) \log ((-4 x+6 x^2) \log (x))}{(-8+12 x) \log (x)+(8-12 x) \log ^2(x)+(-2+3 x) \log ^3(x)} \, dx\) [2603]

Optimal result

Integrand size = 118, antiderivative size = 23 \[ \int \frac {-4 x^2+6 x^3+\left (-2 x^2+9 x^3\right ) \log (x)+\left (2 x^2-6 x^3\right ) \log ^2(x)+\left (\left (-14 x^2+21 x^3\right ) \log (x)+\left (6 x^2-9 x^3\right ) \log ^2(x)\right ) \log \left (\left (-4 x+6 x^2\right ) \log (x)\right )}{(-8+12 x) \log (x)+(8-12 x) \log ^2(x)+(-2+3 x) \log ^3(x)} \, dx=\frac {x^3 \log (2 x (-2+3 x) \log (x))}{2-\log (x)} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-4 x^2+6 x^3+\left (-2 x^2+9 x^3\right ) \log (x)+\left (2 x^2-6 x^3\right ) \log ^2(x)+\left (\left (-14 x^2+21 x^3\right ) \log (x)+\left (6 x^2-9 x^3\right ) \log ^2(x)\right ) \log \left (\left (-4 x+6 x^2\right ) \log (x)\right )}{(-8+12 x) \log (x)+(8-12 x) \log ^2(x)+(-2+3 x) \log ^3(x)} \, dx=-\frac {x^3 \log (2 x (-2+3 x) \log (x))}{-2+\log (x)} \] Input:

Integrate[(-4*x^2 + 6*x^3 + (-2*x^2 + 9*x^3)*Log[x] + (2*x^2 - 6*x^3)*Log[ 
x]^2 + ((-14*x^2 + 21*x^3)*Log[x] + (6*x^2 - 9*x^3)*Log[x]^2)*Log[(-4*x + 
6*x^2)*Log[x]])/((-8 + 12*x)*Log[x] + (8 - 12*x)*Log[x]^2 + (-2 + 3*x)*Log 
[x]^3),x]
 

Output:

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

\(\displaystyle \int \frac {6 x^3-4 x^2+\left (2 x^2-6 x^3\right ) \log ^2(x)+\left (\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (21 x^3-14 x^2\right ) \log (x)\right ) \log \left (\left (6 x^2-4 x\right ) \log (x)\right )+\left (9 x^3-2 x^2\right ) \log (x)}{(3 x-2) \log ^3(x)+(8-12 x) \log ^2(x)+(12 x-8) \log (x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-6 x^3+4 x^2-\left (2 x^2-6 x^3\right ) \log ^2(x)-\left (\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (21 x^3-14 x^2\right ) \log (x)\right ) \log \left (\left (6 x^2-4 x\right ) \log (x)\right )-\left (9 x^3-2 x^2\right ) \log (x)}{(2-3 x) (2-\log (x))^2 \log (x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {6 x^3}{(3 x-2) (\log (x)-2)^2 \log (x)}-\frac {2 (3 x-1) x^2 \log (x)}{(3 x-2) (\log (x)-2)^2}-\frac {x^2 (3 \log (x)-7) \log (2 x (3 x-2) \log (x))}{(\log (x)-2)^2}-\frac {4 x^2}{(3 x-2) (\log (x)-2)^2 \log (x)}+\frac {(9 x-2) x^2}{(3 x-2) (\log (x)-2)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \int \frac {x^3}{(3 x-2) (\log (x)-2)^2}dx-\frac {3}{2} \int \frac {x^3}{(3 x-2) (\log (x)-2)}dx+\frac {3}{2} \int \frac {x^3}{(3 x-2) \log (x)}dx-2 \int \frac {x^2}{(3 x-2) (\log (x)-2)^2}dx-4 \int \frac {x^2 (3 x-1)}{(3 x-2) (\log (x)-2)^2}dx+\int \frac {x^2 (9 x-2)}{(3 x-2) (\log (x)-2)^2}dx+\int \frac {x^2}{(3 x-2) (\log (x)-2)}dx-2 \int \frac {x^2 (3 x-1)}{(3 x-2) (\log (x)-2)}dx-\int \frac {x^2}{(3 x-2) \log (x)}dx+7 \int \frac {x^2 \log (2 x (3 x-2) \log (x))}{(\log (x)-2)^2}dx-3 \int \frac {x^2 \log (x) \log (2 x (3 x-2) \log (x))}{(\log (x)-2)^2}dx\)

Input:

Int[(-4*x^2 + 6*x^3 + (-2*x^2 + 9*x^3)*Log[x] + (2*x^2 - 6*x^3)*Log[x]^2 + 
 ((-14*x^2 + 21*x^3)*Log[x] + (6*x^2 - 9*x^3)*Log[x]^2)*Log[(-4*x + 6*x^2) 
*Log[x]])/((-8 + 12*x)*Log[x] + (8 - 12*x)*Log[x]^2 + (-2 + 3*x)*Log[x]^3) 
,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09

method	result	size
parallelrisch	\(-\frac {x^{3} \ln \left (\left (6 x^{2}-4 x \right ) \ln \left (x \right )\right )}{\ln \left (x \right )-2}\)	\(25\)
risch	\(-\frac {x^{3} \ln \left (-\frac {2}{3}+x \right )}{\ln \left (x \right )-2}-\frac {x^{3} \left (i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \left (-\frac {2}{3}+x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )\right )+i \pi \operatorname {csgn}\left (i x \ln \left (x \right ) \left (-\frac {2}{3}+x \right )\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i x \ln \left (x \right ) \left (-\frac {2}{3}+x \right )\right )^{3}+i \pi \,\operatorname {csgn}\left (i \ln \left (x \right ) \left (-\frac {2}{3}+x \right )\right ) \operatorname {csgn}\left (i x \ln \left (x \right ) \left (-\frac {2}{3}+x \right )\right )^{2}-i \pi \,\operatorname {csgn}\left (i \ln \left (x \right ) \left (-\frac {2}{3}+x \right )\right ) \operatorname {csgn}\left (i x \ln \left (x \right ) \left (-\frac {2}{3}+x \right )\right ) \operatorname {csgn}\left (i x \right )+i \pi \,\operatorname {csgn}\left (i \left (-\frac {2}{3}+x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \left (-\frac {2}{3}+x \right )\right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (-\frac {2}{3}+x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \left (-\frac {2}{3}+x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right )-i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \left (-\frac {2}{3}+x \right )\right )^{3}+2 \ln \left (2\right )+2 \ln \left (3\right )+2 \ln \left (x \right )+2 \ln \left (\ln \left (x \right )\right )\right )}{2 \left (\ln \left (x \right )-2\right )}\)	\(220\)

Input:

int((((-9*x^3+6*x^2)*ln(x)^2+(21*x^3-14*x^2)*ln(x))*ln((6*x^2-4*x)*ln(x))+ 
(-6*x^3+2*x^2)*ln(x)^2+(9*x^3-2*x^2)*ln(x)+6*x^3-4*x^2)/((-2+3*x)*ln(x)^3+ 
(-12*x+8)*ln(x)^2+(12*x-8)*ln(x)),x,method=_RETURNVERBOSE)
 

Output:

-x^3*ln((6*x^2-4*x)*ln(x))/(ln(x)-2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-4 x^2+6 x^3+\left (-2 x^2+9 x^3\right ) \log (x)+\left (2 x^2-6 x^3\right ) \log ^2(x)+\left (\left (-14 x^2+21 x^3\right ) \log (x)+\left (6 x^2-9 x^3\right ) \log ^2(x)\right ) \log \left (\left (-4 x+6 x^2\right ) \log (x)\right )}{(-8+12 x) \log (x)+(8-12 x) \log ^2(x)+(-2+3 x) \log ^3(x)} \, dx=-\frac {x^{3} \log \left (2 \, {\left (3 \, x^{2} - 2 \, x\right )} \log \left (x\right )\right )}{\log \left (x\right ) - 2} \] Input:

integrate((((-9*x^3+6*x^2)*log(x)^2+(21*x^3-14*x^2)*log(x))*log((6*x^2-4*x 
)*log(x))+(-6*x^3+2*x^2)*log(x)^2+(9*x^3-2*x^2)*log(x)+6*x^3-4*x^2)/((-2+3 
*x)*log(x)^3+(-12*x+8)*log(x)^2+(12*x-8)*log(x)),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-4 x^2+6 x^3+\left (-2 x^2+9 x^3\right ) \log (x)+\left (2 x^2-6 x^3\right ) \log ^2(x)+\left (\left (-14 x^2+21 x^3\right ) \log (x)+\left (6 x^2-9 x^3\right ) \log ^2(x)\right ) \log \left (\left (-4 x+6 x^2\right ) \log (x)\right )}{(-8+12 x) \log (x)+(8-12 x) \log ^2(x)+(-2+3 x) \log ^3(x)} \, dx=- \frac {x^{3} \log {\left (\left (6 x^{2} - 4 x\right ) \log {\left (x \right )} \right )}}{\log {\left (x \right )} - 2} \] Input:

integrate((((-9*x**3+6*x**2)*ln(x)**2+(21*x**3-14*x**2)*ln(x))*ln((6*x**2- 
4*x)*ln(x))+(-6*x**3+2*x**2)*ln(x)**2+(9*x**3-2*x**2)*ln(x)+6*x**3-4*x**2) 
/((-2+3*x)*ln(x)**3+(-12*x+8)*ln(x)**2+(12*x-8)*ln(x)),x)
 

Output:

-x**3*log((6*x**2 - 4*x)*log(x))/(log(x) - 2)
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {-4 x^2+6 x^3+\left (-2 x^2+9 x^3\right ) \log (x)+\left (2 x^2-6 x^3\right ) \log ^2(x)+\left (\left (-14 x^2+21 x^3\right ) \log (x)+\left (6 x^2-9 x^3\right ) \log ^2(x)\right ) \log \left (\left (-4 x+6 x^2\right ) \log (x)\right )}{(-8+12 x) \log (x)+(8-12 x) \log ^2(x)+(-2+3 x) \log ^3(x)} \, dx=-\frac {x^{3} \log \left (2\right ) + x^{3} \log \left (3 \, x - 2\right ) + x^{3} \log \left (x\right ) + x^{3} \log \left (\log \left (x\right )\right )}{\log \left (x\right ) - 2} \] Input:

integrate((((-9*x^3+6*x^2)*log(x)^2+(21*x^3-14*x^2)*log(x))*log((6*x^2-4*x 
)*log(x))+(-6*x^3+2*x^2)*log(x)^2+(9*x^3-2*x^2)*log(x)+6*x^3-4*x^2)/((-2+3 
*x)*log(x)^3+(-12*x+8)*log(x)^2+(12*x-8)*log(x)),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {-4 x^2+6 x^3+\left (-2 x^2+9 x^3\right ) \log (x)+\left (2 x^2-6 x^3\right ) \log ^2(x)+\left (\left (-14 x^2+21 x^3\right ) \log (x)+\left (6 x^2-9 x^3\right ) \log ^2(x)\right ) \log \left (\left (-4 x+6 x^2\right ) \log (x)\right )}{(-8+12 x) \log (x)+(8-12 x) \log ^2(x)+(-2+3 x) \log ^3(x)} \, dx=-x^{3} - \frac {x^{3} \log \left (6 \, x \log \left (x\right ) - 4 \, \log \left (x\right )\right )}{\log \left (x\right ) - 2} - \frac {2 \, x^{3}}{\log \left (x\right ) - 2} \] Input:

integrate((((-9*x^3+6*x^2)*log(x)^2+(21*x^3-14*x^2)*log(x))*log((6*x^2-4*x 
)*log(x))+(-6*x^3+2*x^2)*log(x)^2+(9*x^3-2*x^2)*log(x)+6*x^3-4*x^2)/((-2+3 
*x)*log(x)^3+(-12*x+8)*log(x)^2+(12*x-8)*log(x)),x, algorithm="giac")
 

Output:

-x^3 - x^3*log(6*x*log(x) - 4*log(x))/(log(x) - 2) - 2*x^3/(log(x) - 2)
 

Mupad [B] (verification not implemented)

Time = 2.91 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-4 x^2+6 x^3+\left (-2 x^2+9 x^3\right ) \log (x)+\left (2 x^2-6 x^3\right ) \log ^2(x)+\left (\left (-14 x^2+21 x^3\right ) \log (x)+\left (6 x^2-9 x^3\right ) \log ^2(x)\right ) \log \left (\left (-4 x+6 x^2\right ) \log (x)\right )}{(-8+12 x) \log (x)+(8-12 x) \log ^2(x)+(-2+3 x) \log ^3(x)} \, dx=-\frac {x^3\,\ln \left (-2\,\ln \left (x\right )\,\left (2\,x-3\,x^2\right )\right )}{\ln \left (x\right )-2} \] Input:

int(-(log(x)*(2*x^2 - 9*x^3) + log(-log(x)*(4*x - 6*x^2))*(log(x)*(14*x^2 
- 21*x^3) - log(x)^2*(6*x^2 - 9*x^3)) - log(x)^2*(2*x^2 - 6*x^3) + 4*x^2 - 
 6*x^3)/(log(x)*(12*x - 8) + log(x)^3*(3*x - 2) - log(x)^2*(12*x - 8)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-4 x^2+6 x^3+\left (-2 x^2+9 x^3\right ) \log (x)+\left (2 x^2-6 x^3\right ) \log ^2(x)+\left (\left (-14 x^2+21 x^3\right ) \log (x)+\left (6 x^2-9 x^3\right ) \log ^2(x)\right ) \log \left (\left (-4 x+6 x^2\right ) \log (x)\right )}{(-8+12 x) \log (x)+(8-12 x) \log ^2(x)+(-2+3 x) \log ^3(x)} \, dx=-\frac {\mathrm {log}\left (6 \,\mathrm {log}\left (x \right ) x^{2}-4 \,\mathrm {log}\left (x \right ) x \right ) x^{3}}{\mathrm {log}\left (x \right )-2} \] Input:

int((((-9*x^3+6*x^2)*log(x)^2+(21*x^3-14*x^2)*log(x))*log((6*x^2-4*x)*log( 
x))+(-6*x^3+2*x^2)*log(x)^2+(9*x^3-2*x^2)*log(x)+6*x^3-4*x^2)/((-2+3*x)*lo 
g(x)^3+(-12*x+8)*log(x)^2+(12*x-8)*log(x)),x)
 

Output:

( - log(6*log(x)*x**2 - 4*log(x)*x)*x**3)/(log(x) - 2)