\(\int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log (\log (\frac {3}{x}))}} ((6+3 x) \log (\frac {3}{x})+6 \log (\frac {3}{x}) \log (x)-3 x \log ^2(x))}{(4+4 x+x^2) \log (\frac {3}{x})+(-4 x-2 x^2) \log (\frac {3}{x}) \log (x) \log (\log (\frac {3}{x}))+x^2 \log (\frac {3}{x}) \log ^2(x) \log ^2(\log (\frac {3}{x}))} \, dx\) [2824]

Optimal result

Integrand size = 120, antiderivative size = 27 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{\frac {3}{\frac {2+x}{x \log (x)}-\log \left (\log \left (\frac {3}{x}\right )\right )}} \] Output:

exp(3/((2+x)/x/ln(x)-ln(ln(3/x))))
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

\[x^{-\frac {3 x}{x \ln \left (x \right ) \ln \left (\ln \left (3\right )-\ln \left (x \right )\right )-x -2}}\]

Input:

int((-3*x*ln(x)^2+6*ln(3/x)*ln(x)+(6+3*x)*ln(3/x))*exp(-3*x*ln(x)/(x*ln(x) 
*ln(ln(3/x))-x-2))/(x^2*ln(3/x)*ln(x)^2*ln(ln(3/x))^2+(-2*x^2-4*x)*ln(3/x) 
*ln(x)*ln(ln(3/x))+(x^2+4*x+4)*ln(3/x)),x)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{\left (-\frac {3 \, {\left (x \log \left (3\right ) - x \log \left (\frac {3}{x}\right )\right )}}{{\left (x \log \left (3\right ) - x \log \left (\frac {3}{x}\right )\right )} \log \left (\log \left (\frac {3}{x}\right )\right ) - x - 2}\right )} \] Input:

integrate((-3*x*log(x)^2+6*log(3/x)*log(x)+(6+3*x)*log(3/x))*exp(-3*x*log( 
x)/(x*log(x)*log(log(3/x))-x-2))/(x^2*log(3/x)*log(x)^2*log(log(3/x))^2+(- 
2*x^2-4*x)*log(3/x)*log(x)*log(log(3/x))+(x^2+4*x+4)*log(3/x)),x, algorith 
m="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 6.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{- \frac {3 x \log {\left (x \right )}}{x \log {\left (x \right )} \log {\left (- \log {\left (x \right )} + \log {\left (3 \right )} \right )} - x - 2}} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (26) = 52\).

Time = 0.36 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.56 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{\left (-\frac {6 \, \log \left (x\right )}{x \log \left (x\right )^{2} \log \left (\log \left (3\right ) - \log \left (x\right )\right )^{2} - 2 \, {\left (x \log \left (\log \left (3\right ) - \log \left (x\right )\right ) + \log \left (\log \left (3\right ) - \log \left (x\right )\right )\right )} \log \left (x\right ) + x + 2} - \frac {3 \, \log \left (x\right )}{\log \left (x\right ) \log \left (\log \left (3\right ) - \log \left (x\right )\right ) - 1}\right )} \] Input:

integrate((-3*x*log(x)^2+6*log(3/x)*log(x)+(6+3*x)*log(3/x))*exp(-3*x*log( 
x)/(x*log(x)*log(log(3/x))-x-2))/(x^2*log(3/x)*log(x)^2*log(log(3/x))^2+(- 
2*x^2-4*x)*log(3/x)*log(x)*log(log(3/x))+(x^2+4*x+4)*log(3/x)),x, algorith 
m="maxima")
 

Output:

e^(-6*log(x)/(x*log(x)^2*log(log(3) - log(x))^2 - 2*(x*log(log(3) - log(x) 
) + log(log(3) - log(x)))*log(x) + x + 2) - 3*log(x)/(log(x)*log(log(3) - 
log(x)) - 1))
 

Giac [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\frac {1}{x^{\frac {3 \, x}{x \log \left (x\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - x - 2}}} \] Input:

integrate((-3*x*log(x)^2+6*log(3/x)*log(x)+(6+3*x)*log(3/x))*exp(-3*x*log( 
x)/(x*log(x)*log(log(3/x))-x-2))/(x^2*log(3/x)*log(x)^2*log(log(3/x))^2+(- 
2*x^2-4*x)*log(3/x)*log(x)*log(log(3/x))+(x^2+4*x+4)*log(3/x)),x, algorith 
m="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx={\mathrm {e}}^{\frac {3\,x\,\ln \left (x\right )}{x-x\,\ln \left (\ln \left (\frac {1}{x}\right )+\ln \left (3\right )\right )\,\ln \left (x\right )+2}} \] Input:

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

Output:

exp((3*x*log(x))/(x - x*log(log(1/x) + log(3))*log(x) + 2))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\frac {1}{e^{\frac {3 \,\mathrm {log}\left (x \right ) x}{\mathrm {log}\left (\mathrm {log}\left (\frac {3}{x}\right )\right ) \mathrm {log}\left (x \right ) x -x -2}}} \] Input:

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

Output:

1/e**((3*log(x)*x)/(log(log(3/x))*log(x)*x - x - 2))