Integrand size = 120, antiderivative size = 27 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{\frac {3}{\frac {2+x}{x \log (x)}-\log \left (\log \left (\frac {3}{x}\right )\right )}} \]
Time = 0.70 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \]
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]
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 |
\(\displaystyle -3 \int \frac {x^{\frac {3 x}{-\log (x) \log \left (\log \left (\frac {3}{x}\right )\right ) x+x+2}+1} \log ^2(x)}{\log \left (\frac {3}{x}\right ) \left (\log (x) \log \left (\log \left (\frac {3}{x}\right )\right ) x-x-2\right )^2}dx+3 \int \frac {x^{\frac {3 x}{-\log (x) \log \left (\log \left (\frac {3}{x}\right )\right ) x+x+2}+1}}{\left (\log (x) \log \left (\log \left (\frac {3}{x}\right )\right ) x-x-2\right )^2}dx+6 \int \frac {x^{\frac {3 x}{-\log (x) \log \left (\log \left (\frac {3}{x}\right )\right ) x+x+2}}}{\left (\log (x) \log \left (\log \left (\frac {3}{x}\right )\right ) x-x-2\right )^2}dx+6 \int \frac {x^{\frac {3 x}{-\log (x) \log \left (\log \left (\frac {3}{x}\right )\right ) x+x+2}} \log (x)}{\left (\log (x) \log \left (\log \left (\frac {3}{x}\right )\right ) x-x-2\right )^2}dx\) |
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]
3.29.24.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
\[x^{-\frac {3 x}{x \ln \left (x \right ) \ln \left (-\ln \left (x \right )+\ln \left (3\right )\right )-x -2}}\]
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)
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{\left (-\frac {3 \, {\left (x \log \left (3\right ) - x \log \left (\frac {3}{x}\right )\right )}}{{\left (x \log \left (3\right ) - x \log \left (\frac {3}{x}\right )\right )} \log \left (\log \left (\frac {3}{x}\right )\right ) - x - 2}\right )} \]
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=\
Time = 2.56 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{- \frac {3 x \log {\left (x \right )}}{x \log {\left (x \right )} \log {\left (- \log {\left (x \right )} + \log {\left (3 \right )} \right )} - x - 2}} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (26) = 52\).
Time = 0.52 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.56 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{\left (-\frac {6 \, \log \left (x\right )}{x \log \left (x\right )^{2} \log \left (\log \left (3\right ) - \log \left (x\right )\right )^{2} - 2 \, {\left (x \log \left (\log \left (3\right ) - \log \left (x\right )\right ) + \log \left (\log \left (3\right ) - \log \left (x\right )\right )\right )} \log \left (x\right ) + x + 2} - \frac {3 \, \log \left (x\right )}{\log \left (x\right ) \log \left (\log \left (3\right ) - \log \left (x\right )\right ) - 1}\right )} \]
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=\
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))
Time = 0.71 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\frac {1}{x^{\frac {3 \, x}{x \log \left (x\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - x - 2}}} \]
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=\
Time = 12.65 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx={\mathrm {e}}^{\frac {3\,x\,\ln \left (x\right )}{x-x\,\ln \left (\ln \left (\frac {1}{x}\right )+\ln \left (3\right )\right )\,\ln \left (x\right )+2}} \]