Integrand size = 62, antiderivative size = 18 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=2+\frac {1}{3+\frac {3 \log (x) \log (\log (x))}{2+x}} \] Output:
4/(12*ln(ln(x))*ln(x)/(2+x)+12)+2
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=-\frac {-2-x}{3 (2+x+\log (x) \log (\log (x)))} \] Input:
Integrate[(-2 - x + (-2 - x + x*Log[x])*Log[Log[x]])/(12*x + 12*x^2 + 3*x^ 3 + (12*x + 6*x^2)*Log[x]*Log[Log[x]] + 3*x*Log[x]^2*Log[Log[x]]^2),x]
Output:
-1/3*(-2 - x)/(2 + x + Log[x]*Log[Log[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.
\(\displaystyle \int \frac {-x+(-x+x \log (x)-2) \log (\log (x))-2}{3 x^3+12 x^2+\left (6 x^2+12 x\right ) \log (x) \log (\log (x))+12 x+3 x \log ^2(x) \log ^2(\log (x))} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {-x-(x+x (-\log (x))+2) \log (\log (x))-2}{3 x (x+\log (x) \log (\log (x))+2)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int -\frac {x+(-\log (x) x+x+2) \log (\log (x))+2}{x (x+\log (x) \log (\log (x))+2)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {x+(-\log (x) x+x+2) \log (\log (x))+2}{x (x+\log (x) \log (\log (x))+2)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{3} \int \left (\frac {-\log (x) x+x+2}{x \log (x) (x+\log (x) \log (\log (x))+2)}+\frac {(x+2) (\log (x) x-x+\log (x)-2)}{x \log (x) (x+\log (x) \log (\log (x))+2)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\int \frac {1}{-x-\log (x) \log (\log (x))-2}dx-3 \int \frac {1}{(x+\log (x) \log (\log (x))+2)^2}dx-2 \int \frac {1}{x (x+\log (x) \log (\log (x))+2)^2}dx-\int \frac {x}{(x+\log (x) \log (\log (x))+2)^2}dx+4 \int \frac {1}{\log (x) (x+\log (x) \log (\log (x))+2)^2}dx+4 \int \frac {1}{x \log (x) (x+\log (x) \log (\log (x))+2)^2}dx+\int \frac {x}{\log (x) (x+\log (x) \log (\log (x))+2)^2}dx-\int \frac {1}{\log (x) (x+\log (x) \log (\log (x))+2)}dx-2 \int \frac {1}{x \log (x) (x+\log (x) \log (\log (x))+2)}dx\right )\) |
Input:
Int[(-2 - x + (-2 - x + x*Log[x])*Log[Log[x]])/(12*x + 12*x^2 + 3*x^3 + (1 2*x + 6*x^2)*Log[x]*Log[Log[x]] + 3*x*Log[x]^2*Log[Log[x]]^2),x]
Output:
$Aborted
Time = 0.33 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {2+x}{3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+3 x +6}\) | \(17\) |
risch | \(\frac {2+x}{3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+3 x +6}\) | \(17\) |
parallelrisch | \(\frac {2+x}{3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+3 x +6}\) | \(17\) |
Input:
int(((x*ln(x)-x-2)*ln(ln(x))-x-2)/(3*x*ln(x)^2*ln(ln(x))^2+(6*x^2+12*x)*ln (x)*ln(ln(x))+3*x^3+12*x^2+12*x),x,method=_RETURNVERBOSE)
Output:
1/3*(2+x)/(ln(x)*ln(ln(x))+x+2)
Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {x + 2}{3 \, {\left (\log \left (x\right ) \log \left (\log \left (x\right )\right ) + x + 2\right )}} \] Input:
integrate(((x*log(x)-x-2)*log(log(x))-x-2)/(3*x*log(x)^2*log(log(x))^2+(6* x^2+12*x)*log(x)*log(log(x))+3*x^3+12*x^2+12*x),x, algorithm="fricas")
Output:
1/3*(x + 2)/(log(x)*log(log(x)) + x + 2)
Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {x + 2}{3 x + 3 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )} + 6} \] Input:
integrate(((x*ln(x)-x-2)*ln(ln(x))-x-2)/(3*x*ln(x)**2*ln(ln(x))**2+(6*x**2 +12*x)*ln(x)*ln(ln(x))+3*x**3+12*x**2+12*x),x)
Output:
(x + 2)/(3*x + 3*log(x)*log(log(x)) + 6)
Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {x + 2}{3 \, {\left (\log \left (x\right ) \log \left (\log \left (x\right )\right ) + x + 2\right )}} \] Input:
integrate(((x*log(x)-x-2)*log(log(x))-x-2)/(3*x*log(x)^2*log(log(x))^2+(6* x^2+12*x)*log(x)*log(log(x))+3*x^3+12*x^2+12*x),x, algorithm="maxima")
Output:
1/3*(x + 2)/(log(x)*log(log(x)) + x + 2)
Time = 0.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {x + 2}{3 \, {\left (\log \left (x\right ) \log \left (\log \left (x\right )\right ) + x + 2\right )}} \] Input:
integrate(((x*log(x)-x-2)*log(log(x))-x-2)/(3*x*log(x)^2*log(log(x))^2+(6* x^2+12*x)*log(x)*log(log(x))+3*x^3+12*x^2+12*x),x, algorithm="giac")
Output:
1/3*(x + 2)/(log(x)*log(log(x)) + x + 2)
Time = 3.79 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.50 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {\frac {4\,x}{3}-\ln \left (x\right )\,\left (\frac {x^3}{3}+x^2+\frac {2\,x}{3}\right )+\frac {4\,x^2}{3}+\frac {x^3}{3}}{\left (x+\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )+2\right )\,\left (2\,x-x^2\,\ln \left (x\right )-x\,\ln \left (x\right )+x^2\right )} \] Input:
int(-(x + log(log(x))*(x - x*log(x) + 2) + 2)/(12*x + 12*x^2 + 3*x^3 + log (log(x))*log(x)*(12*x + 6*x^2) + 3*x*log(log(x))^2*log(x)^2),x)
Output:
((4*x)/3 - log(x)*((2*x)/3 + x^2 + x^3/3) + (4*x^2)/3 + x^3/3)/((x + log(l og(x))*log(x) + 2)*(2*x - x^2*log(x) - x*log(x) + x^2))
Time = 0.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=-\frac {\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (x \right )}{3 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (x \right )+3 x +6} \] Input:
int(((x*log(x)-x-2)*log(log(x))-x-2)/(3*x*log(x)^2*log(log(x))^2+(6*x^2+12 *x)*log(x)*log(log(x))+3*x^3+12*x^2+12*x),x)
Output:
( - log(log(x))*log(x))/(3*(log(log(x))*log(x) + x + 2))