Integrand size = 94, antiderivative size = 23 \[ \int \frac {-3 x^2+3 x^3 \log (x)+\left (-3 x^3 \log (x)+3 x^2 \log (x) \log (\log (x))\right ) \log (-x+\log (\log (x)))+(-12 x \log (x)+12 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))}{\left (-x^3 \log (x)+x^2 \log (x) \log (\log (x))\right ) \log ^2(-x+\log (\log (x)))} \, dx=3 \left (-\frac {4+x}{x}+\frac {x}{\log (-x+\log (\log (x)))}\right ) \] Output:
3*x/ln(ln(ln(x))-x)-3*(4+x)/x
Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-3 x^2+3 x^3 \log (x)+\left (-3 x^3 \log (x)+3 x^2 \log (x) \log (\log (x))\right ) \log (-x+\log (\log (x)))+(-12 x \log (x)+12 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))}{\left (-x^3 \log (x)+x^2 \log (x) \log (\log (x))\right ) \log ^2(-x+\log (\log (x)))} \, dx=3 \left (-\frac {4}{x}+\frac {x}{\log (-x+\log (\log (x)))}\right ) \] Input:
Integrate[(-3*x^2 + 3*x^3*Log[x] + (-3*x^3*Log[x] + 3*x^2*Log[x]*Log[Log[x ]])*Log[-x + Log[Log[x]]] + (-12*x*Log[x] + 12*Log[x]*Log[Log[x]])*Log[-x + Log[Log[x]]]^2)/((-(x^3*Log[x]) + x^2*Log[x]*Log[Log[x]])*Log[-x + Log[L og[x]]]^2),x]
Output:
3*(-4/x + 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.
\(\displaystyle \int \frac {3 x^3 \log (x)-3 x^2+\left (3 x^2 \log (x) \log (\log (x))-3 x^3 \log (x)\right ) \log (\log (\log (x))-x)+(12 \log (x) \log (\log (x))-12 x \log (x)) \log ^2(\log (\log (x))-x)}{\left (x^2 \log (x) \log (\log (x))-x^3 \log (x)\right ) \log ^2(\log (\log (x))-x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-3 x^3 \log (x)+3 x^2-\left (3 x^2 \log (x) \log (\log (x))-3 x^3 \log (x)\right ) \log (\log (\log (x))-x)-(12 \log (x) \log (\log (x))-12 x \log (x)) \log ^2(\log (\log (x))-x)}{x^2 \log (x) (x-\log (\log (x))) \log ^2(\log (\log (x))-x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {12}{x^2}-\frac {3 (x \log (x)-1)}{\log (x) (x-\log (\log (x))) \log ^2(\log (\log (x))-x)}+\frac {3}{\log (\log (\log (x))-x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 \int \frac {x}{(x-\log (\log (x))) \log ^2(\log (\log (x))-x)}dx+3 \int \frac {1}{\log (x) (x-\log (\log (x))) \log ^2(\log (\log (x))-x)}dx+3 \int \frac {1}{\log (\log (\log (x))-x)}dx-\frac {12}{x}\) |
Input:
Int[(-3*x^2 + 3*x^3*Log[x] + (-3*x^3*Log[x] + 3*x^2*Log[x]*Log[Log[x]])*Lo g[-x + Log[Log[x]]] + (-12*x*Log[x] + 12*Log[x]*Log[Log[x]])*Log[-x + Log[ Log[x]]]^2)/((-(x^3*Log[x]) + x^2*Log[x]*Log[Log[x]])*Log[-x + Log[Log[x]] ]^2),x]
Output:
$Aborted
Time = 1.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {12}{x}+\frac {3 x}{\ln \left (\ln \left (\ln \left (x \right )\right )-x \right )}\) | \(20\) |
parallelrisch | \(\frac {6 x^{2}-24 \ln \left (\ln \left (\ln \left (x \right )\right )-x \right )}{2 x \ln \left (\ln \left (\ln \left (x \right )\right )-x \right )}\) | \(32\) |
Input:
int(((12*ln(x)*ln(ln(x))-12*x*ln(x))*ln(ln(ln(x))-x)^2+(3*x^2*ln(x)*ln(ln( x))-3*x^3*ln(x))*ln(ln(ln(x))-x)+3*x^3*ln(x)-3*x^2)/(x^2*ln(x)*ln(ln(x))-x ^3*ln(x))/ln(ln(ln(x))-x)^2,x,method=_RETURNVERBOSE)
Output:
-12/x+3*x/ln(ln(ln(x))-x)
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {-3 x^2+3 x^3 \log (x)+\left (-3 x^3 \log (x)+3 x^2 \log (x) \log (\log (x))\right ) \log (-x+\log (\log (x)))+(-12 x \log (x)+12 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))}{\left (-x^3 \log (x)+x^2 \log (x) \log (\log (x))\right ) \log ^2(-x+\log (\log (x)))} \, dx=\frac {3 \, {\left (x^{2} - 4 \, \log \left (-x + \log \left (\log \left (x\right )\right )\right )\right )}}{x \log \left (-x + \log \left (\log \left (x\right )\right )\right )} \] Input:
integrate(((12*log(x)*log(log(x))-12*x*log(x))*log(log(log(x))-x)^2+(3*x^2 *log(x)*log(log(x))-3*x^3*log(x))*log(log(log(x))-x)+3*x^3*log(x)-3*x^2)/( x^2*log(x)*log(log(x))-x^3*log(x))/log(log(log(x))-x)^2,x, algorithm="fric as")
Output:
3*(x^2 - 4*log(-x + log(log(x))))/(x*log(-x + log(log(x))))
Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {-3 x^2+3 x^3 \log (x)+\left (-3 x^3 \log (x)+3 x^2 \log (x) \log (\log (x))\right ) \log (-x+\log (\log (x)))+(-12 x \log (x)+12 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))}{\left (-x^3 \log (x)+x^2 \log (x) \log (\log (x))\right ) \log ^2(-x+\log (\log (x)))} \, dx=\frac {3 x}{\log {\left (- x + \log {\left (\log {\left (x \right )} \right )} \right )}} - \frac {12}{x} \] Input:
integrate(((12*ln(x)*ln(ln(x))-12*x*ln(x))*ln(ln(ln(x))-x)**2+(3*x**2*ln(x )*ln(ln(x))-3*x**3*ln(x))*ln(ln(ln(x))-x)+3*x**3*ln(x)-3*x**2)/(x**2*ln(x) *ln(ln(x))-x**3*ln(x))/ln(ln(ln(x))-x)**2,x)
Output:
3*x/log(-x + log(log(x))) - 12/x
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {-3 x^2+3 x^3 \log (x)+\left (-3 x^3 \log (x)+3 x^2 \log (x) \log (\log (x))\right ) \log (-x+\log (\log (x)))+(-12 x \log (x)+12 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))}{\left (-x^3 \log (x)+x^2 \log (x) \log (\log (x))\right ) \log ^2(-x+\log (\log (x)))} \, dx=\frac {3 \, {\left (x^{2} - 4 \, \log \left (-x + \log \left (\log \left (x\right )\right )\right )\right )}}{x \log \left (-x + \log \left (\log \left (x\right )\right )\right )} \] Input:
integrate(((12*log(x)*log(log(x))-12*x*log(x))*log(log(log(x))-x)^2+(3*x^2 *log(x)*log(log(x))-3*x^3*log(x))*log(log(log(x))-x)+3*x^3*log(x)-3*x^2)/( x^2*log(x)*log(log(x))-x^3*log(x))/log(log(log(x))-x)^2,x, algorithm="maxi ma")
Output:
3*(x^2 - 4*log(-x + log(log(x))))/(x*log(-x + log(log(x))))
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-3 x^2+3 x^3 \log (x)+\left (-3 x^3 \log (x)+3 x^2 \log (x) \log (\log (x))\right ) \log (-x+\log (\log (x)))+(-12 x \log (x)+12 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))}{\left (-x^3 \log (x)+x^2 \log (x) \log (\log (x))\right ) \log ^2(-x+\log (\log (x)))} \, dx=\frac {3 \, x}{\log \left (-x + \log \left (\log \left (x\right )\right )\right )} - \frac {12}{x} \] Input:
integrate(((12*log(x)*log(log(x))-12*x*log(x))*log(log(log(x))-x)^2+(3*x^2 *log(x)*log(log(x))-3*x^3*log(x))*log(log(log(x))-x)+3*x^3*log(x)-3*x^2)/( x^2*log(x)*log(log(x))-x^3*log(x))/log(log(log(x))-x)^2,x, algorithm="giac ")
Output:
3*x/log(-x + log(log(x))) - 12/x
Time = 2.52 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.91 \[ \int \frac {-3 x^2+3 x^3 \log (x)+\left (-3 x^3 \log (x)+3 x^2 \log (x) \log (\log (x))\right ) \log (-x+\log (\log (x)))+(-12 x \log (x)+12 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))}{\left (-x^3 \log (x)+x^2 \log (x) \log (\log (x))\right ) \log ^2(-x+\log (\log (x)))} \, dx=3\,x-3\,\ln \left (\ln \left (x\right )\right )-\frac {3\,\ln \left (\ln \left (x\right )\right )}{x\,\ln \left (x\right )-1}+\frac {3\,x-\frac {3\,x\,\ln \left (x\right )\,\ln \left (\ln \left (\ln \left (x\right )\right )-x\right )\,\left (x-\ln \left (\ln \left (x\right )\right )\right )}{x\,\ln \left (x\right )-1}}{\ln \left (\ln \left (\ln \left (x\right )\right )-x\right )}-\frac {12}{x}+\frac {3\,\left (x^2+x\right )}{\left (x\,\ln \left (x\right )-1\right )\,\left (x+1\right )} \] Input:
int(-(3*x^3*log(x) - log(log(log(x)) - x)*(3*x^3*log(x) - 3*x^2*log(log(x) )*log(x)) + log(log(log(x)) - x)^2*(12*log(log(x))*log(x) - 12*x*log(x)) - 3*x^2)/(log(log(log(x)) - x)^2*(x^3*log(x) - x^2*log(log(x))*log(x))),x)
Output:
3*x - 3*log(log(x)) - (3*log(log(x)))/(x*log(x) - 1) + (3*x - (3*x*log(x)* log(log(log(x)) - x)*(x - log(log(x))))/(x*log(x) - 1))/log(log(log(x)) - x) - 12/x + (3*(x + x^2))/((x*log(x) - 1)*(x + 1))
Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {-3 x^2+3 x^3 \log (x)+\left (-3 x^3 \log (x)+3 x^2 \log (x) \log (\log (x))\right ) \log (-x+\log (\log (x)))+(-12 x \log (x)+12 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))}{\left (-x^3 \log (x)+x^2 \log (x) \log (\log (x))\right ) \log ^2(-x+\log (\log (x)))} \, dx=\frac {-12 \,\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right )-x \right )+3 x^{2}}{\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right )-x \right ) x} \] Input:
int(((12*log(x)*log(log(x))-12*x*log(x))*log(log(log(x))-x)^2+(3*x^2*log(x )*log(log(x))-3*x^3*log(x))*log(log(log(x))-x)+3*x^3*log(x)-3*x^2)/(x^2*lo g(x)*log(log(x))-x^3*log(x))/log(log(log(x))-x)^2,x)
Output:
(3*( - 4*log(log(log(x)) - x) + x**2))/(log(log(log(x)) - x)*x)