Integrand size = 42, antiderivative size = 18 \[ \int \frac {-\log (3)+\log (3) \log (x) \log (\log (x))+(-1+4 \log (3)+\log (3) \log (4)) \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=-5-x+x \log (3) \left (4+\log (4)+\frac {1}{\log (\log (x))}\right ) \] Output:
ln(3)*x*(4+2*ln(2)+1/ln(ln(x)))-5-x
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {-\log (3)+\log (3) \log (x) \log (\log (x))+(-1+4 \log (3)+\log (3) \log (4)) \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=x (-1+\log (3) \log (4)+\log (81))+\frac {x \log (3)}{\log (\log (x))} \] Input:
Integrate[(-Log[3] + Log[3]*Log[x]*Log[Log[x]] + (-1 + 4*Log[3] + Log[3]*L og[4])*Log[x]*Log[Log[x]]^2)/(Log[x]*Log[Log[x]]^2),x]
Output:
x*(-1 + Log[3]*Log[4] + Log[81]) + (x*Log[3])/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.
\(\displaystyle \int \frac {(-1+4 \log (3)+\log (3) \log (4)) \log (x) \log ^2(\log (x))+\log (3) \log (x) \log (\log (x))-\log (3)}{\log (x) \log ^2(\log (x))} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {\log (3)}{\log ^2(\log (x)) \log (x)}+\frac {\log (3)}{\log (\log (x))}-1+\log (81)+\log (3) \log (4)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\log (3) \int \frac {1}{\log (x) \log ^2(\log (x))}dx+\log (3) \int \frac {1}{\log (\log (x))}dx-(x (1-\log (3) \log (4)-\log (81)))\) |
Input:
Int[(-Log[3] + Log[3]*Log[x]*Log[Log[x]] + (-1 + 4*Log[3] + Log[3]*Log[4]) *Log[x]*Log[Log[x]]^2)/(Log[x]*Log[Log[x]]^2),x]
Output:
$Aborted
Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44
method | result | size |
risch | \(2 x \ln \left (2\right ) \ln \left (3\right )+4 x \ln \left (3\right )-x +\frac {x \ln \left (3\right )}{\ln \left (\ln \left (x \right )\right )}\) | \(26\) |
norman | \(\frac {x \ln \left (3\right )+\left (2 \ln \left (2\right ) \ln \left (3\right )+4 \ln \left (3\right )-1\right ) x \ln \left (\ln \left (x \right )\right )}{\ln \left (\ln \left (x \right )\right )}\) | \(29\) |
default | \(-x +\frac {x \ln \left (3\right )+4 \ln \left (3\right ) x \ln \left (\ln \left (x \right )\right )}{\ln \left (\ln \left (x \right )\right )}+2 x \ln \left (2\right ) \ln \left (3\right )\) | \(31\) |
parallelrisch | \(\frac {2 \ln \left (2\right ) \ln \left (3\right ) x \ln \left (\ln \left (x \right )\right )+4 \ln \left (3\right ) x \ln \left (\ln \left (x \right )\right )+x \ln \left (3\right )-x \ln \left (\ln \left (x \right )\right )}{\ln \left (\ln \left (x \right )\right )}\) | \(36\) |
Input:
int(((2*ln(2)*ln(3)+4*ln(3)-1)*ln(x)*ln(ln(x))^2+ln(3)*ln(x)*ln(ln(x))-ln( 3))/ln(x)/ln(ln(x))^2,x,method=_RETURNVERBOSE)
Output:
2*x*ln(2)*ln(3)+4*x*ln(3)-x+x*ln(3)/ln(ln(x))
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {-\log (3)+\log (3) \log (x) \log (\log (x))+(-1+4 \log (3)+\log (3) \log (4)) \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=\frac {x \log \left (3\right ) + {\left (2 \, {\left (x \log \left (2\right ) + 2 \, x\right )} \log \left (3\right ) - x\right )} \log \left (\log \left (x\right )\right )}{\log \left (\log \left (x\right )\right )} \] Input:
integrate(((2*log(2)*log(3)+4*log(3)-1)*log(x)*log(log(x))^2+log(3)*log(x) *log(log(x))-log(3))/log(x)/log(log(x))^2,x, algorithm="fricas")
Output:
(x*log(3) + (2*(x*log(2) + 2*x)*log(3) - x)*log(log(x)))/log(log(x))
Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {-\log (3)+\log (3) \log (x) \log (\log (x))+(-1+4 \log (3)+\log (3) \log (4)) \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=x \left (-1 + 2 \log {\left (2 \right )} \log {\left (3 \right )} + 4 \log {\left (3 \right )}\right ) + \frac {x \log {\left (3 \right )}}{\log {\left (\log {\left (x \right )} \right )}} \] Input:
integrate(((2*ln(2)*ln(3)+4*ln(3)-1)*ln(x)*ln(ln(x))**2+ln(3)*ln(x)*ln(ln( x))-ln(3))/ln(x)/ln(ln(x))**2,x)
Output:
x*(-1 + 2*log(2)*log(3) + 4*log(3)) + x*log(3)/log(log(x))
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {-\log (3)+\log (3) \log (x) \log (\log (x))+(-1+4 \log (3)+\log (3) \log (4)) \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=2 \, x \log \left (3\right ) \log \left (2\right ) + 4 \, x \log \left (3\right ) - x + \frac {x \log \left (3\right )}{\log \left (\log \left (x\right )\right )} \] Input:
integrate(((2*log(2)*log(3)+4*log(3)-1)*log(x)*log(log(x))^2+log(3)*log(x) *log(log(x))-log(3))/log(x)/log(log(x))^2,x, algorithm="maxima")
Output:
2*x*log(3)*log(2) + 4*x*log(3) - x + x*log(3)/log(log(x))
Time = 0.13 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {-\log (3)+\log (3) \log (x) \log (\log (x))+(-1+4 \log (3)+\log (3) \log (4)) \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=2 \, x \log \left (3\right ) \log \left (2\right ) + 4 \, x \log \left (3\right ) - x + \frac {x \log \left (3\right )}{\log \left (\log \left (x\right )\right )} \] Input:
integrate(((2*log(2)*log(3)+4*log(3)-1)*log(x)*log(log(x))^2+log(3)*log(x) *log(log(x))-log(3))/log(x)/log(log(x))^2,x, algorithm="giac")
Output:
2*x*log(3)*log(2) + 4*x*log(3) - x + x*log(3)/log(log(x))
Time = 7.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {-\log (3)+\log (3) \log (x) \log (\log (x))+(-1+4 \log (3)+\log (3) \log (4)) \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=x\,\left (4\,\ln \left (3\right )+2\,\ln \left (2\right )\,\ln \left (3\right )-1\right )+\frac {x\,\ln \left (3\right )}{\ln \left (\ln \left (x\right )\right )} \] Input:
int((log(log(x))*log(3)*log(x) - log(3) + log(log(x))^2*log(x)*(4*log(3) + 2*log(2)*log(3) - 1))/(log(log(x))^2*log(x)),x)
Output:
x*(4*log(3) + 2*log(2)*log(3) - 1) + (x*log(3))/log(log(x))
Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {-\log (3)+\log (3) \log (x) \log (\log (x))+(-1+4 \log (3)+\log (3) \log (4)) \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=\frac {x \left (2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (3\right ) \mathrm {log}\left (2\right )+4 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (3\right )-\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\mathrm {log}\left (3\right )\right )}{\mathrm {log}\left (\mathrm {log}\left (x \right )\right )} \] Input:
int(((2*log(2)*log(3)+4*log(3)-1)*log(x)*log(log(x))^2+log(3)*log(x)*log(l og(x))-log(3))/log(x)/log(log(x))^2,x)
Output:
(x*(2*log(log(x))*log(3)*log(2) + 4*log(log(x))*log(3) - log(log(x)) + log (3)))/log(log(x))