Integrand size = 112, antiderivative size = 26 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 x-\log \left (\frac {x \log \left (\frac {3+x}{3}\right )}{\log ^2(\log (2-x))}\right ) \] Output:
2*x-ln(ln(1+1/3*x)/ln(ln(2-x))^2*x)
Time = 0.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 x-\log (x)-\log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x))) \] Input:
Integrate[((6*x + 2*x^2)*Log[(3 + x)/3] + ((2*x - x^2)*Log[2 - x] + (6 - 1 3*x + x^2 + 2*x^3)*Log[2 - x]*Log[(3 + x)/3])*Log[Log[2 - x]])/((-6*x + x^ 2 + x^3)*Log[2 - x]*Log[(3 + x)/3]*Log[Log[2 - x]]),x]
Output:
2*x - Log[x] - Log[Log[(3 + x)/3]] + 2*Log[Log[Log[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.
\(\displaystyle \int \frac {\left (2 x^2+6 x\right ) \log \left (\frac {x+3}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (2 x^3+x^2-13 x+6\right ) \log \left (\frac {x+3}{3}\right ) \log (2-x)\right ) \log (\log (2-x))}{\left (x^3+x^2-6 x\right ) \log (2-x) \log \left (\frac {x+3}{3}\right ) \log (\log (2-x))} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (2 x^2+6 x\right ) \log \left (\frac {x+3}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (2 x^3+x^2-13 x+6\right ) \log \left (\frac {x+3}{3}\right ) \log (2-x)\right ) \log (\log (2-x))}{x \left (x^2+x-6\right ) \log (2-x) \log \left (\frac {x+3}{3}\right ) \log (\log (2-x))}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {2 x^2 \log \left (\frac {x+3}{3}\right )-x+5 x \log \left (\frac {x+3}{3}\right )-3 \log (x+3)+\log (27)}{x (x+3) \log \left (\frac {x}{3}+1\right )}+\frac {2}{(x-2) \log (2-x) \log (\log (2-x))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \log (27) \int \frac {1}{x \log \left (\frac {x}{3}+1\right )}dx-\int \frac {\log (x+3)}{x \log \left (\frac {x}{3}+1\right )}dx+2 x-\log \left (\frac {x+3}{3}\right ) \log \left (\log \left (\frac {x+3}{3}\right )\right )+\log (x+3) \log \left (\log \left (\frac {x+3}{3}\right )\right )-\frac {1}{3} (3+\log (27)) \log \left (\log \left (\frac {x+3}{3}\right )\right )+2 \log (\log (\log (2-x)))\) |
Input:
Int[((6*x + 2*x^2)*Log[(3 + x)/3] + ((2*x - x^2)*Log[2 - x] + (6 - 13*x + x^2 + 2*x^3)*Log[2 - x]*Log[(3 + x)/3])*Log[Log[2 - x]])/((-6*x + x^2 + x^ 3)*Log[2 - x]*Log[(3 + x)/3]*Log[Log[2 - x]]),x]
Output:
$Aborted
Time = 3.60 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08
method | result | size |
risch | \(2 x -\ln \left (x \right )-\ln \left (\ln \left (1+\frac {x}{3}\right )\right )+2 \ln \left (\ln \left (\ln \left (2-x \right )\right )\right )\) | \(28\) |
parts | \(2 x -\ln \left (x \right )-\ln \left (\ln \left (1+\frac {x}{3}\right )\right )+2 \ln \left (\ln \left (\ln \left (2-x \right )\right )\right )\) | \(28\) |
parallelrisch | \(11-\ln \left (x \right )-\ln \left (\ln \left (1+\frac {x}{3}\right )\right )+2 \ln \left (\ln \left (\ln \left (2-x \right )\right )\right )+2 x\) | \(29\) |
default | \(6+2 x -\ln \left (x \right )-\ln \left (\ln \left (3\right )-\ln \left (3+x \right )\right )+2 \ln \left (\ln \left (\ln \left (2-x \right )\right )\right )\) | \(32\) |
Input:
int((((2*x^3+x^2-13*x+6)*ln(2-x)*ln(1+1/3*x)+(-x^2+2*x)*ln(2-x))*ln(ln(2-x ))+(2*x^2+6*x)*ln(1+1/3*x))/(x^3+x^2-6*x)/ln(2-x)/ln(1+1/3*x)/ln(ln(2-x)), x,method=_RETURNVERBOSE)
Output:
2*x-ln(x)-ln(ln(1+1/3*x))+2*ln(ln(ln(2-x)))
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 \, x - \log \left (x\right ) - \log \left (\log \left (\frac {1}{3} \, x + 1\right )\right ) + 2 \, \log \left (\log \left (\log \left (-x + 2\right )\right )\right ) \] Input:
integrate((((2*x^3+x^2-13*x+6)*log(2-x)*log(1+1/3*x)+(-x^2+2*x)*log(2-x))* log(log(2-x))+(2*x^2+6*x)*log(1+1/3*x))/(x^3+x^2-6*x)/log(2-x)/log(1+1/3*x )/log(log(2-x)),x, algorithm="fricas")
Output:
2*x - log(x) - log(log(1/3*x + 1)) + 2*log(log(log(-x + 2)))
Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 x - \log {\left (x \right )} - \log {\left (\log {\left (\frac {x}{3} + 1 \right )} \right )} + 2 \log {\left (\log {\left (\log {\left (2 - x \right )} \right )} \right )} \] Input:
integrate((((2*x**3+x**2-13*x+6)*ln(2-x)*ln(1+1/3*x)+(-x**2+2*x)*ln(2-x))* ln(ln(2-x))+(2*x**2+6*x)*ln(1+1/3*x))/(x**3+x**2-6*x)/ln(2-x)/ln(1+1/3*x)/ ln(ln(2-x)),x)
Output:
2*x - log(x) - log(log(x/3 + 1)) + 2*log(log(log(2 - x)))
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 \, x - \log \left (x\right ) - \log \left (-\log \left (3\right ) + \log \left (x + 3\right )\right ) + 2 \, \log \left (\log \left (\log \left (-x + 2\right )\right )\right ) \] Input:
integrate((((2*x^3+x^2-13*x+6)*log(2-x)*log(1+1/3*x)+(-x^2+2*x)*log(2-x))* log(log(2-x))+(2*x^2+6*x)*log(1+1/3*x))/(x^3+x^2-6*x)/log(2-x)/log(1+1/3*x )/log(log(2-x)),x, algorithm="maxima")
Output:
2*x - log(x) - log(-log(3) + log(x + 3)) + 2*log(log(log(-x + 2)))
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 \, x - \log \left (-x\right ) - \log \left (-\log \left (3\right ) + \log \left (x + 3\right )\right ) + 2 \, \log \left (\log \left (\log \left (-x + 2\right )\right )\right ) - 4 \] Input:
integrate((((2*x^3+x^2-13*x+6)*log(2-x)*log(1+1/3*x)+(-x^2+2*x)*log(2-x))* log(log(2-x))+(2*x^2+6*x)*log(1+1/3*x))/(x^3+x^2-6*x)/log(2-x)/log(1+1/3*x )/log(log(2-x)),x, algorithm="giac")
Output:
2*x - log(-x) - log(-log(3) + log(x + 3)) + 2*log(log(log(-x + 2))) - 4
Time = 4.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2\,x-\ln \left (\ln \left (\frac {x}{3}+1\right )\right )+2\,\ln \left (\ln \left (\ln \left (2-x\right )\right )\right )-\ln \left (x\right ) \] Input:
int((log(log(2 - x))*(log(2 - x)*(2*x - x^2) + log(2 - x)*log(x/3 + 1)*(x^ 2 - 13*x + 2*x^3 + 6)) + log(x/3 + 1)*(6*x + 2*x^2))/(log(log(2 - x))*log( 2 - x)*log(x/3 + 1)*(x^2 - 6*x + x^3)),x)
Output:
2*x - log(log(x/3 + 1)) + 2*log(log(log(2 - x))) - log(x)
Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 \,\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (-x +2\right )\right )\right )-\mathrm {log}\left (\mathrm {log}\left (\frac {x}{3}+1\right )\right )-\mathrm {log}\left (x \right )+2 x \] Input:
int((((2*x^3+x^2-13*x+6)*log(2-x)*log(1+1/3*x)+(-x^2+2*x)*log(2-x))*log(lo g(2-x))+(2*x^2+6*x)*log(1+1/3*x))/(x^3+x^2-6*x)/log(2-x)/log(1+1/3*x)/log( log(2-x)),x)
Output:
2*log(log(log( - x + 2))) - log(log((x + 3)/3)) - log(x) + 2*x