Optimal. Leaf size=26 \[ 2 x-\log \left (\frac {x \log \left (\frac {3+x}{3}\right )}{\log ^2(\log (2-x))}\right ) \]
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Rubi [F] time = 2.56, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\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))}{x \left (-6+x+x^2\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx\\ &=\int \left (\frac {-x+\log (27)+5 x \log \left (\frac {3+x}{3}\right )+2 x^2 \log \left (\frac {3+x}{3}\right )-3 \log (3+x)}{x (3+x) \log \left (1+\frac {x}{3}\right )}+\frac {2}{(-2+x) \log (2-x) \log (\log (2-x))}\right ) \, dx\\ &=2 \int \frac {1}{(-2+x) \log (2-x) \log (\log (2-x))} \, dx+\int \frac {-x+\log (27)+5 x \log \left (\frac {3+x}{3}\right )+2 x^2 \log \left (\frac {3+x}{3}\right )-3 \log (3+x)}{x (3+x) \log \left (1+\frac {x}{3}\right )} \, dx\\ &=2 \log (\log (\log (2-x)))+\int \left (\frac {-x+\log (27)+5 x \log \left (\frac {3+x}{3}\right )+2 x^2 \log \left (\frac {3+x}{3}\right )}{x (3+x) \log \left (1+\frac {x}{3}\right )}+\frac {3 \log (3+x)}{(-3-x) x \log \left (1+\frac {x}{3}\right )}\right ) \, dx\\ &=2 \log (\log (\log (2-x)))+3 \int \frac {\log (3+x)}{(-3-x) x \log \left (1+\frac {x}{3}\right )} \, dx+\int \frac {-x+\log (27)+5 x \log \left (\frac {3+x}{3}\right )+2 x^2 \log \left (\frac {3+x}{3}\right )}{x (3+x) \log \left (1+\frac {x}{3}\right )} \, dx\\ &=2 \log (\log (\log (2-x)))+3 \int \left (-\frac {\log (3+x)}{3 x \log \left (1+\frac {x}{3}\right )}+\frac {\log (3+x)}{3 (3+x) \log \left (1+\frac {x}{3}\right )}\right ) \, dx+\int \frac {-x+\log (27)+x (5+2 x) \log \left (\frac {3+x}{3}\right )}{x (3+x) \log \left (1+\frac {x}{3}\right )} \, dx\\ &=2 \log (\log (\log (2-x)))+\int \left (\frac {5+2 x}{3+x}+\frac {-x+\log (27)}{x (3+x) \log \left (1+\frac {x}{3}\right )}\right ) \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx+\int \frac {\log (3+x)}{(3+x) \log \left (1+\frac {x}{3}\right )} \, dx\\ &=2 \log (\log (\log (2-x)))+3 \operatorname {Subst}\left (\int \frac {\log (3 x)}{3 x \log (x)} \, dx,x,1+\frac {x}{3}\right )+\int \frac {5+2 x}{3+x} \, dx+\int \frac {-x+\log (27)}{x (3+x) \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx\\ &=2 \log (\log (\log (2-x)))+\int \left (2+\frac {1}{-3-x}\right ) \, dx+\int \left (\frac {-3-\log (27)}{3 (3+x) \log \left (1+\frac {x}{3}\right )}+\frac {\log (27)}{3 x \log \left (1+\frac {x}{3}\right )}\right ) \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx+\operatorname {Subst}\left (\int \frac {\log (3 x)}{x \log (x)} \, dx,x,1+\frac {x}{3}\right )\\ &=2 x-\log (3+x)+\log (3+x) \log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x)))+\frac {1}{3} (-3-\log (27)) \int \frac {1}{(3+x) \log \left (1+\frac {x}{3}\right )} \, dx+\frac {1}{3} \log (27) \int \frac {1}{x \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx-\operatorname {Subst}\left (\int \frac {\log (\log (x))}{x} \, dx,x,1+\frac {x}{3}\right )\\ &=2 x-\log \left (\frac {3+x}{3}\right ) \log \left (\log \left (\frac {3+x}{3}\right )\right )+\log (3+x) \log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x)))+(-3-\log (27)) \operatorname {Subst}\left (\int \frac {1}{3 x \log (x)} \, dx,x,1+\frac {x}{3}\right )+\frac {1}{3} \log (27) \int \frac {1}{x \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx\\ &=2 x-\log \left (\frac {3+x}{3}\right ) \log \left (\log \left (\frac {3+x}{3}\right )\right )+\log (3+x) \log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x)))+\frac {1}{3} (-3-\log (27)) \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,1+\frac {x}{3}\right )+\frac {1}{3} \log (27) \int \frac {1}{x \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx\\ &=2 x-\log \left (\frac {3+x}{3}\right ) \log \left (\log \left (\frac {3+x}{3}\right )\right )+\log (3+x) \log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x)))+\frac {1}{3} (-3-\log (27)) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {3+x}{3}\right )\right )+\frac {1}{3} \log (27) \int \frac {1}{x \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx\\ &=2 x-\frac {1}{3} (3+\log (27)) \log \left (\log \left (\frac {3+x}{3}\right )\right )-\log \left (\frac {3+x}{3}\right ) \log \left (\log \left (\frac {3+x}{3}\right )\right )+\log (3+x) \log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x)))+\frac {1}{3} \log (27) \int \frac {1}{x \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 29, normalized size = 1.12 \begin {gather*} 2 x-\log (x)-\log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x))) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 27, normalized size = 1.04 \begin {gather*} 2 \, x - \log \relax (x) - \log \left (\log \left (\frac {1}{3} \, x + 1\right )\right ) + 2 \, \log \left (\log \left (\log \left (-x + 2\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 30, normalized size = 1.15 \begin {gather*} 2 \, x - \log \relax (x) - \log \left (\log \relax (3) - \log \left (x + 3\right )\right ) + 2 \, \log \left (\log \left (\log \left (-x + 2\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 28, normalized size = 1.08
method | result | size |
risch | \(2 x -\ln \relax (x )-\ln \left (\ln \left (1+\frac {x}{3}\right )\right )+2 \ln \left (\ln \left (\ln \left (2-x \right )\right )\right )\) | \(28\) |
default | \(2 x -\ln \relax (x )-\ln \left (\ln \relax (3)-\ln \left (3+x \right )\right )+2 \ln \left (\ln \left (\ln \left (2-x \right )\right )\right )\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 30, normalized size = 1.15 \begin {gather*} 2 \, x - \log \relax (x) - \log \left (-\log \relax (3) + \log \left (x + 3\right )\right ) + 2 \, \log \left (\log \left (\log \left (-x + 2\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.67, size = 27, normalized size = 1.04 \begin {gather*} 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 \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 24, normalized size = 0.92 \begin {gather*} 2 x - \log {\relax (x )} - \log {\left (\log {\left (\frac {x}{3} + 1 \right )} \right )} + 2 \log {\left (\log {\left (\log {\left (2 - x \right )} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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