Optimal. Leaf size=27 \[ \frac {x+\log \left (\log \left (\log \left (\frac {\frac {1}{x}+\log (\log (x))}{2-x}\right )\right )\right )}{\log (3 x)} \]
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Rubi [F]
time = 29.88, 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 (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+\log (3 x)+\frac {((-2+x) x-2 (-1+x) \log (x)) \log (3 x)}{(-2+x) x \log (x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}-\frac {x \log (3 x) \log (\log (x))}{(-2+x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}-\frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x}}{\log ^2(3 x)} \, dx\\ &=\int \left (\frac {-2 x \log (3 x)+x^2 \log (3 x)+2 \log (x) \log (3 x)-2 x \log (x) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+2 x \log (x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-x^2 \log (x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-2 x \log (x) \log (3 x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+x^2 \log (x) \log (3 x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+2 x^2 \log (x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-x^3 \log (x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-2 x^2 \log (x) \log (3 x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+x^3 \log (x) \log (3 x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}{(-2+x) x \log (x) \log ^2(3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}-\frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)}\right ) \, dx\\ &=\int \frac {-2 x \log (3 x)+x^2 \log (3 x)+2 \log (x) \log (3 x)-2 x \log (x) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+2 x \log (x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-x^2 \log (x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-2 x \log (x) \log (3 x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+x^2 \log (x) \log (3 x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+2 x^2 \log (x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-x^3 \log (x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )-2 x^2 \log (x) \log (3 x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+x^3 \log (x) \log (3 x) \log (\log (x)) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}{(-2+x) x \log (x) \log ^2(3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx\\ &=\int \frac {-1+\frac {\log (3 x) \left ((-2+x) x+\log (x) \left (2-2 x+(-2+x) x \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )+x^2 \log (\log (x)) \left (-1+(-2+x) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )\right )\right )}{(-2+x) x \log (x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}}{\log ^2(3 x)} \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx\\ &=\int \left (\frac {-1+\log (3 x)}{\log ^2(3 x)}+\frac {-2 x+x^2+2 \log (x)-2 x \log (x)-x^2 \log (x) \log (\log (x))}{(-2+x) x \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}\right ) \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx\\ &=\int \frac {-1+\log (3 x)}{\log ^2(3 x)} \, dx+\int \frac {-2 x+x^2+2 \log (x)-2 x \log (x)-x^2 \log (x) \log (\log (x))}{(-2+x) x \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx\\ &=\int \left (-\frac {1}{\log ^2(3 x)}+\frac {1}{\log (3 x)}\right ) \, dx+\int \left (\frac {-2 x+x^2+2 \log (x)-2 x \log (x)-x^2 \log (x) \log (\log (x))}{2 (-2+x) \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}+\frac {2 x-x^2-2 \log (x)+2 x \log (x)+x^2 \log (x) \log (\log (x))}{2 x \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )}\right ) \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx\\ &=\frac {1}{2} \int \frac {-2 x+x^2+2 \log (x)-2 x \log (x)-x^2 \log (x) \log (\log (x))}{(-2+x) \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx+\frac {1}{2} \int \frac {2 x-x^2-2 \log (x)+2 x \log (x)+x^2 \log (x) \log (\log (x))}{x \log (x) \log (3 x) (1+x \log (\log (x))) \log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right ) \log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )} \, dx-\int \frac {1}{\log ^2(3 x)} \, dx+\int \frac {1}{\log (3 x)} \, dx-\int \frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{x \log ^2(3 x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.32, size = 36, normalized size = 1.33 \begin {gather*} \frac {x}{\log (3 x)}+\frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{\log (3 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.08, size = 268, normalized size = 9.93
method | result | size |
risch | \(\frac {2 \ln \left (\ln \left (i \pi -\ln \left (x \right )-\ln \left (x -2\right )+\ln \left (x \ln \left (\ln \left (x \right )\right )+1\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x -2}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x -2}\right )+\mathrm {csgn}\left (\frac {i}{x -2}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x -2}\right )+\mathrm {csgn}\left (i \left (x \ln \left (\ln \left (x \right )\right )+1\right )\right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x \left (x -2\right )}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x \left (x -2\right )}\right )+\mathrm {csgn}\left (\frac {i}{x}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x \left (x -2\right )}\right )+\mathrm {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x -2}\right )\right )}{2}+i \pi \mathrm {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x \left (x -2\right )}\right )^{2} \left (\mathrm {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x \right )\right )+1\right )}{x \left (x -2\right )}\right )-1\right )\right )\right )}{2 \ln \left (3\right )+2 \ln \left (x \right )}+\frac {2 x}{2 \ln \left (3\right )+2 \ln \left (x \right )}\) | \(268\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.64, size = 33, normalized size = 1.22 \begin {gather*} \frac {x + \log \left (\log \left (\log \left (x \log \left (\log \left (x\right )\right ) + 1\right ) - \log \left (x\right ) - \log \left (-x + 2\right )\right )\right )}{\log \left (3\right ) + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 31, normalized size = 1.15 \begin {gather*} \frac {x + \log \left (\log \left (\log \left (-\frac {x \log \left (\log \left (x\right )\right ) + 1}{x^{2} - 2 \, x}\right )\right )\right )}{\log \left (3\right ) + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.85, size = 40, normalized size = 1.48 \begin {gather*} \frac {x}{\log \left (3\right ) + \log \left (x\right )} + \frac {\log \left (\log \left (\log \left (-x \log \left (\log \left (x\right )\right ) - 1\right ) - \log \left (x - 2\right ) - \log \left (x\right )\right )\right )}{\log \left (3\right ) + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.19, size = 31, normalized size = 1.15 \begin {gather*} \frac {x+\ln \left (\ln \left (\ln \left (\frac {x\,\ln \left (\ln \left (x\right )\right )+1}{2\,x-x^2}\right )\right )\right )}{\ln \left (3\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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