Optimal. Leaf size=27 \[ x \left (x+\log \left (3+(2-2 x) x \left (x+\log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )\right )\right ) \]
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Rubi [F] time = 6.31, 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 {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x-4 x^2-\left (-2 x+2 x^2\right ) \log (x)-\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )-\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (3+2 x^2-2 x^3+2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx\\ &=\int \left (\frac {2 x \left (-3-2 x+x^2+2 x^3\right )}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}+\frac {2 (-1+x) x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}-\frac {4 x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}+\frac {4 x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}+\frac {2 x \left (-1+2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}+\log \left (3+2 x^2-2 x^3-2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )\right ) \, dx\\ &=2 \int \frac {x \left (-3-2 x+x^2+2 x^3\right )}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+2 \int \frac {(-1+x) x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+2 \int \frac {x \left (-1+2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-4 \int \frac {x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+4 \int \frac {x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+\int \log \left (3+2 x^2-2 x^3-2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \, dx\\ &=2 \int \left (-\frac {3 x}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}-\frac {2 x^2}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}+\frac {x^3}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}+\frac {2 x^4}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}\right ) \, dx+2 \int \left (\frac {-1+2 x^2}{2 (-1+x)}+\frac {-3+4 x^2+2 x^3+4 x^4-4 x^5}{2 (-1+x) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}\right ) \, dx+2 \int \left (-\frac {x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}+\frac {x^2}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}\right ) \, dx-4 \int \frac {x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+4 \int \frac {x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+\int \log \left (3+2 x^2-2 x^3-2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \, dx\\ &=2 \int \frac {x^3}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-2 \int \frac {x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+2 \int \frac {x^2}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx-4 \int \frac {x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+4 \int \frac {x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx-4 \int \frac {x^2}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+4 \int \frac {x^4}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-6 \int \frac {x}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+\int \frac {-1+2 x^2}{-1+x} \, dx+\int \frac {-3+4 x^2+2 x^3+4 x^4-4 x^5}{(-1+x) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+\int \log \left (3+2 x^2-2 x^3-2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.19, size = 36, normalized size = 1.33 \begin {gather*} x^2+x \log \left (3+2 x^2-2 x^3-2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 37, normalized size = 1.37 \begin {gather*} x^{2} + x \log \left (-2 \, x^{3} + 2 \, x^{2} - 2 \, {\left (x^{2} - x\right )} \log \left (\log \left (\frac {1}{3} \, x \log \relax (x)^{2}\right )\right ) + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 16.98, size = 53, normalized size = 1.96 \begin {gather*} x^{2} + x \log \left (-2 \, x^{3} - 2 \, x^{2} \log \left (-\log \relax (3) + \log \left (\log \relax (x)^{2}\right ) + \log \relax (x)\right ) + 2 \, x^{2} + 2 \, x \log \left (-\log \relax (3) + \log \left (\log \relax (x)^{2}\right ) + \log \relax (x)\right ) + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 124, normalized size = 4.59
method | result | size |
risch | \(x^{2}+x \ln \left (\left (-2 x^{2}+2 x \right ) \ln \left (-\ln \relax (3)+\ln \relax (x )+2 \ln \left (\ln \relax (x )\right )-\frac {i \pi \,\mathrm {csgn}\left (i \ln \relax (x )^{2}\right ) \left (-\mathrm {csgn}\left (i \ln \relax (x )^{2}\right )+\mathrm {csgn}\left (i \ln \relax (x )\right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )^{2}\right ) \left (-\mathrm {csgn}\left (i x \ln \relax (x )^{2}\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \ln \relax (x )^{2}\right )+\mathrm {csgn}\left (i \ln \relax (x )^{2}\right )\right )}{2}\right )-2 x^{3}+2 x^{2}+3\right )\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 50, normalized size = 1.85 \begin {gather*} x^{2} + x \log \left (-2 \, x^{3} - 2 \, x^{2} {\left (\log \left (-\log \relax (3) + \log \relax (x) + 2 \, \log \left (\log \relax (x)\right )\right ) - 1\right )} + 2 \, x \log \left (-\log \relax (3) + \log \relax (x) + 2 \, \log \left (\log \relax (x)\right )\right ) + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.25, size = 36, normalized size = 1.33 \begin {gather*} x\,\left (x+\ln \left (\ln \left (\ln \left (\frac {x\,{\ln \relax (x)}^2}{3}\right )\right )\,\left (2\,x-2\,x^2\right )+2\,x^2-2\,x^3+3\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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