Optimal. Leaf size=23 \[ e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}}+\log \left (\log ^2(x)\right ) \]
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Rubi [A]
time = 1.62, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 147, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6820, 6874,
2339, 29, 6838} \begin {gather*} 2 \log (\log (x))+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2339
Rule 6820
Rule 6838
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \log (x)+\log \left (\frac {2}{x}\right ) \left (-e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} x \log (x)+2 \left (x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )\right )^2\right )}{x \log \left (\frac {2}{x}\right ) \log (x) \left (x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )\right )^2} \, dx\\ &=\int \left (\frac {2}{x \log (x)}-\frac {e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (1+x \log \left (\frac {2}{x}\right )\right )}{x \log \left (\frac {2}{x}\right ) \left (x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {1}{x \log (x)} \, dx-\int \frac {e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (1+x \log \left (\frac {2}{x}\right )\right )}{x \log \left (\frac {2}{x}\right ) \left (x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )\right )^2} \, dx\\ &=e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}}+2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}}+2 \log (\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 23, normalized size = 1.00 \begin {gather*} e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}}+2 \log (\log (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 = 38.63, size = 150, normalized size = 6.52
method | result | size |
risch | \(2 \ln \left (\ln \left (x \right )\right )+{\mathrm e}^{\frac {2}{i \pi \mathrm {csgn}\left (\frac {1}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right )^{3}+i \pi \mathrm {csgn}\left (\frac {1}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right )^{2} \mathrm {csgn}\left (\frac {i}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right )-i \pi \mathrm {csgn}\left (\frac {1}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {1}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right ) \mathrm {csgn}\left (\frac {i}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right )+i \pi +2 \ln \left (6\right )-2 \ln \left (2 i \ln \left (2\right )-2 i \ln \left (x \right )\right )+2 x}}\) | \(150\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 31, normalized size = 1.35 \begin {gather*} e^{\left (\frac {1}{x + \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}\right )} + 2 \, \log \left (-\log \left (2\right ) + \log \left (\frac {2}{x}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.02, size = 20, normalized size = 0.87 \begin {gather*} e^{\frac {1}{x + \log {\left (\frac {3}{- \log {\left (x \right )} + \log {\left (2 \right )}} \right )}}} + 2 \log {\left (\log {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 23, normalized size = 1.00 \begin {gather*} e^{\left (\frac {1}{x + \log \left (3\right ) - \log \left (\log \left (2\right ) - \log \left (x\right )\right )}\right )} + 2 \, \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.55, size = 22, normalized size = 0.96 \begin {gather*} 2\,\ln \left (\ln \left (x\right )\right )+{\mathrm {e}}^{\frac {1}{x+\ln \left (\frac {3}{\ln \left (\frac {2}{x}\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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