Optimal. Leaf size=20 \[ \log (x) \log ^2(\log (x))-2 \log (x) \log (\log (x))+2 \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2296, 2295} \[ \log (x) \log ^2(\log (x))-2 \log (x) \log (\log (x))+2 \log (x) \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2296
Rubi steps
\begin {align*} \int \frac {\log ^2(\log (x))}{x} \, dx &=\operatorname {Subst}\left (\int \log ^2(x) \, dx,x,\log (x)\right )\\ &=\log (x) \log ^2(\log (x))-2 \operatorname {Subst}(\int \log (x) \, dx,x,\log (x))\\ &=2 \log (x)-2 \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \[ \log (x) \log ^2(\log (x))-2 \log (x) \log (\log (x))+2 \log (x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log ^2(\log (x))}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.65, size = 20, normalized size = 1.00 \[ \log \relax (x) \log \left (\log \relax (x)\right )^{2} - 2 \, \log \relax (x) \log \left (\log \relax (x)\right ) + 2 \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 20, normalized size = 1.00 \[ \log \relax (x) \log \left (\log \relax (x)\right )^{2} - 2 \, \log \relax (x) \log \left (\log \relax (x)\right ) + 2 \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 21, normalized size = 1.05
method | result | size |
derivativedivides | \(2 \ln \relax (x )-2 \ln \relax (x ) \ln \left (\ln \relax (x )\right )+\ln \relax (x ) \ln \left (\ln \relax (x )\right )^{2}\) | \(21\) |
default | \(2 \ln \relax (x )-2 \ln \relax (x ) \ln \left (\ln \relax (x )\right )+\ln \relax (x ) \ln \left (\ln \relax (x )\right )^{2}\) | \(21\) |
norman | \(2 \ln \relax (x )-2 \ln \relax (x ) \ln \left (\ln \relax (x )\right )+\ln \relax (x ) \ln \left (\ln \relax (x )\right )^{2}\) | \(21\) |
risch | \(2 \ln \relax (x )-2 \ln \relax (x ) \ln \left (\ln \relax (x )\right )+\ln \relax (x ) \ln \left (\ln \relax (x )\right )^{2}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 15, normalized size = 0.75 \[ {\left (\log \left (\log \relax (x)\right )^{2} - 2 \, \log \left (\log \relax (x)\right ) + 2\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 15, normalized size = 0.75 \[ \ln \relax (x)\,\left ({\ln \left (\ln \relax (x)\right )}^2-2\,\ln \left (\ln \relax (x)\right )+2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 24, normalized size = 1.20 \[ \log {\relax (x )} \log {\left (\log {\relax (x )} \right )}^{2} - 2 \log {\relax (x )} \log {\left (\log {\relax (x )} \right )} + 2 \log {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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