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 = 0.0177781, antiderivative size = 20, normalized size of antiderivative = 1., 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 2296
Rule 2295
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*}
Mathematica [A] time = 0.0059768, size = 20, normalized size = 1. \[ \log (x) \log ^2(\log (x))-2 \log (x) \log (\log (x))+2 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 1.1 \begin{align*} 2\,\ln \left ( x \right ) -2\,\ln \left ( x \right ) \ln \left ( \ln \left ( x \right ) \right ) +\ln \left ( x \right ) \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.958777, size = 20, normalized size = 1. \begin{align*}{\left (\log \left (\log \left (x\right )\right )^{2} - 2 \, \log \left (\log \left (x\right )\right ) + 2\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41174, size = 76, normalized size = 3.8 \begin{align*} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 2 \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.316487, size = 24, normalized size = 1.2 \begin{align*} \log{\left (x \right )} \log{\left (\log{\left (x \right )} \right )}^{2} - 2 \log{\left (x \right )} \log{\left (\log{\left (x \right )} \right )} + 2 \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06349, size = 27, normalized size = 1.35 \begin{align*} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 2 \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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