Optimal. Leaf size=38 \[ 24 \log (x)-24 \log (x) \log (\log (x))+12 \log (x) \log ^2(\log (x))-4 \log (x) \log ^3(\log (x))+\log (x) \log ^4(\log (x)) \]
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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2333, 2332}
\begin {gather*} \log (x) \log ^4(\log (x))-4 \log (x) \log ^3(\log (x))+12 \log (x) \log ^2(\log (x))-24 \log (x) \log (\log (x))+24 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2333
Rubi steps
\begin {align*} \int \frac {\log ^4(\log (x))}{x} \, dx &=\text {Subst}\left (\int \log ^4(x) \, dx,x,\log (x)\right )\\ &=\log (x) \log ^4(\log (x))-4 \text {Subst}\left (\int \log ^3(x) \, dx,x,\log (x)\right )\\ &=-4 \log (x) \log ^3(\log (x))+\log (x) \log ^4(\log (x))+12 \text {Subst}\left (\int \log ^2(x) \, dx,x,\log (x)\right )\\ &=12 \log (x) \log ^2(\log (x))-4 \log (x) \log ^3(\log (x))+\log (x) \log ^4(\log (x))-24 \text {Subst}(\int \log (x) \, dx,x,\log (x))\\ &=24 \log (x)-24 \log (x) \log (\log (x))+12 \log (x) \log ^2(\log (x))-4 \log (x) \log ^3(\log (x))+\log (x) \log ^4(\log (x))\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 38, normalized size = 1.00 \begin {gather*} 24 \log (x)-24 \log (x) \log (\log (x))+12 \log (x) \log ^2(\log (x))-4 \log (x) \log ^3(\log (x))+\log (x) \log ^4(\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 39, normalized size = 1.03
method | result | size |
derivativedivides | \(24 \ln \left (x \right )-24 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+12 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}-4 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{4}\) | \(39\) |
default | \(24 \ln \left (x \right )-24 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+12 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}-4 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{4}\) | \(39\) |
norman | \(24 \ln \left (x \right )-24 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+12 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}-4 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{4}\) | \(39\) |
risch | \(24 \ln \left (x \right )-24 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+12 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}-4 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{4}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.39, size = 29, normalized size = 0.76 \begin {gather*} {\left (\log \left (\log \left (x\right )\right )^{4} - 4 \, \log \left (\log \left (x\right )\right )^{3} + 12 \, \log \left (\log \left (x\right )\right )^{2} - 24 \, \log \left (\log \left (x\right )\right ) + 24\right )} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 38, normalized size = 1.00 \begin {gather*} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{4} - 4 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} + 12 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 24 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 24 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 48, normalized size = 1.26 \begin {gather*} \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{4} - 4 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{3} + 12 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{2} - 24 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )} + 24 \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.01, size = 38, normalized size = 1.00 \begin {gather*} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{4} - 4 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} + 12 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 24 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 24 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 38, normalized size = 1.00 \begin {gather*} \ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^4-4\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^3+12\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^2-24\,\ln \left (x\right )\,\ln \left (\ln \left (x\right )\right )+24\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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