Optimal. Leaf size=29 \[ -6 \log (x)+6 \log (x) \log (\log (x))-3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x)) \]
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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 ^3(\log (x))-3 \log (x) \log ^2(\log (x))+6 \log (x) \log (\log (x))-6 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2333
Rubi steps
\begin {align*} \int \frac {\log ^3(\log (x))}{x} \, dx &=\text {Subst}\left (\int \log ^3(x) \, dx,x,\log (x)\right )\\ &=\log (x) \log ^3(\log (x))-3 \text {Subst}\left (\int \log ^2(x) \, dx,x,\log (x)\right )\\ &=-3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x))+6 \text {Subst}(\int \log (x) \, dx,x,\log (x))\\ &=-6 \log (x)+6 \log (x) \log (\log (x))-3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x))\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} -6 \log (x)+6 \log (x) \log (\log (x))-3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 30, normalized size = 1.03
method | result | size |
derivativedivides | \(-6 \ln \left (x \right )+6 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}\) | \(30\) |
default | \(-6 \ln \left (x \right )+6 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}\) | \(30\) |
norman | \(-6 \ln \left (x \right )+6 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}\) | \(30\) |
risch | \(-6 \ln \left (x \right )+6 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.07, size = 22, normalized size = 0.76 \begin {gather*} {\left (\log \left (\log \left (x\right )\right )^{3} - 3 \, \log \left (\log \left (x\right )\right )^{2} + 6 \, \log \left (\log \left (x\right )\right ) - 6\right )} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.82, size = 29, normalized size = 1.00 \begin {gather*} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} - 3 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + 6 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 6 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 36, normalized size = 1.24 \begin {gather*} \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{3} - 3 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{2} + 6 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )} - 6 \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.91, size = 29, normalized size = 1.00 \begin {gather*} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} - 3 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + 6 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 6 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 29, normalized size = 1.00 \begin {gather*} \ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^3-3\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^2+6\,\ln \left (x\right )\,\ln \left (\ln \left (x\right )\right )-6\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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