Optimal. Leaf size=9 \[ -\log (1-\log (x)) \]
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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2339, 29}
\begin {gather*} -\log (1-\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2339
Rubi steps
\begin {align*} \int \frac {1}{x (1-\log (x))} \, dx &=-\text {Subst}\left (\int \frac {1}{x} \, dx,x,1-\log (x)\right )\\ &=-\log (1-\log (x))\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 7, normalized size = 0.78 \begin {gather*} -\log (-1+\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.72, size = 7, normalized size = 0.78 \begin {gather*} -\text {Log}\left [-1+\text {Log}\left [x\right ]\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 10, normalized size = 1.11
method | result | size |
norman | \(-\ln \left (-1+\ln \left (x \right )\right )\) | \(8\) |
risch | \(-\ln \left (-1+\ln \left (x \right )\right )\) | \(8\) |
derivativedivides | \(-\ln \left (1-\ln \left (x \right )\right )\) | \(10\) |
default | \(-\ln \left (1-\ln \left (x \right )\right )\) | \(10\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 7, normalized size = 0.78 \begin {gather*} -\log \left (\log \left (x\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 7, normalized size = 0.78 \begin {gather*} -\log \left (\log \left (x\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 7, normalized size = 0.78 \begin {gather*} - \log {\left (\log {\left (x \right )} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs.
\(2 (9) = 18\).
time = 0.00, size = 24, normalized size = 2.67 \begin {gather*} -\frac {\ln \left (\left (\ln \left |x\right |-1\right )^{2}+\left (\frac {1}{2} \pi \left (1-\mathrm {sign}\left (x\right )\right )\right )^{2}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 7, normalized size = 0.78 \begin {gather*} -\ln \left (\ln \left (x\right )-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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