Optimal. Leaf size=12 \[ -\frac {1}{2} \log ^2(1-t) \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2437, 2338}
\begin {gather*} -\frac {1}{2} \log ^2(1-t) \end {gather*}
Antiderivative was successfully verified.
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Rule 2338
Rule 2437
Rubi steps
\begin {align*} \int \frac {\log (1-t)}{1-t} \, dt &=-\text {Subst}\left (\int \frac {\log (t)}{t} \, dt,t,1-t\right )\\ &=-\frac {1}{2} \log ^2(1-t)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \log ^2(1-t) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 11, normalized size = 0.92
method | result | size |
derivativedivides | \(-\frac {\ln \left (1-t \right )^{2}}{2}\) | \(11\) |
default | \(-\frac {\ln \left (1-t \right )^{2}}{2}\) | \(11\) |
norman | \(-\frac {\ln \left (1-t \right )^{2}}{2}\) | \(11\) |
risch | \(-\frac {\ln \left (1-t \right )^{2}}{2}\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.53, size = 10, normalized size = 0.83 \begin {gather*} -\frac {1}{2} \, \log \left (-t + 1\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.08, size = 10, normalized size = 0.83 \begin {gather*} -\frac {1}{2} \, \log \left (-t + 1\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 0.67 \begin {gather*} - \frac {\log {\left (1 - t \right )}^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 10, normalized size = 0.83 \begin {gather*} -\frac {1}{2} \, \log \left (-t + 1\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 10, normalized size = 0.83 \begin {gather*} -\frac {{\ln \left (1-t\right )}^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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