Optimal. Leaf size=35 \[ \frac {1}{2 x}+\frac {1}{2} \log (1-x)-\frac {\log (-1+x)}{2 x^2}-\frac {\log (x)}{2} \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2442, 46}
\begin {gather*} -\frac {\log (x-1)}{2 x^2}+\frac {1}{2 x}+\frac {1}{2} \log (1-x)-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2442
Rubi steps
\begin {align*} \int \frac {\log (-1+x)}{x^3} \, dx &=-\frac {\log (-1+x)}{2 x^2}+\frac {1}{2} \int \frac {1}{(-1+x) x^2} \, dx\\ &=-\frac {\log (-1+x)}{2 x^2}+\frac {1}{2} \int \left (\frac {1}{-1+x}-\frac {1}{x^2}-\frac {1}{x}\right ) \, dx\\ &=\frac {1}{2 x}+\frac {1}{2} \log (1-x)-\frac {\log (-1+x)}{2 x^2}-\frac {\log (x)}{2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.77 \begin {gather*} \frac {1}{2} \left (\frac {1}{x}+\log (1-x)-\frac {\log (-1+x)}{x^2}-\log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 26, normalized size = 0.74
method | result | size |
derivativedivides | \(-\frac {\ln \left (x \right )}{2}+\frac {1}{2 x}+\frac {\ln \left (-1+x \right ) \left (-1+x \right ) \left (1+x \right )}{2 x^{2}}\) | \(26\) |
default | \(-\frac {\ln \left (x \right )}{2}+\frac {1}{2 x}+\frac {\ln \left (-1+x \right ) \left (-1+x \right ) \left (1+x \right )}{2 x^{2}}\) | \(26\) |
norman | \(\frac {\frac {x}{2}+\frac {x^{2} \ln \left (-1+x \right )}{2}-\frac {\ln \left (-1+x \right )}{2}}{x^{2}}-\frac {\ln \left (x \right )}{2}\) | \(29\) |
risch | \(-\frac {\ln \left (-1+x \right )}{2 x^{2}}+\frac {\ln \left (-1+x \right ) x -x \ln \left (x \right )+1}{2 x}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.91, size = 25, normalized size = 0.71 \begin {gather*} \frac {1}{2 \, x} - \frac {\log \left (x - 1\right )}{2 \, x^{2}} + \frac {1}{2} \, \log \left (x - 1\right ) - \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.72, size = 26, normalized size = 0.74 \begin {gather*} -\frac {x^{2} \log \left (x\right ) - {\left (x^{2} - 1\right )} \log \left (x - 1\right ) - x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 26, normalized size = 0.74 \begin {gather*} - \frac {\log {\left (x \right )}}{2} + \frac {\log {\left (x - 1 \right )}}{2} + \frac {1}{2 x} - \frac {\log {\left (x - 1 \right )}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.36, size = 27, normalized size = 0.77 \begin {gather*} \frac {1}{2 \, x} - \frac {\log \left (x - 1\right )}{2 \, x^{2}} + \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 25, normalized size = 0.71 \begin {gather*} \frac {x-\ln \left (x-1\right )+x^2\,\ln \left (1-\frac {1}{x}\right )}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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