Optimal. Leaf size=35 \[ -\frac {\log (x-1)}{2 x^2}+\frac {1}{2 x}+\frac {1}{2} \log (1-x)-\frac {\log (x)}{2} \]
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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 = {2395, 44} \[ -\frac {\log (x-1)}{2 x^2}+\frac {1}{2 x}+\frac {1}{2} \log (1-x)-\frac {\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2395
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 \[ \frac {1}{2} \left (-\frac {\log (x-1)}{x^2}+\frac {1}{x}+\log (1-x)-\log (x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log (-1+x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.78, size = 26, normalized size = 0.74 \[ -\frac {x^{2} \log \relax (x) - {\left (x^{2} - 1\right )} \log \left (x - 1\right ) - x}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.79, size = 27, normalized size = 0.77 \[ \frac {1}{2 \, x} - \frac {\log \left (x - 1\right )}{2 \, x^{2}} + \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 26, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {1}{2 x}-\frac {\ln \relax (x )}{2}+\frac {\ln \left (-1+x \right ) \left (-1+x \right ) \left (1+x \right )}{2 x^{2}}\) | \(26\) |
default | \(\frac {1}{2 x}-\frac {\ln \relax (x )}{2}+\frac {\ln \left (-1+x \right ) \left (-1+x \right ) \left (1+x \right )}{2 x^{2}}\) | \(26\) |
norman | \(\frac {\frac {x}{2}+\frac {x^{2} \ln \left (-1+x \right )}{2}-\frac {\ln \left (-1+x \right )}{2}}{x^{2}}-\frac {\ln \relax (x )}{2}\) | \(29\) |
risch | \(-\frac {\ln \left (-1+x \right )}{2 x^{2}}+\frac {\ln \left (-1+x \right ) x -x \ln \relax (x )+1}{2 x}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 25, normalized size = 0.71 \[ \frac {1}{2 \, x} - \frac {\log \left (x - 1\right )}{2 \, x^{2}} + \frac {1}{2} \, \log \left (x - 1\right ) - \frac {1}{2} \, \log \relax (x) \]
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 \[ \frac {x-\ln \left (x-1\right )+x^2\,\ln \left (1-\frac {1}{x}\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 26, normalized size = 0.74 \[ - \frac {\log {\relax (x )}}{2} + \frac {\log {\left (x - 1 \right )}}{2} + \frac {1}{2 x} - \frac {\log {\left (x - 1 \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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