Optimal. Leaf size=36 \[ \frac {1}{3} x^3 \log \left (\frac {x-1}{x}\right )-\frac {x^2}{6}-\frac {x}{3}-\frac {1}{3} \log (x-1) \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2461, 2455, 263, 43} \[ -\frac {x^2}{6}+\frac {1}{3} x^3 \log \left (1-\frac {1}{x}\right )-\frac {x}{3}-\frac {1}{3} \log (1-x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 2455
Rule 2461
Rubi steps
\begin {align*} \int x^2 \log \left (\frac {-1+x}{x}\right ) \, dx &=\int x^2 \log \left (1-\frac {1}{x}\right ) \, dx\\ &=\frac {1}{3} x^3 \log \left (1-\frac {1}{x}\right )-\frac {1}{3} \int \frac {x}{1-\frac {1}{x}} \, dx\\ &=\frac {1}{3} x^3 \log \left (1-\frac {1}{x}\right )-\frac {1}{3} \int \frac {x^2}{-1+x} \, dx\\ &=\frac {1}{3} x^3 \log \left (1-\frac {1}{x}\right )-\frac {1}{3} \int \left (1+\frac {1}{-1+x}+x\right ) \, dx\\ &=-\frac {x}{3}-\frac {x^2}{6}+\frac {1}{3} x^3 \log \left (1-\frac {1}{x}\right )-\frac {1}{3} \log (1-x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.06 \[ \frac {1}{3} x^3 \log \left (\frac {x-1}{x}\right )-\frac {x^2}{6}-\frac {x}{3}-\frac {1}{3} \log (1-x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^2 \log \left (\frac {-1+x}{x}\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.93, size = 28, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} \log \left (\frac {x - 1}{x}\right ) - \frac {1}{6} \, x^{2} - \frac {1}{3} \, x - \frac {1}{3} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.05, size = 70, normalized size = 1.94 \[ \frac {\frac {2 \, {\left (x - 1\right )}}{x} - 3}{6 \, {\left (\frac {x - 1}{x} - 1\right )}^{2}} - \frac {\log \left (\frac {x - 1}{x}\right )}{3 \, {\left (\frac {x - 1}{x} - 1\right )}^{3}} - \frac {1}{3} \, \log \left (\frac {{\left | x - 1 \right |}}{{\left | x \right |}}\right ) + \frac {1}{3} \, \log \left ({\left | \frac {x - 1}{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 0.81
method | result | size |
risch | \(-\frac {x}{3}-\frac {x^{2}}{6}-\frac {\ln \left (-1+x \right )}{3}+\frac {x^{3} \ln \left (\frac {-1+x}{x}\right )}{3}\) | \(29\) |
derivativedivides | \(\frac {\ln \left (-\frac {1}{x}\right )}{3}-\frac {x}{3}-\frac {x^{2}}{6}+\frac {\ln \left (1-\frac {1}{x}\right ) \left (1-\frac {1}{x}\right ) \left (\left (1-\frac {1}{x}\right )^{2}+\frac {3}{x}\right ) x^{3}}{3}\) | \(53\) |
default | \(\frac {\ln \left (-\frac {1}{x}\right )}{3}-\frac {x}{3}-\frac {x^{2}}{6}+\frac {\ln \left (1-\frac {1}{x}\right ) \left (1-\frac {1}{x}\right ) \left (\left (1-\frac {1}{x}\right )^{2}+\frac {3}{x}\right ) x^{3}}{3}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 28, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} \log \left (\frac {x - 1}{x}\right ) - \frac {1}{6} \, x^{2} - \frac {1}{3} \, x - \frac {1}{3} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 40, normalized size = 1.11 \[ \frac {x^3\,\ln \left (\frac {x-1}{x}\right )}{3}-\frac {\ln \left (x\,\left (x-1\right )\right )}{6}-\frac {\ln \left (\frac {x-1}{x}\right )}{6}-\frac {x}{3}-\frac {x^2}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 26, normalized size = 0.72 \[ \frac {x^{3} \log {\left (\frac {x - 1}{x} \right )}}{3} - \frac {x^{2}}{6} - \frac {x}{3} - \frac {\log {\left (x - 1 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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