Optimal. Leaf size=36 \[ -\frac {x}{3}-\frac {x^2}{6}-\frac {1}{3} \log (-1+x)+\frac {1}{3} x^3 \log \left (\frac {-1+x}{x}\right ) \]
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Rubi [A]
time = 0.02, 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 = {2511, 2505,
269, 45} \begin {gather*} \frac {1}{3} x^3 \log \left (1-\frac {1}{x}\right )-\frac {x^2}{6}-\frac {x}{3}-\frac {1}{3} \log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rule 2505
Rule 2511
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 \begin {gather*} -\frac {x}{3}-\frac {x^2}{6}-\frac {1}{3} \log (1-x)+\frac {1}{3} x^3 \log \left (\frac {-1+x}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 53, normalized size = 1.47
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 | \(-\frac {x^{2}}{6}+\frac {\ln \left (-\frac {1}{x}\right )}{3}-\frac {x}{3}+\frac {\ln \left (1-\frac {1}{x}\right ) \left (1-\frac {1}{x}\right ) \left (\left (1-\frac {1}{x}\right )^{2}+\frac {3}{x}\right ) x^{3}}{3}\) | \(53\) |
default | \(-\frac {x^{2}}{6}+\frac {\ln \left (-\frac {1}{x}\right )}{3}-\frac {x}{3}+\frac {\ln \left (1-\frac {1}{x}\right ) \left (1-\frac {1}{x}\right ) \left (\left (1-\frac {1}{x}\right )^{2}+\frac {3}{x}\right ) x^{3}}{3}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.85, size = 28, normalized size = 0.78 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 28, normalized size = 0.78 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 26, normalized size = 0.72 \begin {gather*} \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} \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. 70 vs.
\(2 (28) = 56\).
time = 0.88, size = 70, normalized size = 1.94 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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