Integrand size = 12, antiderivative size = 36 \[ \int x^2 \log \left (\frac {-1+x}{x}\right ) \, dx=-\frac {x}{3}-\frac {x^2}{6}-\frac {1}{3} \log (-1+x)+\frac {1}{3} x^3 \log \left (\frac {-1+x}{x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2511, 2505, 269, 45} \[ \int x^2 \log \left (\frac {-1+x}{x}\right ) \, dx=\frac {1}{3} x^3 \log \left (1-\frac {1}{x}\right )-\frac {x^2}{6}-\frac {x}{3}-\frac {1}{3} \log (1-x) \]
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Rule 45
Rule 269
Rule 2505
Rule 2511
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int x^2 \log \left (\frac {-1+x}{x}\right ) \, dx=-\frac {x}{3}-\frac {x^2}{6}-\frac {1}{3} \log (1-x)+\frac {1}{3} x^3 \log \left (\frac {-1+x}{x}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81
method | result | size |
risch | \(-\frac {x}{3}-\frac {x^{2}}{6}-\frac {\ln \left (-1+x \right )}{3}+\frac {x^{3} \ln \left (\frac {-1+x}{x}\right )}{3}\) | \(29\) |
parts | \(-\frac {x}{3}-\frac {x^{2}}{6}-\frac {\ln \left (-1+x \right )}{3}+\frac {x^{3} \ln \left (\frac {-1+x}{x}\right )}{3}\) | \(29\) |
parallelrisch | \(\frac {x^{3} \ln \left (\frac {-1+x}{x}\right )}{3}-\frac {1}{3}-\frac {x^{2}}{6}-\frac {\ln \left (x \right )}{3}-\frac {x}{3}-\frac {\ln \left (\frac {-1+x}{x}\right )}{3}\) | \(38\) |
derivativedivides | \(-\frac {x^{2}}{6}-\frac {x}{3}+\frac {\ln \left (-\frac {1}{x}\right )}{3}+\frac {\ln \left (1-\frac {1}{x}\right ) \left (1-\frac {1}{x}\right ) \left (\left (1-\frac {1}{x}\right )^{2}+\frac {3}{x}\right ) x^{3}}{3}\) | \(53\) |
default | \(-\frac {x^{2}}{6}-\frac {x}{3}+\frac {\ln \left (-\frac {1}{x}\right )}{3}+\frac {\ln \left (1-\frac {1}{x}\right ) \left (1-\frac {1}{x}\right ) \left (\left (1-\frac {1}{x}\right )^{2}+\frac {3}{x}\right ) x^{3}}{3}\) | \(53\) |
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Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int x^2 \log \left (\frac {-1+x}{x}\right ) \, dx=\frac {1}{3} \, x^{3} \log \left (\frac {x - 1}{x}\right ) - \frac {1}{6} \, x^{2} - \frac {1}{3} \, x - \frac {1}{3} \, \log \left (x - 1\right ) \]
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Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int x^2 \log \left (\frac {-1+x}{x}\right ) \, dx=\frac {x^{3} \log {\left (\frac {x - 1}{x} \right )}}{3} - \frac {x^{2}}{6} - \frac {x}{3} - \frac {\log {\left (x - 1 \right )}}{3} \]
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Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int x^2 \log \left (\frac {-1+x}{x}\right ) \, dx=\frac {1}{3} \, x^{3} \log \left (\frac {x - 1}{x}\right ) - \frac {1}{6} \, x^{2} - \frac {1}{3} \, x - \frac {1}{3} \, \log \left (x - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.94 \[ \int x^2 \log \left (\frac {-1+x}{x}\right ) \, dx=\frac {\frac {2 \, {\left (x - 1\right )}}{x} - 3}{6 \, {\left (\frac {x - 1}{x} - 1\right )}^{2}} - \frac {\log \left (\frac {x - 1}{x}\right )}{3 \, {\left (\frac {x - 1}{x} - 1\right )}^{3}} - \frac {1}{3} \, \log \left (\frac {{\left | x - 1 \right |}}{{\left | x \right |}}\right ) + \frac {1}{3} \, \log \left ({\left | \frac {x - 1}{x} - 1 \right |}\right ) \]
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Time = 0.41 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int x^2 \log \left (\frac {-1+x}{x}\right ) \, dx=\frac {x^3\,\ln \left (\frac {x-1}{x}\right )}{3}-\frac {\ln \left (x\,\left (x-1\right )\right )}{6}-\frac {\ln \left (\frac {x-1}{x}\right )}{6}-\frac {x}{3}-\frac {x^2}{6} \]
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