Integrand size = 35, antiderivative size = 22 \[ \int \frac {1-6 x+2 x^2+\left (-2+2 x^2\right ) \log \left (-\frac {x}{-6+6 x}\right )}{-1+x} \, dx=-1+(2+x) \left (-5+x \left (1+\log \left (\frac {x}{6-6 x}\right )\right )\right ) \]
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Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6874, 712, 2547, 84} \[ \int \frac {1-6 x+2 x^2+\left (-2+2 x^2\right ) \log \left (-\frac {x}{-6+6 x}\right )}{-1+x} \, dx=x^2-3 x+\log (1-x)-\log (x)+(x+1)^2 \log \left (\frac {x}{6 (1-x)}\right ) \]
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Rule 84
Rule 712
Rule 2547
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1-6 x+2 x^2}{-1+x}+2 (1+x) \log \left (\frac {x}{6-6 x}\right )\right ) \, dx \\ & = 2 \int (1+x) \log \left (\frac {x}{6-6 x}\right ) \, dx+\int \frac {1-6 x+2 x^2}{-1+x} \, dx \\ & = (1+x)^2 \log \left (\frac {x}{6 (1-x)}\right )-6 \int \frac {(1+x)^2}{(6-6 x) x} \, dx+\int \left (-4-\frac {3}{-1+x}+2 x\right ) \, dx \\ & = -4 x+x^2-3 \log (1-x)+(1+x)^2 \log \left (\frac {x}{6 (1-x)}\right )-6 \int \left (-\frac {1}{6}-\frac {2}{3 (-1+x)}+\frac {1}{6 x}\right ) \, dx \\ & = -3 x+x^2+\log (1-x)-\log (x)+(1+x)^2 \log \left (\frac {x}{6 (1-x)}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1-6 x+2 x^2+\left (-2+2 x^2\right ) \log \left (-\frac {x}{-6+6 x}\right )}{-1+x} \, dx=x \left (-3+x+(2+x) \log \left (\frac {x}{6-6 x}\right )\right ) \]
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Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23
method | result | size |
risch | \(\left (x^{2}+2 x \right ) \ln \left (-\frac {x}{6 x -6}\right )+x^{2}-3 x\) | \(27\) |
parallelrisch | \(x^{2} \ln \left (-\frac {x}{6 \left (-1+x \right )}\right )-7+x^{2}+2 x \ln \left (-\frac {x}{6 \left (-1+x \right )}\right )-3 x\) | \(34\) |
norman | \(x^{2}+x^{2} \ln \left (-\frac {x}{6 x -6}\right )-3 x +2 x \ln \left (-\frac {x}{6 x -6}\right )\) | \(37\) |
derivativedivides | \(\left (-1+x \right )^{2}+1-x -12 \ln \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (\frac {1}{2}-\frac {1}{2 \left (-1+x \right )}\right ) \left (-1+x \right )^{2}-24 \ln \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (-1+x \right )\) | \(70\) |
default | \(\left (-1+x \right )^{2}+1-x -12 \ln \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (\frac {1}{2}-\frac {1}{2 \left (-1+x \right )}\right ) \left (-1+x \right )^{2}-24 \ln \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (-1+x \right )\) | \(70\) |
parts | \(-3 \ln \left (-\frac {1}{-1+x}\right )-1-3 x -12 \ln \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (\frac {1}{2}-\frac {1}{2 \left (-1+x \right )}\right ) \left (-1+x \right )^{2}-24 \ln \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (-\frac {1}{6}-\frac {1}{6 \left (-1+x \right )}\right ) \left (-1+x \right )+x^{2}-3 \ln \left (-1+x \right )\) | \(84\) |
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1-6 x+2 x^2+\left (-2+2 x^2\right ) \log \left (-\frac {x}{-6+6 x}\right )}{-1+x} \, dx=x^{2} + {\left (x^{2} + 2 \, x\right )} \log \left (-\frac {x}{6 \, {\left (x - 1\right )}}\right ) - 3 \, x \]
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Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1-6 x+2 x^2+\left (-2+2 x^2\right ) \log \left (-\frac {x}{-6+6 x}\right )}{-1+x} \, dx=x^{2} - 3 x + \left (x^{2} + 2 x\right ) \log {\left (- \frac {x}{6 x - 6} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (21) = 42\).
Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \frac {1-6 x+2 x^2+\left (-2+2 x^2\right ) \log \left (-\frac {x}{-6+6 x}\right )}{-1+x} \, dx=-x^{2} {\left (\log \left (3\right ) + \log \left (2\right )\right )} + x^{2} - x {\left (2 \, \log \left (3\right ) + 2 \, \log \left (2\right ) - 1\right )} + {\left (x^{2} + 2 \, x\right )} \log \left (x\right ) - {\left (x^{2} + 2 \, x - 3\right )} \log \left (-x + 1\right ) - 4 \, x - 3 \, \log \left (x - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.95 \[ \int \frac {1-6 x+2 x^2+\left (-2+2 x^2\right ) \log \left (-\frac {x}{-6+6 x}\right )}{-1+x} \, dx=-\frac {{\left (\frac {4 \, x}{x - 1} - 3\right )} \log \left (-\frac {x}{6 \, {\left (x - 1\right )}}\right )}{\frac {2 \, x}{x - 1} - \frac {x^{2}}{{\left (x - 1\right )}^{2}} - 1} + \frac {\frac {x}{x - 1} - 2}{\frac {2 \, x}{x - 1} - \frac {x^{2}}{{\left (x - 1\right )}^{2}} - 1} + 3 \, \log \left (-\frac {x}{6 \, {\left (x - 1\right )}}\right ) \]
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Time = 9.36 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {1-6 x+2 x^2+\left (-2+2 x^2\right ) \log \left (-\frac {x}{-6+6 x}\right )}{-1+x} \, dx=x\,\left (2\,\ln \left (-\frac {x}{6\,x-6}\right )-3\right )+x^2\,\left (\ln \left (-\frac {x}{6\,x-6}\right )+1\right ) \]
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