Optimal. Leaf size=22 \[ -1+(2+x) \left (-5+x \left (1+\log \left (\frac {x}{6-6 x}\right )\right )\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 36, normalized size of antiderivative = 1.64, number of steps used = 7, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6742, 698, 2492, 72} \begin {gather*} x^2-3 x+\log (1-x)-\log (x)+(x+1)^2 \log \left (\frac {x}{6 (1-x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rule 698
Rule 2492
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 19, normalized size = 0.86 \begin {gather*} x \left (-3+x+(2+x) \log \left (\frac {x}{6-6 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 24, normalized size = 1.09 \begin {gather*} x^{2} + {\left (x^{2} + 2 \, x\right )} \log \left (-\frac {x}{6 \, {\left (x - 1\right )}}\right ) - 3 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 87, normalized size = 3.95 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 27, normalized size = 1.23
method | result | size |
risch | \(\left (x^{2}+2 x \right ) \ln \left (-\frac {x}{6 x -6}\right )+x^{2}-3 x\) | \(27\) |
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 | \(-x +1+\left (x -1\right )^{2}-24 \ln \left (-\frac {1}{6}-\frac {1}{6 \left (x -1\right )}\right ) \left (-\frac {1}{6}-\frac {1}{6 \left (x -1\right )}\right ) \left (x -1\right )-12 \ln \left (-\frac {1}{6}-\frac {1}{6 \left (x -1\right )}\right ) \left (-\frac {1}{6}-\frac {1}{6 \left (x -1\right )}\right ) \left (-\frac {1}{2 \left (x -1\right )}+\frac {1}{2}\right ) \left (x -1\right )^{2}\) | \(70\) |
default | \(-x +1+\left (x -1\right )^{2}-24 \ln \left (-\frac {1}{6}-\frac {1}{6 \left (x -1\right )}\right ) \left (-\frac {1}{6}-\frac {1}{6 \left (x -1\right )}\right ) \left (x -1\right )-12 \ln \left (-\frac {1}{6}-\frac {1}{6 \left (x -1\right )}\right ) \left (-\frac {1}{6}-\frac {1}{6 \left (x -1\right )}\right ) \left (-\frac {1}{2 \left (x -1\right )}+\frac {1}{2}\right ) \left (x -1\right )^{2}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.78, size = 62, normalized size = 2.82 \begin {gather*} -x^{2} {\left (\log \relax (3) + \log \relax (2)\right )} + x^{2} - x {\left (2 \, \log \relax (3) + 2 \, \log \relax (2) - 1\right )} + {\left (x^{2} + 2 \, x\right )} \log \relax (x) - {\left (x^{2} + 2 \, x - 3\right )} \log \left (-x + 1\right ) - 4 \, x - 3 \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 35, normalized size = 1.59 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 22, normalized size = 1.00 \begin {gather*} x^{2} - 3 x + \left (x^{2} + 2 x\right ) \log {\left (- \frac {x}{6 x - 6} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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