Optimal. Leaf size=34 \[ -2+\log (4)+\frac {\left (2 x-x^2\right ) \log \left (\frac {x^2}{3 \left (x+2 x^2\right )}\right )}{x} \]
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Rubi [A] time = 0.15, antiderivative size = 29, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1593, 6742, 72, 2486, 31} \begin {gather*} 2 \log (x)-x \log \left (\frac {x}{3 (2 x+1)}\right )-2 \log (2 x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 72
Rule 1593
Rule 2486
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2-x+\left (-x-2 x^2\right ) \log \left (\frac {x}{3+6 x}\right )}{x (1+2 x)} \, dx\\ &=\int \left (\frac {2-x}{x (1+2 x)}-\log \left (\frac {x}{3+6 x}\right )\right ) \, dx\\ &=\int \frac {2-x}{x (1+2 x)} \, dx-\int \log \left (\frac {x}{3+6 x}\right ) \, dx\\ &=-x \log \left (\frac {x}{3 (1+2 x)}\right )+3 \int \frac {1}{3+6 x} \, dx+\int \left (\frac {2}{x}-\frac {5}{1+2 x}\right ) \, dx\\ &=2 \log (x)-x \log \left (\frac {x}{3 (1+2 x)}\right )-2 \log (1+2 x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 26, normalized size = 0.76 \begin {gather*} 2 \log (x)-2 \log (1+2 x)-x \log \left (\frac {x}{3+6 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 16, normalized size = 0.47 \begin {gather*} -{\left (x - 2\right )} \log \left (\frac {x}{3 \, {\left (2 \, x + 1\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 41, normalized size = 1.21 \begin {gather*} \frac {\log \left (\frac {x}{3 \, {\left (2 \, x + 1\right )}}\right )}{2 \, {\left (\frac {2 \, x}{2 \, x + 1} - 1\right )}} + \frac {5}{2} \, \log \left (\frac {x}{3 \, {\left (2 \, x + 1\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 27, normalized size = 0.79
method | result | size |
norman | \(2 \ln \left (\frac {x}{6 x +3}\right )-x \ln \left (\frac {x}{6 x +3}\right )\) | \(27\) |
risch | \(-x \ln \left (\frac {x}{6 x +3}\right )+2 \ln \relax (x )-2 \ln \left (2 x +1\right )\) | \(27\) |
derivativedivides | \(-3 \ln \left (\frac {1}{6}-\frac {1}{6 \left (2 x +1\right )}\right ) \left (\frac {1}{6}-\frac {1}{6 \left (2 x +1\right )}\right ) \left (2 x +1\right )+2 \ln \left (\frac {1}{6}-\frac {1}{6 \left (2 x +1\right )}\right )\) | \(46\) |
default | \(-3 \ln \left (\frac {1}{6}-\frac {1}{6 \left (2 x +1\right )}\right ) \left (\frac {1}{6}-\frac {1}{6 \left (2 x +1\right )}\right ) \left (2 x +1\right )+2 \ln \left (\frac {1}{6}-\frac {1}{6 \left (2 x +1\right )}\right )\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 35, normalized size = 1.03 \begin {gather*} x \log \relax (3) + \frac {1}{2} \, {\left (2 \, x + 1\right )} \log \left (2 \, x + 1\right ) - x \log \relax (x) - \frac {5}{2} \, \log \left (2 \, x + 1\right ) + 2 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.39, size = 15, normalized size = 0.44 \begin {gather*} -\ln \left (\frac {x}{6\,x+3}\right )\,\left (x-2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 22, normalized size = 0.65 \begin {gather*} - x \log {\left (\frac {x}{6 x + 3} \right )} + 2 \log {\relax (x )} - 2 \log {\left (x + \frac {1}{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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