Integrand size = 43, antiderivative size = 23 \[ \int \frac {4-x^2+\left (4-4 x+x^2+x^3\right ) \log \left (\frac {x}{3}\right )}{\left (-4 x+x^3\right ) \log \left (\frac {x}{3}\right )} \, dx=x+\log (3)-\log (x)+\log \left (-4+x^2\right )-\log \left (\log \left (\frac {x}{3}\right )\right ) \]
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Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {1607, 6857, 1816, 2339, 29} \[ \int \frac {4-x^2+\left (4-4 x+x^2+x^3\right ) \log \left (\frac {x}{3}\right )}{\left (-4 x+x^3\right ) \log \left (\frac {x}{3}\right )} \, dx=x+\log (2-x)-\log (x)+\log (x+2)-\log \left (\log \left (\frac {x}{3}\right )\right ) \]
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Rule 29
Rule 1607
Rule 1816
Rule 2339
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {4-x^2+\left (4-4 x+x^2+x^3\right ) \log \left (\frac {x}{3}\right )}{x \left (-4+x^2\right ) \log \left (\frac {x}{3}\right )} \, dx \\ & = \int \left (\frac {4-4 x+x^2+x^3}{x \left (-4+x^2\right )}-\frac {1}{x \log \left (\frac {x}{3}\right )}\right ) \, dx \\ & = \int \frac {4-4 x+x^2+x^3}{x \left (-4+x^2\right )} \, dx-\int \frac {1}{x \log \left (\frac {x}{3}\right )} \, dx \\ & = \int \left (1+\frac {1}{-2+x}-\frac {1}{x}+\frac {1}{2+x}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {x}{3}\right )\right ) \\ & = x+\log (2-x)-\log (x)+\log (2+x)-\log \left (\log \left (\frac {x}{3}\right )\right ) \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {4-x^2+\left (4-4 x+x^2+x^3\right ) \log \left (\frac {x}{3}\right )}{\left (-4 x+x^3\right ) \log \left (\frac {x}{3}\right )} \, dx=x-\log (x)+\log \left (4-x^2\right )-\log \left (\log \left (\frac {x}{3}\right )\right ) \]
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Time = 0.57 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
risch | \(x -\ln \left (x \right )+\ln \left (x^{2}-4\right )-\ln \left (\ln \left (\frac {x}{3}\right )\right )\) | \(20\) |
parts | \(x -\ln \left (x \right )+\ln \left (-2+x \right )+\ln \left (2+x \right )-\ln \left (\ln \left (\frac {x}{3}\right )\right )\) | \(22\) |
derivativedivides | \(x +\ln \left (2+x \right )+\ln \left (-2+x \right )-\ln \left (\frac {x}{3}\right )-\ln \left (\ln \left (\frac {x}{3}\right )\right )\) | \(24\) |
default | \(x +\ln \left (2+x \right )+\ln \left (-2+x \right )-\ln \left (\frac {x}{3}\right )-\ln \left (\ln \left (\frac {x}{3}\right )\right )\) | \(24\) |
norman | \(x +\ln \left (2+x \right )+\ln \left (-2+x \right )-\ln \left (\frac {x}{3}\right )-\ln \left (\ln \left (\frac {x}{3}\right )\right )\) | \(24\) |
parallelrisch | \(x +\ln \left (2+x \right )+\ln \left (-2+x \right )-\ln \left (\frac {x}{3}\right )-\ln \left (\ln \left (\frac {x}{3}\right )\right )\) | \(24\) |
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Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {4-x^2+\left (4-4 x+x^2+x^3\right ) \log \left (\frac {x}{3}\right )}{\left (-4 x+x^3\right ) \log \left (\frac {x}{3}\right )} \, dx=x + \log \left (x^{2} - 4\right ) - \log \left (x\right ) - \log \left (\log \left (\frac {1}{3} \, x\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {4-x^2+\left (4-4 x+x^2+x^3\right ) \log \left (\frac {x}{3}\right )}{\left (-4 x+x^3\right ) \log \left (\frac {x}{3}\right )} \, dx=x - \log {\left (x \right )} + \log {\left (x^{2} - 4 \right )} - \log {\left (\log {\left (\frac {x}{3} \right )} \right )} \]
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Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {4-x^2+\left (4-4 x+x^2+x^3\right ) \log \left (\frac {x}{3}\right )}{\left (-4 x+x^3\right ) \log \left (\frac {x}{3}\right )} \, dx=x + \log \left (x + 2\right ) + \log \left (x - 2\right ) - \log \left (x\right ) - \log \left (-\log \left (3\right ) + \log \left (x\right )\right ) \]
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Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {4-x^2+\left (4-4 x+x^2+x^3\right ) \log \left (\frac {x}{3}\right )}{\left (-4 x+x^3\right ) \log \left (\frac {x}{3}\right )} \, dx=x + \log \left (x^{2} - 4\right ) - \log \left (\frac {1}{3} \, x\right ) - \log \left (\log \left (\frac {1}{3} \, x\right )\right ) \]
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Time = 14.72 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {4-x^2+\left (4-4 x+x^2+x^3\right ) \log \left (\frac {x}{3}\right )}{\left (-4 x+x^3\right ) \log \left (\frac {x}{3}\right )} \, dx=x+\ln \left (x+2\right )+\ln \left (\frac {x-2}{x}\right )-\ln \left (\ln \left (\frac {x}{3}\right )\right ) \]
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