Integrand size = 66, antiderivative size = 28 \[ \int \frac {9 x-3 x^3+\left (-3+12 x-10 x^2+2 x^3+3 x^4\right ) \log (-1+3 x)}{\left (3 x-9 x^2-x^3+3 x^4\right ) \log (-1+3 x)} \, dx=x+\log \left (\frac {\frac {3}{x}-x}{8 \log (5) \log (-1+3 x)}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {6873, 6874, 1816, 266, 2437, 2339, 29} \[ \int \frac {9 x-3 x^3+\left (-3+12 x-10 x^2+2 x^3+3 x^4\right ) \log (-1+3 x)}{\left (3 x-9 x^2-x^3+3 x^4\right ) \log (-1+3 x)} \, dx=\log \left (3-x^2\right )+x-\log (x)-\log (\log (3 x-1)) \]
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Rule 29
Rule 266
Rule 1816
Rule 2339
Rule 2437
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {9 x-3 x^3+\left (-3+12 x-10 x^2+2 x^3+3 x^4\right ) \log (-1+3 x)}{x \left (3-9 x-x^2+3 x^3\right ) \log (-1+3 x)} \, dx \\ & = \int \left (\frac {3-3 x+x^2+x^3}{x \left (-3+x^2\right )}-\frac {3}{(-1+3 x) \log (-1+3 x)}\right ) \, dx \\ & = -\left (3 \int \frac {1}{(-1+3 x) \log (-1+3 x)} \, dx\right )+\int \frac {3-3 x+x^2+x^3}{x \left (-3+x^2\right )} \, dx \\ & = \int \left (1-\frac {1}{x}+\frac {2 x}{-3+x^2}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,-1+3 x\right ) \\ & = x-\log (x)+2 \int \frac {x}{-3+x^2} \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (-1+3 x)\right ) \\ & = x-\log (x)+\log \left (3-x^2\right )-\log (\log (-1+3 x)) \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {9 x-3 x^3+\left (-3+12 x-10 x^2+2 x^3+3 x^4\right ) \log (-1+3 x)}{\left (3 x-9 x^2-x^3+3 x^4\right ) \log (-1+3 x)} \, dx=x-\log (x)+\log \left (3-x^2\right )-\log (\log (-1+3 x)) \]
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Time = 1.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79
method | result | size |
norman | \(x -\ln \left (x \right )-\ln \left (\ln \left (-1+3 x \right )\right )+\ln \left (x^{2}-3\right )\) | \(22\) |
risch | \(x -\ln \left (x \right )-\ln \left (\ln \left (-1+3 x \right )\right )+\ln \left (x^{2}-3\right )\) | \(22\) |
parts | \(x -\ln \left (x \right )-\ln \left (\ln \left (-1+3 x \right )\right )+\ln \left (x^{2}-3\right )\) | \(22\) |
parallelrisch | \(\frac {2}{3}-\ln \left (x \right )-\ln \left (\ln \left (-1+3 x \right )\right )+\ln \left (x^{2}-3\right )+x\) | \(23\) |
derivativedivides | \(-\frac {1}{3}+x +\ln \left (\left (-1+3 x \right )^{2}-28+6 x \right )-\ln \left (3 x \right )-\ln \left (\ln \left (-1+3 x \right )\right )\) | \(32\) |
default | \(-\frac {1}{3}+x +\ln \left (\left (-1+3 x \right )^{2}-28+6 x \right )-\ln \left (3 x \right )-\ln \left (\ln \left (-1+3 x \right )\right )\) | \(32\) |
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {9 x-3 x^3+\left (-3+12 x-10 x^2+2 x^3+3 x^4\right ) \log (-1+3 x)}{\left (3 x-9 x^2-x^3+3 x^4\right ) \log (-1+3 x)} \, dx=x + \log \left (x^{2} - 3\right ) - \log \left (x\right ) - \log \left (\log \left (3 \, x - 1\right )\right ) \]
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Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {9 x-3 x^3+\left (-3+12 x-10 x^2+2 x^3+3 x^4\right ) \log (-1+3 x)}{\left (3 x-9 x^2-x^3+3 x^4\right ) \log (-1+3 x)} \, dx=x - \log {\left (x \right )} + \log {\left (x^{2} - 3 \right )} - \log {\left (\log {\left (3 x - 1 \right )} \right )} \]
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Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {9 x-3 x^3+\left (-3+12 x-10 x^2+2 x^3+3 x^4\right ) \log (-1+3 x)}{\left (3 x-9 x^2-x^3+3 x^4\right ) \log (-1+3 x)} \, dx=x + \log \left (x^{2} - 3\right ) - \log \left (x\right ) - \log \left (\log \left (3 \, x - 1\right )\right ) \]
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {9 x-3 x^3+\left (-3+12 x-10 x^2+2 x^3+3 x^4\right ) \log (-1+3 x)}{\left (3 x-9 x^2-x^3+3 x^4\right ) \log (-1+3 x)} \, dx=x + \log \left (x^{2} - 3\right ) - \log \left (x\right ) - \log \left (\log \left (3 \, x - 1\right )\right ) \]
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Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {9 x-3 x^3+\left (-3+12 x-10 x^2+2 x^3+3 x^4\right ) \log (-1+3 x)}{\left (3 x-9 x^2-x^3+3 x^4\right ) \log (-1+3 x)} \, dx=x-\ln \left (\ln \left (3\,x-1\right )\right )+\ln \left (x^2-3\right )-\ln \left (x\right ) \]
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