Integrand size = 110, antiderivative size = 36 \[ \int \frac {2+4 x-2 x^2-4 x^3+\left (-1-2 x+x^2+2 x^3\right ) \log (4)}{-4 x^3+4 x^4+4 x^5+\left (4 x^2-4 x^3-4 x^4\right ) \log \left (\frac {6-6 x-6 x^2}{x}\right )+\left (-x+x^2+x^3\right ) \log ^2\left (\frac {6-6 x-6 x^2}{x}\right )} \, dx=\frac {-2+\log (4)}{x \left (-2+\frac {\log \left (3 \left (-3-x+\frac {2+x-x^2}{x}\right )\right )}{x}\right )} \]
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Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6820, 12, 6818} \[ \int \frac {2+4 x-2 x^2-4 x^3+\left (-1-2 x+x^2+2 x^3\right ) \log (4)}{-4 x^3+4 x^4+4 x^5+\left (4 x^2-4 x^3-4 x^4\right ) \log \left (\frac {6-6 x-6 x^2}{x}\right )+\left (-x+x^2+x^3\right ) \log ^2\left (\frac {6-6 x-6 x^2}{x}\right )} \, dx=\frac {2-\log (4)}{2 x-\log \left (-6 x+\frac {6}{x}-6\right )} \]
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Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (1+2 x-x^2-2 x^3\right ) (-2+\log (4))}{x \left (1-x-x^2\right ) \left (2 x-\log \left (-6+\frac {6}{x}-6 x\right )\right )^2} \, dx \\ & = (-2+\log (4)) \int \frac {1+2 x-x^2-2 x^3}{x \left (1-x-x^2\right ) \left (2 x-\log \left (-6+\frac {6}{x}-6 x\right )\right )^2} \, dx \\ & = \frac {2-\log (4)}{2 x-\log \left (-6+\frac {6}{x}-6 x\right )} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.61 \[ \int \frac {2+4 x-2 x^2-4 x^3+\left (-1-2 x+x^2+2 x^3\right ) \log (4)}{-4 x^3+4 x^4+4 x^5+\left (4 x^2-4 x^3-4 x^4\right ) \log \left (\frac {6-6 x-6 x^2}{x}\right )+\left (-x+x^2+x^3\right ) \log ^2\left (\frac {6-6 x-6 x^2}{x}\right )} \, dx=\frac {-2+\log (4)}{-2 x+\log \left (-6+\frac {6}{x}-6 x\right )} \]
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Time = 4.69 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81
method | result | size |
parallelrisch | \(-\frac {2 \ln \left (2\right )-2}{-\ln \left (-\frac {6 \left (x^{2}+x -1\right )}{x}\right )+2 x}\) | \(29\) |
norman | \(\frac {2-2 \ln \left (2\right )}{2 x -\ln \left (\frac {-6 x^{2}-6 x +6}{x}\right )}\) | \(31\) |
default | \(\frac {2 \ln \left (2\right )}{\ln \left (6\right )-2 x +\ln \left (-\frac {x^{2}+x -1}{x}\right )}-\frac {2}{\ln \left (6\right )-2 x +\ln \left (-\frac {x^{2}+x -1}{x}\right )}\) | \(48\) |
risch | \(-\frac {2 \ln \left (2\right )}{2 x -\ln \left (\frac {-6 x^{2}-6 x +6}{x}\right )}+\frac {2}{2 x -\ln \left (\frac {-6 x^{2}-6 x +6}{x}\right )}\) | \(54\) |
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none
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {2+4 x-2 x^2-4 x^3+\left (-1-2 x+x^2+2 x^3\right ) \log (4)}{-4 x^3+4 x^4+4 x^5+\left (4 x^2-4 x^3-4 x^4\right ) \log \left (\frac {6-6 x-6 x^2}{x}\right )+\left (-x+x^2+x^3\right ) \log ^2\left (\frac {6-6 x-6 x^2}{x}\right )} \, dx=-\frac {2 \, {\left (\log \left (2\right ) - 1\right )}}{2 \, x - \log \left (-\frac {6 \, {\left (x^{2} + x - 1\right )}}{x}\right )} \]
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Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.61 \[ \int \frac {2+4 x-2 x^2-4 x^3+\left (-1-2 x+x^2+2 x^3\right ) \log (4)}{-4 x^3+4 x^4+4 x^5+\left (4 x^2-4 x^3-4 x^4\right ) \log \left (\frac {6-6 x-6 x^2}{x}\right )+\left (-x+x^2+x^3\right ) \log ^2\left (\frac {6-6 x-6 x^2}{x}\right )} \, dx=\frac {-2 + 2 \log {\left (2 \right )}}{- 2 x + \log {\left (\frac {- 6 x^{2} - 6 x + 6}{x} \right )}} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {2+4 x-2 x^2-4 x^3+\left (-1-2 x+x^2+2 x^3\right ) \log (4)}{-4 x^3+4 x^4+4 x^5+\left (4 x^2-4 x^3-4 x^4\right ) \log \left (\frac {6-6 x-6 x^2}{x}\right )+\left (-x+x^2+x^3\right ) \log ^2\left (\frac {6-6 x-6 x^2}{x}\right )} \, dx=\frac {2 \, {\left (\log \left (2\right ) - 1\right )}}{i \, \pi - 2 \, x + \log \left (3\right ) + \log \left (2\right ) + \log \left (x^{2} + x - 1\right ) - \log \left (x\right )} \]
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none
Time = 0.34 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {2+4 x-2 x^2-4 x^3+\left (-1-2 x+x^2+2 x^3\right ) \log (4)}{-4 x^3+4 x^4+4 x^5+\left (4 x^2-4 x^3-4 x^4\right ) \log \left (\frac {6-6 x-6 x^2}{x}\right )+\left (-x+x^2+x^3\right ) \log ^2\left (\frac {6-6 x-6 x^2}{x}\right )} \, dx=-\frac {2 \, {\left (\log \left (2\right ) - 1\right )}}{2 \, x - \log \left (2\right ) - \log \left (-3 \, x^{2} - 3 \, x + 3\right ) + \log \left (x\right )} \]
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Time = 9.71 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19 \[ \int \frac {2+4 x-2 x^2-4 x^3+\left (-1-2 x+x^2+2 x^3\right ) \log (4)}{-4 x^3+4 x^4+4 x^5+\left (4 x^2-4 x^3-4 x^4\right ) \log \left (\frac {6-6 x-6 x^2}{x}\right )+\left (-x+x^2+x^3\right ) \log ^2\left (\frac {6-6 x-6 x^2}{x}\right )} \, dx=-\frac {\ln \left (4\right )+x\,\left (\ln \left (16\right )-4\right )-2}{\left (2\,x+1\right )\,\left (2\,x-\ln \left (-\frac {6\,x^2+6\,x-6}{x}\right )\right )} \]
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