Integrand size = 72, antiderivative size = 20 \[ \int \frac {-90+84 x-27 x^2+4 x^3+\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{-36 x+18 x^2-3 x^3+\left (108-54 x+9 x^2\right ) \log \left (12-6 x+x^2\right )} \, dx=-255-\frac {4 x}{3}+\log \left (x-3 \log \left (3+(-3+x)^2\right )\right ) \]
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Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6873, 12, 6860, 6816} \[ \int \frac {-90+84 x-27 x^2+4 x^3+\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{-36 x+18 x^2-3 x^3+\left (108-54 x+9 x^2\right ) \log \left (12-6 x+x^2\right )} \, dx=\log \left (x-3 \log \left (x^2-6 x+12\right )\right )-\frac {4 x}{3} \]
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Rule 12
Rule 6816
Rule 6860
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \int \frac {90-84 x+27 x^2-4 x^3-\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{3 \left (12-6 x+x^2\right ) \left (x-3 \log \left (12-6 x+x^2\right )\right )} \, dx \\ & = \frac {1}{3} \int \frac {90-84 x+27 x^2-4 x^3-\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{\left (12-6 x+x^2\right ) \left (x-3 \log \left (12-6 x+x^2\right )\right )} \, dx \\ & = \frac {1}{3} \int \left (-4+\frac {3 \left (30-12 x+x^2\right )}{\left (12-6 x+x^2\right ) \left (x-3 \log \left (12-6 x+x^2\right )\right )}\right ) \, dx \\ & = -\frac {4 x}{3}+\int \frac {30-12 x+x^2}{\left (12-6 x+x^2\right ) \left (x-3 \log \left (12-6 x+x^2\right )\right )} \, dx \\ & = -\frac {4 x}{3}+\log \left (x-3 \log \left (12-6 x+x^2\right )\right ) \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {-90+84 x-27 x^2+4 x^3+\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{-36 x+18 x^2-3 x^3+\left (108-54 x+9 x^2\right ) \log \left (12-6 x+x^2\right )} \, dx=\frac {1}{3} \left (-4 x+3 \log \left (x-3 \log \left (12-6 x+x^2\right )\right )\right ) \]
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Time = 2.45 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
norman | \(-\frac {4 x}{3}+\ln \left (x -3 \ln \left (x^{2}-6 x +12\right )\right )\) | \(19\) |
risch | \(-\frac {4 x}{3}+\ln \left (\ln \left (x^{2}-6 x +12\right )-\frac {x}{3}\right )\) | \(19\) |
parallelrisch | \(\ln \left (x -3 \ln \left (x^{2}-6 x +12\right )\right )-\frac {4 x}{3}-16\) | \(20\) |
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Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-90+84 x-27 x^2+4 x^3+\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{-36 x+18 x^2-3 x^3+\left (108-54 x+9 x^2\right ) \log \left (12-6 x+x^2\right )} \, dx=-\frac {4}{3} \, x + \log \left (-x + 3 \, \log \left (x^{2} - 6 \, x + 12\right )\right ) \]
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Time = 0.15 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {-90+84 x-27 x^2+4 x^3+\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{-36 x+18 x^2-3 x^3+\left (108-54 x+9 x^2\right ) \log \left (12-6 x+x^2\right )} \, dx=- \frac {4 x}{3} + \log {\left (- \frac {x}{3} + \log {\left (x^{2} - 6 x + 12 \right )} \right )} \]
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Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-90+84 x-27 x^2+4 x^3+\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{-36 x+18 x^2-3 x^3+\left (108-54 x+9 x^2\right ) \log \left (12-6 x+x^2\right )} \, dx=-\frac {4}{3} \, x + \log \left (-\frac {1}{3} \, x + \log \left (x^{2} - 6 \, x + 12\right )\right ) \]
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Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-90+84 x-27 x^2+4 x^3+\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{-36 x+18 x^2-3 x^3+\left (108-54 x+9 x^2\right ) \log \left (12-6 x+x^2\right )} \, dx=-\frac {4}{3} \, x + \log \left (x - 3 \, \log \left (x^{2} - 6 \, x + 12\right )\right ) \]
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Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-90+84 x-27 x^2+4 x^3+\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{-36 x+18 x^2-3 x^3+\left (108-54 x+9 x^2\right ) \log \left (12-6 x+x^2\right )} \, dx=\ln \left (\ln \left (x^2-6\,x+12\right )-\frac {x}{3}\right )-\frac {4\,x}{3} \]
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