Integrand size = 69, antiderivative size = 20 \[ \int \frac {-x-6 x \log (3 x)+2 x \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )}{45 \log (3 x)-30 \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )+5 \log (3 x) \log ^2\left (\frac {1}{4} \log (3 x)\right )} \, dx=\frac {x^2}{5 \left (-3+\log \left (\frac {1}{4} \log (3 x)\right )\right )} \]
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\[ \int \frac {-x-6 x \log (3 x)+2 x \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )}{45 \log (3 x)-30 \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )+5 \log (3 x) \log ^2\left (\frac {1}{4} \log (3 x)\right )} \, dx=\int \frac {-x-6 x \log (3 x)+2 x \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )}{45 \log (3 x)-30 \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )+5 \log (3 x) \log ^2\left (\frac {1}{4} \log (3 x)\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (-1+2 \log (3 x) \left (-3+\log \left (\frac {1}{4} \log (3 x)\right )\right )\right )}{5 \log (3 x) \left (3-\log \left (\frac {1}{4} \log (3 x)\right )\right )^2} \, dx \\ & = \frac {1}{5} \int \frac {x \left (-1+2 \log (3 x) \left (-3+\log \left (\frac {1}{4} \log (3 x)\right )\right )\right )}{\log (3 x) \left (3-\log \left (\frac {1}{4} \log (3 x)\right )\right )^2} \, dx \\ & = \frac {1}{5} \int \left (-\frac {x}{\log (3 x) \left (-3+\log \left (\frac {1}{4} \log (3 x)\right )\right )^2}+\frac {2 x}{-3+\log \left (\frac {1}{4} \log (3 x)\right )}\right ) \, dx \\ & = -\left (\frac {1}{5} \int \frac {x}{\log (3 x) \left (-3+\log \left (\frac {1}{4} \log (3 x)\right )\right )^2} \, dx\right )+\frac {2}{5} \int \frac {x}{-3+\log \left (\frac {1}{4} \log (3 x)\right )} \, dx \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-x-6 x \log (3 x)+2 x \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )}{45 \log (3 x)-30 \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )+5 \log (3 x) \log ^2\left (\frac {1}{4} \log (3 x)\right )} \, dx=\frac {x^2}{5 \left (-3+\log \left (\frac {1}{4} \log (3 x)\right )\right )} \]
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Time = 1.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
norman | \(\frac {x^{2}}{5 \ln \left (\frac {\ln \left (3 x \right )}{4}\right )-15}\) | \(17\) |
risch | \(\frac {x^{2}}{5 \ln \left (\frac {\ln \left (3 x \right )}{4}\right )-15}\) | \(17\) |
parallelrisch | \(\frac {x^{2}}{5 \ln \left (\frac {\ln \left (3 x \right )}{4}\right )-15}\) | \(17\) |
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Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {-x-6 x \log (3 x)+2 x \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )}{45 \log (3 x)-30 \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )+5 \log (3 x) \log ^2\left (\frac {1}{4} \log (3 x)\right )} \, dx=\frac {x^{2}}{5 \, {\left (\log \left (\frac {1}{4} \, \log \left (3 \, x\right )\right ) - 3\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {-x-6 x \log (3 x)+2 x \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )}{45 \log (3 x)-30 \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )+5 \log (3 x) \log ^2\left (\frac {1}{4} \log (3 x)\right )} \, dx=\frac {x^{2}}{5 \log {\left (\frac {\log {\left (3 x \right )}}{4} \right )} - 15} \]
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Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {-x-6 x \log (3 x)+2 x \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )}{45 \log (3 x)-30 \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )+5 \log (3 x) \log ^2\left (\frac {1}{4} \log (3 x)\right )} \, dx=-\frac {x^{2}}{5 \, {\left (2 \, \log \left (2\right ) - \log \left (\log \left (3\right ) + \log \left (x\right )\right ) + 3\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-x-6 x \log (3 x)+2 x \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )}{45 \log (3 x)-30 \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )+5 \log (3 x) \log ^2\left (\frac {1}{4} \log (3 x)\right )} \, dx=-\frac {x^{2}}{5 \, {\left (2 \, \log \left (2\right ) - \log \left (\log \left (3 \, x\right )\right ) + 3\right )}} \]
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Time = 8.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-x-6 x \log (3 x)+2 x \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )}{45 \log (3 x)-30 \log (3 x) \log \left (\frac {1}{4} \log (3 x)\right )+5 \log (3 x) \log ^2\left (\frac {1}{4} \log (3 x)\right )} \, dx=\frac {x^2}{5\,\left (\ln \left (\frac {\ln \left (3\,x\right )}{4}\right )-3\right )} \]
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