Integrand size = 54, antiderivative size = 24 \[ \int \frac {-147 x+24 \log ^2(4)+42 x \log (6 x)-3 x \log ^2(6 x)}{49 x \log (4)-14 x \log (4) \log (6 x)+x \log (4) \log ^2(6 x)} \, dx=3 \left (-\frac {x}{\log (4)}+\frac {8 \log (4)}{7-\log (6 x)}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6873, 12, 6874, 2339, 30} \[ \int \frac {-147 x+24 \log ^2(4)+42 x \log (6 x)-3 x \log ^2(6 x)}{49 x \log (4)-14 x \log (4) \log (6 x)+x \log (4) \log ^2(6 x)} \, dx=\frac {24 \log (4)}{7-\log (6 x)}-\frac {3 x}{\log (4)} \]
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Rule 12
Rule 30
Rule 2339
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {3 \left (-49 x+8 \log ^2(4)+14 x \log (6 x)-x \log ^2(6 x)\right )}{x \log (4) (7-\log (6 x))^2} \, dx \\ & = \frac {3 \int \frac {-49 x+8 \log ^2(4)+14 x \log (6 x)-x \log ^2(6 x)}{x (7-\log (6 x))^2} \, dx}{\log (4)} \\ & = \frac {3 \int \left (-1+\frac {8 \log ^2(4)}{x (-7+\log (6 x))^2}\right ) \, dx}{\log (4)} \\ & = -\frac {3 x}{\log (4)}+(24 \log (4)) \int \frac {1}{x (-7+\log (6 x))^2} \, dx \\ & = -\frac {3 x}{\log (4)}+(24 \log (4)) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,-7+\log (6 x)\right ) \\ & = -\frac {3 x}{\log (4)}+\frac {24 \log (4)}{7-\log (6 x)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-147 x+24 \log ^2(4)+42 x \log (6 x)-3 x \log ^2(6 x)}{49 x \log (4)-14 x \log (4) \log (6 x)+x \log (4) \log ^2(6 x)} \, dx=-\frac {3 \left (x-\frac {8 \log ^2(4)}{7-\log (6 x)}\right )}{\log (4)} \]
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Time = 0.12 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(-\frac {3 x}{2 \ln \left (2\right )}-\frac {48 \ln \left (2\right )}{\ln \left (6 x \right )-7}\) | \(21\) |
default | \(-\frac {3 x}{2 \ln \left (2\right )}-\frac {48 \ln \left (2\right )}{\ln \left (6 x \right )-7}\) | \(21\) |
risch | \(-\frac {3 x}{2 \ln \left (2\right )}-\frac {48 \ln \left (2\right )}{\ln \left (6 x \right )-7}\) | \(21\) |
parallelrisch | \(-\frac {96 \ln \left (2\right )^{2}+3 x \ln \left (6 x \right )-21 x}{2 \ln \left (2\right ) \left (\ln \left (6 x \right )-7\right )}\) | \(32\) |
norman | \(\frac {\frac {21 x}{2 \ln \left (2\right )}-\frac {3 x \ln \left (6 x \right )}{2 \ln \left (2\right )}-48 \ln \left (2\right )}{\ln \left (6 x \right )-7}\) | \(33\) |
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {-147 x+24 \log ^2(4)+42 x \log (6 x)-3 x \log ^2(6 x)}{49 x \log (4)-14 x \log (4) \log (6 x)+x \log (4) \log ^2(6 x)} \, dx=-\frac {3 \, {\left (32 \, \log \left (2\right )^{2} + x \log \left (6 \, x\right ) - 7 \, x\right )}}{2 \, {\left (\log \left (2\right ) \log \left (6 \, x\right ) - 7 \, \log \left (2\right )\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-147 x+24 \log ^2(4)+42 x \log (6 x)-3 x \log ^2(6 x)}{49 x \log (4)-14 x \log (4) \log (6 x)+x \log (4) \log ^2(6 x)} \, dx=- \frac {3 x}{2 \log {\left (2 \right )}} - \frac {48 \log {\left (2 \right )}}{\log {\left (6 x \right )} - 7} \]
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Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {-147 x+24 \log ^2(4)+42 x \log (6 x)-3 x \log ^2(6 x)}{49 x \log (4)-14 x \log (4) \log (6 x)+x \log (4) \log ^2(6 x)} \, dx=-\frac {48 \, \log \left (2\right )^{2}}{{\left (\log \left (3\right ) - 7\right )} \log \left (2\right ) + \log \left (2\right )^{2} + \log \left (2\right ) \log \left (x\right )} - \frac {3 \, x}{2 \, \log \left (2\right )} \]
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Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-147 x+24 \log ^2(4)+42 x \log (6 x)-3 x \log ^2(6 x)}{49 x \log (4)-14 x \log (4) \log (6 x)+x \log (4) \log ^2(6 x)} \, dx=-\frac {3 \, x}{2 \, \log \left (2\right )} - \frac {48 \, \log \left (2\right )}{\log \left (2\right ) + \log \left (3 \, x\right ) - 7} \]
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Time = 13.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-147 x+24 \log ^2(4)+42 x \log (6 x)-3 x \log ^2(6 x)}{49 x \log (4)-14 x \log (4) \log (6 x)+x \log (4) \log ^2(6 x)} \, dx=-\frac {3\,x}{2\,\ln \left (2\right )}-\frac {48\,\ln \left (2\right )}{\ln \left (6\,x\right )-7} \]
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