Integrand size = 78, antiderivative size = 22 \[ \int \frac {24+12 x+(4-12 x) \log \left (\frac {3}{-1+3 x}\right )}{-9+27 x+\left (-12+30 x+18 x^2\right ) \log \left (\frac {3}{-1+3 x}\right )+\left (-4+8 x+11 x^2+3 x^3\right ) \log ^2\left (\frac {3}{-1+3 x}\right )} \, dx=2+\frac {4}{3+(2+x) \log \left (\frac {3}{-1+3 x}\right )} \]
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Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6820, 12, 6840, 32} \[ \int \frac {24+12 x+(4-12 x) \log \left (\frac {3}{-1+3 x}\right )}{-9+27 x+\left (-12+30 x+18 x^2\right ) \log \left (\frac {3}{-1+3 x}\right )+\left (-4+8 x+11 x^2+3 x^3\right ) \log ^2\left (\frac {3}{-1+3 x}\right )} \, dx=\frac {4}{(x+2) \log \left (-\frac {3}{1-3 x}\right )+3} \]
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Rule 12
Rule 32
Rule 6820
Rule 6840
Rubi steps \begin{align*} \text {integral}& = \int \frac {4 \left (-3 (2+x)-(1-3 x) \log \left (\frac {3}{-1+3 x}\right )\right )}{(1-3 x) \left (3+(2+x) \log \left (\frac {3}{-1+3 x}\right )\right )^2} \, dx \\ & = 4 \int \frac {-3 (2+x)-(1-3 x) \log \left (\frac {3}{-1+3 x}\right )}{(1-3 x) \left (3+(2+x) \log \left (\frac {3}{-1+3 x}\right )\right )^2} \, dx \\ & = -\left (4 \text {Subst}\left (\int \frac {1}{(3+x)^2} \, dx,x,(2+x) \log \left (\frac {3}{-1+3 x}\right )\right )\right ) \\ & = \frac {4}{3+(2+x) \log \left (-\frac {3}{1-3 x}\right )} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {24+12 x+(4-12 x) \log \left (\frac {3}{-1+3 x}\right )}{-9+27 x+\left (-12+30 x+18 x^2\right ) \log \left (\frac {3}{-1+3 x}\right )+\left (-4+8 x+11 x^2+3 x^3\right ) \log ^2\left (\frac {3}{-1+3 x}\right )} \, dx=\frac {4}{3+(2+x) \log \left (\frac {3}{-1+3 x}\right )} \]
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41
method | result | size |
norman | \(\frac {4}{\ln \left (\frac {3}{-1+3 x}\right ) x +2 \ln \left (\frac {3}{-1+3 x}\right )+3}\) | \(31\) |
risch | \(\frac {4}{\ln \left (\frac {3}{-1+3 x}\right ) x +2 \ln \left (\frac {3}{-1+3 x}\right )+3}\) | \(31\) |
parallelrisch | \(\frac {4}{\ln \left (\frac {3}{-1+3 x}\right ) x +2 \ln \left (\frac {3}{-1+3 x}\right )+3}\) | \(31\) |
derivativedivides | \(\frac {36}{\left (-1+3 x \right ) \left (\frac {21 \ln \left (\frac {3}{-1+3 x}\right )}{-1+3 x}+\frac {27}{-1+3 x}+3 \ln \left (\frac {3}{-1+3 x}\right )\right )}\) | \(53\) |
default | \(\frac {36}{\left (-1+3 x \right ) \left (\frac {21 \ln \left (\frac {3}{-1+3 x}\right )}{-1+3 x}+\frac {27}{-1+3 x}+3 \ln \left (\frac {3}{-1+3 x}\right )\right )}\) | \(53\) |
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Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {24+12 x+(4-12 x) \log \left (\frac {3}{-1+3 x}\right )}{-9+27 x+\left (-12+30 x+18 x^2\right ) \log \left (\frac {3}{-1+3 x}\right )+\left (-4+8 x+11 x^2+3 x^3\right ) \log ^2\left (\frac {3}{-1+3 x}\right )} \, dx=\frac {4}{{\left (x + 2\right )} \log \left (\frac {3}{3 \, x - 1}\right ) + 3} \]
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Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {24+12 x+(4-12 x) \log \left (\frac {3}{-1+3 x}\right )}{-9+27 x+\left (-12+30 x+18 x^2\right ) \log \left (\frac {3}{-1+3 x}\right )+\left (-4+8 x+11 x^2+3 x^3\right ) \log ^2\left (\frac {3}{-1+3 x}\right )} \, dx=\frac {4}{\left (x + 2\right ) \log {\left (\frac {3}{3 x - 1} \right )} + 3} \]
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Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {24+12 x+(4-12 x) \log \left (\frac {3}{-1+3 x}\right )}{-9+27 x+\left (-12+30 x+18 x^2\right ) \log \left (\frac {3}{-1+3 x}\right )+\left (-4+8 x+11 x^2+3 x^3\right ) \log ^2\left (\frac {3}{-1+3 x}\right )} \, dx=\frac {4}{x \log \left (3\right ) - {\left (x + 2\right )} \log \left (3 \, x - 1\right ) + 2 \, \log \left (3\right ) + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.27 \[ \int \frac {24+12 x+(4-12 x) \log \left (\frac {3}{-1+3 x}\right )}{-9+27 x+\left (-12+30 x+18 x^2\right ) \log \left (\frac {3}{-1+3 x}\right )+\left (-4+8 x+11 x^2+3 x^3\right ) \log ^2\left (\frac {3}{-1+3 x}\right )} \, dx=\frac {12}{{\left (3 \, x - 1\right )} {\left (\frac {7 \, \log \left (\frac {3}{3 \, x - 1}\right )}{3 \, x - 1} + \frac {9}{3 \, x - 1} + \log \left (\frac {3}{3 \, x - 1}\right )\right )}} \]
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Time = 0.51 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {24+12 x+(4-12 x) \log \left (\frac {3}{-1+3 x}\right )}{-9+27 x+\left (-12+30 x+18 x^2\right ) \log \left (\frac {3}{-1+3 x}\right )+\left (-4+8 x+11 x^2+3 x^3\right ) \log ^2\left (\frac {3}{-1+3 x}\right )} \, dx=\frac {4}{2\,\ln \left (\frac {3}{3\,x-1}\right )+x\,\ln \left (\frac {3}{3\,x-1}\right )+3} \]
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