Integrand size = 100, antiderivative size = 19 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{16-\log (x)+\log \left (3+\frac {3}{\log (x)}\right )} \]
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Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {12, 6818} \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{-\log (x)+\log \left (\frac {3 (\log (x)+1)}{\log (x)}\right )+16} \]
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Rule 12
Rule 6818
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {2 \left (1+x+x^2\right )}{x (1+x) \left (-16+x-\log \left (\frac {3+3 x}{x}\right )\right )^2} \, dx,x,\log (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {1+x+x^2}{x (1+x) \left (-16+x-\log \left (\frac {3+3 x}{x}\right )\right )^2} \, dx,x,\log (x)\right ) \\ & = \frac {2}{16-\log (x)+\log \left (\frac {3 (1+\log (x))}{\log (x)}\right )} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{16-\log (x)+\log \left (3+\frac {3}{\log (x)}\right )} \]
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Time = 1.47 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16
method | result | size |
parallelrisch | \(-\frac {2}{-16+\ln \left (x \right )-\ln \left (\frac {3 \ln \left (x \right )+3}{\ln \left (x \right )}\right )}\) | \(22\) |
default | \(\frac {2}{\ln \left (x \right ) \left (\ln \left (3\right ) \left (1+\frac {1}{\ln \left (x \right )}\right )+\ln \left (1+\frac {1}{\ln \left (x \right )}\right ) \left (1+\frac {1}{\ln \left (x \right )}\right )-\ln \left (3\right )-1+\frac {16}{\ln \left (x \right )}-\ln \left (1+\frac {1}{\ln \left (x \right )}\right )\right )}\) | \(53\) |
risch | \(\frac {4 i}{\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )+1\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )+1\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )^{3}+2 i \ln \left (3\right )-2 i \ln \left (x \right )-2 i \ln \left (\ln \left (x \right )\right )+2 i \ln \left (\ln \left (x \right )+1\right )+32 i}\) | \(129\) |
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=-\frac {2}{\log \left (x\right ) - \log \left (\frac {3 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )}\right ) - 16} \]
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Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{- \log {\left (x \right )} + \log {\left (\frac {3 \log {\left (x \right )} + 3}{\log {\left (x \right )}} \right )} + 16} \]
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Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{\log \left (3\right ) - \log \left (x\right ) + \log \left (\log \left (x\right ) + 1\right ) - \log \left (\log \left (x\right )\right ) + 16} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.16 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2 \, {\left (\frac {\log \left (x\right ) + 1}{\log \left (x\right )} - 1\right )}}{\frac {{\left (\log \left (x\right ) + 1\right )} \log \left (\frac {3 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )}\right )}{\log \left (x\right )} + \frac {16 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )} - \log \left (\frac {3 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )}\right ) - 17} \]
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Time = 9.97 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{\ln \left (\frac {3\,\ln \left (x\right )+3}{\ln \left (x\right )}\right )-\ln \left (x\right )+16} \]
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