Integrand size = 88, antiderivative size = 18 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 (-4+10 (4+x)+\log (1+\log (x)))} \]
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Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6820, 12, 6818} \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 (10 x+\log (\log (x)+1)+36)} \]
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Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 (-1-10 x-10 x \log (x))}{5 x (1+\log (x)) (36+10 x+\log (1+\log (x)))^2} \, dx \\ & = \frac {2}{5} \int \frac {-1-10 x-10 x \log (x)}{x (1+\log (x)) (36+10 x+\log (1+\log (x)))^2} \, dx \\ & = \frac {2}{5 (36+10 x+\log (1+\log (x)))} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 (36+10 x+\log (1+\log (x)))} \]
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Time = 0.37 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {2}{5 \left (\ln \left (\ln \left (x \right )+1\right )+10 x +36\right )}\) | \(15\) |
risch | \(\frac {2}{5 \left (\ln \left (\ln \left (x \right )+1\right )+10 x +36\right )}\) | \(15\) |
parallelrisch | \(\frac {2}{5 \left (\ln \left (\ln \left (x \right )+1\right )+10 x +36\right )}\) | \(15\) |
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Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 \, {\left (10 \, x + \log \left (\log \left (x\right ) + 1\right ) + 36\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{50 x + 5 \log {\left (\log {\left (x \right )} + 1 \right )} + 180} \]
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Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 \, {\left (10 \, x + \log \left (\log \left (x\right ) + 1\right ) + 36\right )}} \]
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Time = 0.33 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 \, {\left (10 \, x + \log \left (\log \left (x\right ) + 1\right ) + 36\right )}} \]
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Timed out. \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\int -\frac {20\,x+20\,x\,\ln \left (x\right )+2}{6480\,x+\ln \left (\ln \left (x\right )+1\right )\,\left (360\,x+\ln \left (x\right )\,\left (100\,x^2+360\,x\right )+100\,x^2\right )+{\ln \left (\ln \left (x\right )+1\right )}^2\,\left (5\,x+5\,x\,\ln \left (x\right )\right )+3600\,x^2+500\,x^3+\ln \left (x\right )\,\left (500\,x^3+3600\,x^2+6480\,x\right )} \,d x \]
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