Integrand size = 182, antiderivative size = 21 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\frac {10}{x}+\frac {1}{\left (\frac {1}{x}+\log (2 x-\log (2))\right )^2} \]
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\[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (-10 x+2 x^4-2 x^5+5 \log (2)-x^3 \log (2)-15 x (2 x-\log (2)) \log (2 x-\log (2))-15 x^2 (2 x-\log (2)) \log ^2(2 x-\log (2))-5 x^3 (2 x-\log (2)) \log ^3(2 x-\log (2))\right )}{x^2 (2 x-\log (2)) (1+x \log (2 x-\log (2)))^3} \, dx \\ & = 2 \int \frac {-10 x+2 x^4-2 x^5+5 \log (2)-x^3 \log (2)-15 x (2 x-\log (2)) \log (2 x-\log (2))-15 x^2 (2 x-\log (2)) \log ^2(2 x-\log (2))-5 x^3 (2 x-\log (2)) \log ^3(2 x-\log (2))}{x^2 (2 x-\log (2)) (1+x \log (2 x-\log (2)))^3} \, dx \\ & = 2 \int \left (-\frac {5}{x^2}-\frac {x \left (-2 x+2 x^2+\log (2)\right )}{(2 x-\log (2)) (1+x \log (2 x-\log (2)))^3}\right ) \, dx \\ & = \frac {10}{x}-2 \int \frac {x \left (-2 x+2 x^2+\log (2)\right )}{(2 x-\log (2)) (1+x \log (2 x-\log (2)))^3} \, dx \\ & = \frac {10}{x}-2 \int \left (\frac {x^2}{(1+x \log (2 x-\log (2)))^3}+\frac {x (-2+\log (2))}{2 (1+x \log (2 x-\log (2)))^3}+\frac {\log ^2(2)}{4 (1+x \log (2 x-\log (2)))^3}+\frac {\log ^3(2)}{4 (2 x-\log (2)) (1+x \log (2 x-\log (2)))^3}\right ) \, dx \\ & = \frac {10}{x}-2 \int \frac {x^2}{(1+x \log (2 x-\log (2)))^3} \, dx-(-2+\log (2)) \int \frac {x}{(1+x \log (2 x-\log (2)))^3} \, dx-\frac {1}{2} \log ^2(2) \int \frac {1}{(1+x \log (2 x-\log (2)))^3} \, dx-\frac {1}{2} \log ^3(2) \int \frac {1}{(2 x-\log (2)) (1+x \log (2 x-\log (2)))^3} \, dx \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=-2 \left (-\frac {5}{x}-\frac {x^2}{2 (1+x \log (2 x-\log (2)))^2}\right ) \]
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Time = 1.92 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {10}{x}+\frac {x^{2}}{\left (x \ln \left (2 x -\ln \left (2\right )\right )+1\right )^{2}}\) | \(26\) |
parallelrisch | \(\frac {40+40 \ln \left (2 x -\ln \left (2\right )\right )^{2} x^{2}+4 x^{3}+80 x \ln \left (2 x -\ln \left (2\right )\right )}{4 x \left (\ln \left (2 x -\ln \left (2\right )\right )^{2} x^{2}+2 x \ln \left (2 x -\ln \left (2\right )\right )+1\right )}\) | \(72\) |
derivativedivides | \(-\frac {-\frac {3 \ln \left (2\right )^{2} \left (2 x -\ln \left (2\right )\right )}{2}-40 \ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )-10 \ln \left (2\right )^{2} \ln \left (2 x -\ln \left (2\right )\right )^{2}-10 \ln \left (2 x -\ln \left (2\right )\right )^{2} \left (2 x -\ln \left (2\right )\right )^{2}-40 \ln \left (2 x -\ln \left (2\right )\right ) \left (2 x -\ln \left (2\right )\right )-20 \ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )^{2} \left (2 x -\ln \left (2\right )\right )-\frac {\left (2 x -\ln \left (2\right )\right )^{3}}{2}-\frac {3 \ln \left (2\right ) \left (2 x -\ln \left (2\right )\right )^{2}}{2}-40-\frac {\ln \left (2\right )^{3}}{2}}{x \left (\ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )+\ln \left (2 x -\ln \left (2\right )\right ) \left (2 x -\ln \left (2\right )\right )+2\right )^{2}}\) | \(183\) |
default | \(-\frac {-\frac {3 \ln \left (2\right )^{2} \left (2 x -\ln \left (2\right )\right )}{2}-40 \ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )-10 \ln \left (2\right )^{2} \ln \left (2 x -\ln \left (2\right )\right )^{2}-10 \ln \left (2 x -\ln \left (2\right )\right )^{2} \left (2 x -\ln \left (2\right )\right )^{2}-40 \ln \left (2 x -\ln \left (2\right )\right ) \left (2 x -\ln \left (2\right )\right )-20 \ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )^{2} \left (2 x -\ln \left (2\right )\right )-\frac {\left (2 x -\ln \left (2\right )\right )^{3}}{2}-\frac {3 \ln \left (2\right ) \left (2 x -\ln \left (2\right )\right )^{2}}{2}-40-\frac {\ln \left (2\right )^{3}}{2}}{x \left (\ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )+\ln \left (2 x -\ln \left (2\right )\right ) \left (2 x -\ln \left (2\right )\right )+2\right )^{2}}\) | \(183\) |
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.19 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\frac {10 \, x^{2} \log \left (2 \, x - \log \left (2\right )\right )^{2} + x^{3} + 20 \, x \log \left (2 \, x - \log \left (2\right )\right ) + 10}{x^{3} \log \left (2 \, x - \log \left (2\right )\right )^{2} + 2 \, x^{2} \log \left (2 \, x - \log \left (2\right )\right ) + x} \]
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Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\frac {x^{2}}{x^{2} \log {\left (2 x - \log {\left (2 \right )} \right )}^{2} + 2 x \log {\left (2 x - \log {\left (2 \right )} \right )} + 1} + \frac {10}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (21) = 42\).
Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.19 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\frac {10 \, x^{2} \log \left (2 \, x - \log \left (2\right )\right )^{2} + x^{3} + 20 \, x \log \left (2 \, x - \log \left (2\right )\right ) + 10}{x^{3} \log \left (2 \, x - \log \left (2\right )\right )^{2} + 2 \, x^{2} \log \left (2 \, x - \log \left (2\right )\right ) + x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (21) = 42\).
Time = 0.35 (sec) , antiderivative size = 128, normalized size of antiderivative = 6.10 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\frac {2 \, x^{4} - 2 \, x^{3} + x^{2} \log \left (2\right )}{2 \, x^{4} \log \left (2 \, x - \log \left (2\right )\right )^{2} - 2 \, x^{3} \log \left (2 \, x - \log \left (2\right )\right )^{2} + x^{2} \log \left (2\right ) \log \left (2 \, x - \log \left (2\right )\right )^{2} + 4 \, x^{3} \log \left (2 \, x - \log \left (2\right )\right ) - 4 \, x^{2} \log \left (2 \, x - \log \left (2\right )\right ) + 2 \, x \log \left (2\right ) \log \left (2 \, x - \log \left (2\right )\right ) + 2 \, x^{2} - 2 \, x + \log \left (2\right )} + \frac {10}{x} \]
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Timed out. \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\int -\frac {{\ln \left (2\,x-\ln \left (2\right )\right )}^3\,\left (10\,x^3\,\ln \left (2\right )-20\,x^4\right )-\ln \left (2\right )\,\left (2\,x^3-10\right )-20\,x+{\ln \left (2\,x-\ln \left (2\right )\right )}^2\,\left (30\,x^2\,\ln \left (2\right )-60\,x^3\right )+\ln \left (2\,x-\ln \left (2\right )\right )\,\left (30\,x\,\ln \left (2\right )-60\,x^2\right )+4\,x^4-4\,x^5}{\ln \left (2\,x-\ln \left (2\right )\right )\,\left (3\,x^3\,\ln \left (2\right )-6\,x^4\right )+{\ln \left (2\,x-\ln \left (2\right )\right )}^3\,\left (x^5\,\ln \left (2\right )-2\,x^6\right )+{\ln \left (2\,x-\ln \left (2\right )\right )}^2\,\left (3\,x^4\,\ln \left (2\right )-6\,x^5\right )+x^2\,\ln \left (2\right )-2\,x^3} \,d x \]
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