Integrand size = 167, antiderivative size = 32 \[ \int \frac {5+15 x-10 x^2-30 x^3+\left (20+5 x^2+5 \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log \left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )+\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )}{\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )} \, dx=x+\frac {x}{\left (x+\frac {2+x}{5}\right ) \log \left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )} \]
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\[ \int \frac {5+15 x-10 x^2-30 x^3+\left (20+5 x^2+5 \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log \left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )+\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )}{\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )} \, dx=\int \frac {5+15 x-10 x^2-30 x^3+\left (20+5 x^2+5 \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log \left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )+\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )}{\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (1-\frac {5 \left (-1+2 x^2\right )}{2 (1+3 x) \left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right ) \log ^2\left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )}+\frac {5}{2 (1+3 x)^2 \log \left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )}\right ) \, dx \\ & = x-\frac {5}{2} \int \frac {-1+2 x^2}{(1+3 x) \left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right ) \log ^2\left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )} \, dx+\frac {5}{2} \int \frac {1}{(1+3 x)^2 \log \left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )} \, dx \\ & = x-\frac {5}{2} \int \left (\frac {2}{9 \left (-x^2-\log \left (\frac {1}{x}\right )-4 (1+\log (\log (4)))\right ) \log ^2\left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )}+\frac {7}{9 (-1-3 x) \left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right ) \log ^2\left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )}+\frac {2 x}{3 \left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right ) \log ^2\left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )}\right ) \, dx+\frac {5}{2} \int \frac {1}{(1+3 x)^2 \log \left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )} \, dx \\ & = x-\frac {5}{9} \int \frac {1}{\left (-x^2-\log \left (\frac {1}{x}\right )-4 (1+\log (\log (4)))\right ) \log ^2\left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )} \, dx-\frac {5}{3} \int \frac {x}{\left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right ) \log ^2\left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )} \, dx-\frac {35}{18} \int \frac {1}{(-1-3 x) \left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right ) \log ^2\left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )} \, dx+\frac {5}{2} \int \frac {1}{(1+3 x)^2 \log \left (x^2+\log \left (\frac {1}{x}\right )+4 (1+\log (\log (4)))\right )} \, dx \\ \end{align*}
Time = 0.81 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {5+15 x-10 x^2-30 x^3+\left (20+5 x^2+5 \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log \left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )+\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )}{\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )} \, dx=x+\frac {5 x}{2 (1+3 x) \log \left (4+x^2+\log \left (\frac {1}{x}\right )+4 \log (\log (4))\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(89\) vs. \(2(29)=58\).
Time = 190.87 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.81
method | result | size |
parallelrisch | \(\frac {36 \ln \left (\ln \left (\frac {16 \ln \left (2\right )^{4}}{x}\right )+x^{2}+4\right ) x^{2}-420 \ln \left (\ln \left (\frac {16 \ln \left (2\right )^{4}}{x}\right )+x^{2}+4\right ) x +30 x -144 \ln \left (\ln \left (\frac {16 \ln \left (2\right )^{4}}{x}\right )+x^{2}+4\right )}{12 \left (1+3 x \right ) \ln \left (\ln \left (\frac {16 \ln \left (2\right )^{4}}{x}\right )+x^{2}+4\right )}\) | \(90\) |
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Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75 \[ \int \frac {5+15 x-10 x^2-30 x^3+\left (20+5 x^2+5 \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log \left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )+\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )}{\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )} \, dx=\frac {2 \, {\left (3 \, x^{2} + x\right )} \log \left (x^{2} + \log \left (\frac {16 \, \log \left (2\right )^{4}}{x}\right ) + 4\right ) + 5 \, x}{2 \, {\left (3 \, x + 1\right )} \log \left (x^{2} + \log \left (\frac {16 \, \log \left (2\right )^{4}}{x}\right ) + 4\right )} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {5+15 x-10 x^2-30 x^3+\left (20+5 x^2+5 \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log \left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )+\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )}{\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )} \, dx=x + \frac {5 x}{\left (6 x + 2\right ) \log {\left (x^{2} + \log {\left (\frac {16 \log {\left (2 \right )}^{4}}{x} \right )} + 4 \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (30) = 60\).
Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.94 \[ \int \frac {5+15 x-10 x^2-30 x^3+\left (20+5 x^2+5 \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log \left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )+\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )}{\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )} \, dx=\frac {2 \, {\left (3 \, x^{2} + x\right )} \log \left (x^{2} + 4 \, \log \left (2\right ) - \log \left (x\right ) + 4 \, \log \left (\log \left (2\right )\right ) + 4\right ) + 5 \, x}{2 \, {\left (3 \, x + 1\right )} \log \left (x^{2} + 4 \, \log \left (2\right ) - \log \left (x\right ) + 4 \, \log \left (\log \left (2\right )\right ) + 4\right )} \]
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Time = 0.52 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {5+15 x-10 x^2-30 x^3+\left (20+5 x^2+5 \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log \left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )+\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )}{\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )} \, dx=x + \frac {5 \, x}{2 \, {\left (3 \, x \log \left (x^{2} + 4 \, \log \left (2\right ) - \log \left (x\right ) + 4 \, \log \left (\log \left (2\right )\right ) + 4\right ) + \log \left (x^{2} + 4 \, \log \left (2\right ) - \log \left (x\right ) + 4 \, \log \left (\log \left (2\right )\right ) + 4\right )\right )}} \]
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Time = 15.22 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.69 \[ \int \frac {5+15 x-10 x^2-30 x^3+\left (20+5 x^2+5 \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log \left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )+\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )}{\left (8+48 x+74 x^2+12 x^3+18 x^4+\left (2+12 x+18 x^2\right ) \log \left (\frac {\log ^4(4)}{x}\right )\right ) \log ^2\left (4+x^2+\log \left (\frac {\log ^4(4)}{x}\right )\right )} \, dx=x-\frac {\frac {5\,x^3}{36}+\frac {5\,x}{9}}{-x^4-\frac {2\,x^3}{3}+\frac {7\,x^2}{18}+\frac {x}{3}+\frac {1}{18}}+\frac {\frac {5\,x}{2\,\left (3\,x+1\right )}-\frac {5\,x\,\ln \left (\ln \left (\frac {16\,{\ln \left (2\right )}^4}{x}\right )+x^2+4\right )\,\left (\ln \left (\frac {16\,{\ln \left (2\right )}^4}{x}\right )+x^2+4\right )}{2\,{\left (3\,x+1\right )}^2\,\left (2\,x^2-1\right )}}{\ln \left (\ln \left (\frac {16\,{\ln \left (2\right )}^4}{x}\right )+x^2+4\right )}-\frac {5\,x\,\ln \left (\frac {16\,{\ln \left (2\right )}^4}{x}\right )}{36\,\left (-x^4-\frac {2\,x^3}{3}+\frac {7\,x^2}{18}+\frac {x}{3}+\frac {1}{18}\right )} \]
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