Integrand size = 95, antiderivative size = 27 \[ \int \frac {-16 x^2+16 x^3+\left (12 x^2+4 x^3\right ) \log (x \log (4))+8 x^2 \log ^2(x \log (4))+\left (-2+4 x+x^2-4 x^3+2 x^4\right ) \log ^3(x \log (4))}{\left (-2 x+6 x^2-6 x^3+2 x^4\right ) \log ^3(x \log (4))} \, dx=x+\log (x)-\frac {\left (x+\frac {4 x}{\log (x \log (4))}\right )^2}{4 (-1+x)^2} \]
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\[ \int \frac {-16 x^2+16 x^3+\left (12 x^2+4 x^3\right ) \log (x \log (4))+8 x^2 \log ^2(x \log (4))+\left (-2+4 x+x^2-4 x^3+2 x^4\right ) \log ^3(x \log (4))}{\left (-2 x+6 x^2-6 x^3+2 x^4\right ) \log ^3(x \log (4))} \, dx=\int \frac {-16 x^2+16 x^3+\left (12 x^2+4 x^3\right ) \log (x \log (4))+8 x^2 \log ^2(x \log (4))+\left (-2+4 x+x^2-4 x^3+2 x^4\right ) \log ^3(x \log (4))}{\left (-2 x+6 x^2-6 x^3+2 x^4\right ) \log ^3(x \log (4))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {16 x^2-16 x^3-\left (12 x^2+4 x^3\right ) \log (x \log (4))-8 x^2 \log ^2(x \log (4))-\left (-2+4 x+x^2-4 x^3+2 x^4\right ) \log ^3(x \log (4))}{x \left (2-6 x+6 x^2-2 x^3\right ) \log ^3(x \log (4))} \, dx \\ & = \int \left (\frac {-2+4 x+x^2-4 x^3+2 x^4}{2 (-1+x)^3 x}+\frac {8 x}{(-1+x)^2 \log ^3(x \log (4))}+\frac {2 x (3+x)}{(-1+x)^3 \log ^2(x \log (4))}+\frac {4 x}{(-1+x)^3 \log (x \log (4))}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-2+4 x+x^2-4 x^3+2 x^4}{(-1+x)^3 x} \, dx+2 \int \frac {x (3+x)}{(-1+x)^3 \log ^2(x \log (4))} \, dx+4 \int \frac {x}{(-1+x)^3 \log (x \log (4))} \, dx+8 \int \frac {x}{(-1+x)^2 \log ^3(x \log (4))} \, dx \\ & = \frac {1}{2} \int \left (2+\frac {1}{(-1+x)^3}+\frac {1}{(-1+x)^2}+\frac {2}{x}\right ) \, dx+2 \int \frac {x (3+x)}{(-1+x)^3 \log ^2(x \log (4))} \, dx+4 \int \frac {x}{(-1+x)^3 \log (x \log (4))} \, dx+8 \int \frac {x}{(-1+x)^2 \log ^3(x \log (4))} \, dx \\ & = -\frac {1}{4 (1-x)^2}+\frac {1}{2 (1-x)}+x+\log (x)+2 \int \frac {x (3+x)}{(-1+x)^3 \log ^2(x \log (4))} \, dx+4 \int \frac {x}{(-1+x)^3 \log (x \log (4))} \, dx+8 \int \frac {x}{(-1+x)^2 \log ^3(x \log (4))} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(27)=54\).
Time = 0.36 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.30 \[ \int \frac {-16 x^2+16 x^3+\left (12 x^2+4 x^3\right ) \log (x \log (4))+8 x^2 \log ^2(x \log (4))+\left (-2+4 x+x^2-4 x^3+2 x^4\right ) \log ^3(x \log (4))}{\left (-2 x+6 x^2-6 x^3+2 x^4\right ) \log ^3(x \log (4))} \, dx=\frac {1}{2} \left (-\frac {1}{2 (-1+x)^2}-\frac {1}{-1+x}+2 x+2 \log (x)-\frac {8 x^2}{(-1+x)^2 \log ^2(x \log (4))}-\frac {4 x^2}{(-1+x)^2 \log (x \log (4))}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(28)=56\).
Time = 15.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81
method | result | size |
risch | \(\frac {4 x^{2} \ln \left (x \right )+4 x^{3}-8 x \ln \left (x \right )-8 x^{2}+4 \ln \left (x \right )+2 x +1}{4 x^{2}-8 x +4}-\frac {2 \left (\ln \left (2 x \ln \left (2\right )\right )+2\right ) x^{2}}{\left (x^{2}-2 x +1\right ) \ln \left (2 x \ln \left (2\right )\right )^{2}}\) | \(76\) |
norman | \(\frac {\ln \left (2 x \ln \left (2\right )\right )^{3}+\frac {9 \ln \left (2 x \ln \left (2\right )\right )^{2}}{4}+\ln \left (2 x \ln \left (2\right )\right )^{2} x^{3}-2 \ln \left (2 x \ln \left (2\right )\right )^{3} x +\ln \left (2 x \ln \left (2\right )\right )^{3} x^{2}-\frac {7 \ln \left (2 x \ln \left (2\right )\right )^{2} x}{2}-4 x^{2}-2 x^{2} \ln \left (2 x \ln \left (2\right )\right )}{\left (-1+x \right )^{2} \ln \left (2 x \ln \left (2\right )\right )^{2}}\) | \(96\) |
parallelrisch | \(\frac {4 \ln \left (2 x \ln \left (2\right )\right )^{2} x^{3}+4 \ln \left (2 x \ln \left (2\right )\right )^{3} x^{2}-8 \ln \left (2 x \ln \left (2\right )\right )^{3} x -8 x^{2} \ln \left (2 x \ln \left (2\right )\right )-14 \ln \left (2 x \ln \left (2\right )\right )^{2} x +4 \ln \left (2 x \ln \left (2\right )\right )^{3}-16 x^{2}+9 \ln \left (2 x \ln \left (2\right )\right )^{2}}{4 \ln \left (2 x \ln \left (2\right )\right )^{2} \left (x^{2}-2 x +1\right )}\) | \(106\) |
default | \(\frac {\frac {9 \ln \left (x \right )^{2}}{2}+\left (2 \ln \left (2\right )^{2}+4 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+2 \ln \left (\ln \left (2\right )\right )^{2}\right ) x^{3}+\left (-6 \ln \left (2\right )^{2}-12 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )-6 \ln \left (\ln \left (2\right )\right )^{2}+9 \ln \left (2\right )+9 \ln \left (\ln \left (2\right )\right )\right ) \ln \left (x \right )+\left (-4 \ln \left (2\right )^{3}-12 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )-12 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )^{2}-4 \ln \left (\ln \left (2\right )\right )^{3}-4 \ln \left (2\right )-4 \ln \left (\ln \left (2\right )\right )-8\right ) x^{2}+\left (8 \ln \left (2\right )^{3}+24 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )+24 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )^{2}+8 \ln \left (\ln \left (2\right )\right )^{3}-7 \ln \left (2\right )^{2}-14 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )-7 \ln \left (\ln \left (2\right )\right )^{2}\right ) x -7 x \ln \left (x \right )^{2}+\left (4 \ln \left (2\right )+4 \ln \left (\ln \left (2\right )\right )\right ) \ln \left (x \right ) x^{3}+\left (-6 \ln \left (2\right )^{2}-12 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )-6 \ln \left (\ln \left (2\right )\right )^{2}-4\right ) \ln \left (x \right ) x^{2}+\left (12 \ln \left (2\right )^{2}+24 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+12 \ln \left (\ln \left (2\right )\right )^{2}-14 \ln \left (2\right )-14 \ln \left (\ln \left (2\right )\right )\right ) \ln \left (x \right ) x +2 \ln \left (x \right )^{3}-4 x \ln \left (x \right )^{3}+2 x^{2} \ln \left (x \right )^{3}+2 x^{3} \ln \left (x \right )^{2}+\frac {9 \ln \left (2\right )^{2}}{2}+9 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+\frac {9 \ln \left (\ln \left (2\right )\right )^{2}}{2}-4 \ln \left (2\right )^{3}-12 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )-12 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )^{2}-4 \ln \left (\ln \left (2\right )\right )^{3}}{2 \left (-1+x \right )^{2} \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )+\ln \left (x \right )\right )^{2}}\) | \(350\) |
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (26) = 52\).
Time = 0.47 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.93 \[ \int \frac {-16 x^2+16 x^3+\left (12 x^2+4 x^3\right ) \log (x \log (4))+8 x^2 \log ^2(x \log (4))+\left (-2+4 x+x^2-4 x^3+2 x^4\right ) \log ^3(x \log (4))}{\left (-2 x+6 x^2-6 x^3+2 x^4\right ) \log ^3(x \log (4))} \, dx=\frac {4 \, {\left (x^{2} - 2 \, x + 1\right )} \log \left (2 \, x \log \left (2\right )\right )^{3} - 8 \, x^{2} \log \left (2 \, x \log \left (2\right )\right ) + {\left (4 \, x^{3} - 8 \, x^{2} + 2 \, x + 1\right )} \log \left (2 \, x \log \left (2\right )\right )^{2} - 16 \, x^{2}}{4 \, {\left (x^{2} - 2 \, x + 1\right )} \log \left (2 \, x \log \left (2\right )\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.15 \[ \int \frac {-16 x^2+16 x^3+\left (12 x^2+4 x^3\right ) \log (x \log (4))+8 x^2 \log ^2(x \log (4))+\left (-2+4 x+x^2-4 x^3+2 x^4\right ) \log ^3(x \log (4))}{\left (-2 x+6 x^2-6 x^3+2 x^4\right ) \log ^3(x \log (4))} \, dx=x + \frac {1 - 2 x}{4 x^{2} - 8 x + 4} + \frac {- 2 x^{2} \log {\left (2 x \log {\left (2 \right )} \right )} - 4 x^{2}}{\left (x^{2} - 2 x + 1\right ) \log {\left (2 x \log {\left (2 \right )} \right )}^{2}} + \log {\left (x \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (26) = 52\).
Time = 0.31 (sec) , antiderivative size = 261, normalized size of antiderivative = 9.67 \[ \int \frac {-16 x^2+16 x^3+\left (12 x^2+4 x^3\right ) \log (x \log (4))+8 x^2 \log ^2(x \log (4))+\left (-2+4 x+x^2-4 x^3+2 x^4\right ) \log ^3(x \log (4))}{\left (-2 x+6 x^2-6 x^3+2 x^4\right ) \log ^3(x \log (4))} \, dx=\frac {4 \, {\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} x^{3} - 8 \, {\left ({\left (2 \, \log \left (\log \left (2\right )\right ) + 1\right )} \log \left (2\right ) + \log \left (2\right )^{2} + \log \left (\log \left (2\right )\right )^{2} + \log \left (\log \left (2\right )\right ) + 2\right )} x^{2} + {\left (4 \, x^{3} - 8 \, x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} + 2 \, {\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} x + \log \left (2\right )^{2} + 2 \, {\left (4 \, x^{3} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} - 4 \, x^{2} {\left (2 \, \log \left (2\right ) + 2 \, \log \left (\log \left (2\right )\right ) + 1\right )} + 2 \, x {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + \log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} \log \left (x\right ) + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}}{4 \, {\left ({\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} x^{2} + {\left (x^{2} - 2 \, x + 1\right )} \log \left (x\right )^{2} - 2 \, {\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} x + \log \left (2\right )^{2} + 2 \, {\left (x^{2} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} - 2 \, x {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + \log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} \log \left (x\right ) + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )}} + \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (26) = 52\).
Time = 0.35 (sec) , antiderivative size = 125, normalized size of antiderivative = 4.63 \[ \int \frac {-16 x^2+16 x^3+\left (12 x^2+4 x^3\right ) \log (x \log (4))+8 x^2 \log ^2(x \log (4))+\left (-2+4 x+x^2-4 x^3+2 x^4\right ) \log ^3(x \log (4))}{\left (-2 x+6 x^2-6 x^3+2 x^4\right ) \log ^3(x \log (4))} \, dx=x - \frac {2 \, {\left (x^{2} \log \left (2\right ) + x^{2} \log \left (x \log \left (2\right )\right ) + 2 \, x^{2}\right )}}{x^{2} \log \left (2\right )^{2} + 2 \, x^{2} \log \left (2\right ) \log \left (x \log \left (2\right )\right ) + x^{2} \log \left (x \log \left (2\right )\right )^{2} - 2 \, x \log \left (2\right )^{2} - 4 \, x \log \left (2\right ) \log \left (x \log \left (2\right )\right ) - 2 \, x \log \left (x \log \left (2\right )\right )^{2} + \log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (x \log \left (2\right )\right ) + \log \left (x \log \left (2\right )\right )^{2}} - \frac {2 \, x - 1}{4 \, {\left (x^{2} - 2 \, x + 1\right )}} + \log \left (x\right ) \]
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Time = 18.31 (sec) , antiderivative size = 203, normalized size of antiderivative = 7.52 \[ \int \frac {-16 x^2+16 x^3+\left (12 x^2+4 x^3\right ) \log (x \log (4))+8 x^2 \log ^2(x \log (4))+\left (-2+4 x+x^2-4 x^3+2 x^4\right ) \log ^3(x \log (4))}{\left (-2 x+6 x^2-6 x^3+2 x^4\right ) \log ^3(x \log (4))} \, dx=x+\ln \left (x\right )+\frac {\frac {x\,\left (5\,x-x^2\right )}{{\left (x-1\right )}^3}+\frac {2\,x^2\,{\ln \left (2\,x\,\ln \left (2\right )\right )}^2\,\left (x+2\right )}{{\left (x-1\right )}^4}+\frac {2\,x\,\ln \left (2\,x\,\ln \left (2\right )\right )\,\left (x^2+5\,x\right )}{{\left (x-1\right )}^4}}{\ln \left (2\,x\,\ln \left (2\right )\right )}-\frac {\frac {x^3}{2}+\frac {43\,x^2}{4}+x-\frac {1}{4}}{x^4-4\,x^3+6\,x^2-4\,x+1}-\frac {\frac {4\,x^2}{{\left (x-1\right )}^2}+\frac {2\,x^2\,{\ln \left (2\,x\,\ln \left (2\right )\right )}^2}{{\left (x-1\right )}^3}+\frac {x^2\,\ln \left (2\,x\,\ln \left (2\right )\right )\,\left (x+3\right )}{{\left (x-1\right )}^3}}{{\ln \left (2\,x\,\ln \left (2\right )\right )}^2}-\frac {\ln \left (2\,x\,\ln \left (2\right )\right )\,\left (2\,x^3+4\,x^2\right )}{x^4-4\,x^3+6\,x^2-4\,x+1} \]
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