Integrand size = 134, antiderivative size = 30 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log \left (x^2 \log (4)+\frac {\left (x-x^3\right ) \left (x-\log \left (2+\log \left (x^2\right )\right )\right )}{x}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6, 6873, 6816} \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log \left (-x^3+x^2 \log \left (\log \left (x^2\right )+2\right )+x^2 \log (4)-\log \left (\log \left (x^2\right )+2\right )+x\right ) \]
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Rule 6
Rule 6816
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \int \frac {-2+2 x-6 x^3+x^2 (2+4 \log (4))+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx \\ & = \int \frac {-2+2 x-6 x^3+x^2 (2+4 \log (4))+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{x \left (2+\log \left (x^2\right )\right ) \left (x-x^3+x^2 \log (4)-\log \left (2+\log \left (x^2\right )\right )+x^2 \log \left (2+\log \left (x^2\right )\right )\right )} \, dx \\ & = \log \left (x-x^3+x^2 \log (4)-\log \left (2+\log \left (x^2\right )\right )+x^2 \log \left (2+\log \left (x^2\right )\right )\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log \left (-x+x^3-x^2 \log (4)+\log \left (2+\log \left (x^2\right )\right )-x^2 \log \left (2+\log \left (x^2\right )\right )\right ) \]
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Time = 1.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17
method | result | size |
parallelrisch | \(\ln \left (-2 x^{2} \ln \left (2\right )+x^{3}-\ln \left (2+\ln \left (x^{2}\right )\right ) x^{2}-x +\ln \left (2+\ln \left (x^{2}\right )\right )\right )\) | \(35\) |
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Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log \left (x^{2} - 1\right ) + \log \left (-\frac {x^{3} - 2 \, x^{2} \log \left (2\right ) - {\left (x^{2} - 1\right )} \log \left (\log \left (x^{2}\right ) + 2\right ) - x}{x^{2} - 1}\right ) \]
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Time = 0.59 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log {\left (x^{2} - 1 \right )} + \log {\left (\log {\left (\log {\left (x^{2} \right )} + 2 \right )} + \frac {- x^{3} + 2 x^{2} \log {\left (2 \right )} + x}{x^{2} - 1} \right )} \]
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Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log \left (x + 1\right ) + \log \left (x - 1\right ) + \log \left (-\frac {x^{3} - 3 \, x^{2} \log \left (2\right ) - {\left (x^{2} - 1\right )} \log \left (\log \left (x\right ) + 1\right ) - x + \log \left (2\right )}{x^{2} - 1}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log \left (-x^{3} + 2 \, x^{2} \log \left (2\right ) + x^{2} \log \left (\log \left (x^{2}\right ) + 2\right ) + x - \log \left (\log \left (x^{2}\right ) + 2\right )\right ) \]
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Timed out. \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\int \frac {2\,x+\ln \left (x^2\right )\,\left (-3\,x^3+4\,\ln \left (2\right )\,x^2+x\right )+\ln \left (\ln \left (x^2\right )+2\right )\,\left (2\,x^2\,\ln \left (x^2\right )+4\,x^2\right )+8\,x^2\,\ln \left (2\right )+2\,x^2-6\,x^3-2}{4\,x^3\,\ln \left (2\right )-\ln \left (\ln \left (x^2\right )+2\right )\,\left (2\,x+\ln \left (x^2\right )\,\left (x-x^3\right )-2\,x^3\right )+\ln \left (x^2\right )\,\left (-x^4+2\,\ln \left (2\right )\,x^3+x^2\right )+2\,x^2-2\,x^4} \,d x \]
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