Integrand size = 117, antiderivative size = 18 \[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=\left (x+\log \left (4+x+\frac {\log (x)}{x}+\log (4 x)\right )\right )^2 \]
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Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6820, 12, 6818} \[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=\left (x+\log \left (x+\log (4 x)+\frac {\log (x)}{x}+4\right )\right )^2 \]
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Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (1+x+5 x^2+x^3+(-1+x) \log (x)+x^2 \log (4 x)\right ) \left (x+\log \left (4+x+\frac {\log (x)}{x}+\log (4 x)\right )\right )}{x (\log (x)+x (4+x+\log (4 x)))} \, dx \\ & = 2 \int \frac {\left (1+x+5 x^2+x^3+(-1+x) \log (x)+x^2 \log (4 x)\right ) \left (x+\log \left (4+x+\frac {\log (x)}{x}+\log (4 x)\right )\right )}{x (\log (x)+x (4+x+\log (4 x)))} \, dx \\ & = \left (x+\log \left (4+x+\frac {\log (x)}{x}+\log (4 x)\right )\right )^2 \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=\left (x+\log \left (4+x+\frac {\log (x)}{x}+\log (4 x)\right )\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs. \(2(18)=36\).
Time = 10.86 (sec) , antiderivative size = 94, normalized size of antiderivative = 5.22
method | result | size |
parallelrisch | \(x^{2}+2 \ln \left (\frac {x \ln \left (4 x \right )+\ln \left (x \right )+x^{2}+4 x}{x}\right ) x +\ln \left (\frac {x \ln \left (4 x \right )+\ln \left (x \right )+x^{2}+4 x}{x}\right )^{2}+8 \ln \left (x \ln \left (4 x \right )+\ln \left (x \right )+x^{2}+4 x \right )-8 \ln \left (x \right )-8 \ln \left (\frac {x \ln \left (4 x \right )+\ln \left (x \right )+x^{2}+4 x}{x}\right )\) | \(94\) |
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (18) = 36\).
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.06 \[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=x^{2} + 2 \, x \log \left (\frac {x^{2} + 2 \, x \log \left (2\right ) + {\left (x + 1\right )} \log \left (x\right ) + 4 \, x}{x}\right ) + \log \left (\frac {x^{2} + 2 \, x \log \left (2\right ) + {\left (x + 1\right )} \log \left (x\right ) + 4 \, x}{x}\right )^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (17) = 34\).
Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.83 \[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=x^{2} + 2 x \log {\left (\frac {x^{2} + x \left (\log {\left (x \right )} + \log {\left (4 \right )}\right ) + 4 x + \log {\left (x \right )}}{x} \right )} + \log {\left (\frac {x^{2} + x \left (\log {\left (x \right )} + \log {\left (4 \right )}\right ) + 4 x + \log {\left (x \right )}}{x} \right )}^{2} \]
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\[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=\int { \frac {2 \, {\left (x^{4} + x^{3} \log \left (4 \, x\right ) + 5 \, x^{3} + x^{2} + {\left (x^{2} - x\right )} \log \left (x\right ) + {\left (x^{3} + x^{2} \log \left (4 \, x\right ) + 5 \, x^{2} + {\left (x - 1\right )} \log \left (x\right ) + x + 1\right )} \log \left (\frac {x^{2} + x \log \left (4 \, x\right ) + 4 \, x + \log \left (x\right )}{x}\right ) + x\right )}}{x^{3} + x^{2} \log \left (4 \, x\right ) + 4 \, x^{2} + x \log \left (x\right )} \,d x } \]
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\[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=\int { \frac {2 \, {\left (x^{4} + x^{3} \log \left (4 \, x\right ) + 5 \, x^{3} + x^{2} + {\left (x^{2} - x\right )} \log \left (x\right ) + {\left (x^{3} + x^{2} \log \left (4 \, x\right ) + 5 \, x^{2} + {\left (x - 1\right )} \log \left (x\right ) + x + 1\right )} \log \left (\frac {x^{2} + x \log \left (4 \, x\right ) + 4 \, x + \log \left (x\right )}{x}\right ) + x\right )}}{x^{3} + x^{2} \log \left (4 \, x\right ) + 4 \, x^{2} + x \log \left (x\right )} \,d x } \]
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Time = 11.88 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx={\left (x+\ln \left (\frac {4\,x+\ln \left (x\right )+x\,\ln \left (4\,x\right )+x^2}{x}\right )\right )}^2 \]
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