Integrand size = 34, antiderivative size = 11 \[ \int \frac {4+x^4+\left (4 x+4 x^4+x^5\right ) \log (x)}{\left (4 x+x^5\right ) \log (x)} \, dx=x+\log \left (\left (4+x^4\right ) \log (x)\right ) \]
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Time = 0.16 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1607, 6857, 1899, 266, 2339, 29} \[ \int \frac {4+x^4+\left (4 x+4 x^4+x^5\right ) \log (x)}{\left (4 x+x^5\right ) \log (x)} \, dx=\log \left (x^4+4\right )+x+\log (\log (x)) \]
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Rule 29
Rule 266
Rule 1607
Rule 1899
Rule 2339
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {4+x^4+\left (4 x+4 x^4+x^5\right ) \log (x)}{x \left (4+x^4\right ) \log (x)} \, dx \\ & = \int \left (\frac {4+4 x^3+x^4}{4+x^4}+\frac {1}{x \log (x)}\right ) \, dx \\ & = \int \frac {4+4 x^3+x^4}{4+x^4} \, dx+\int \frac {1}{x \log (x)} \, dx \\ & = \int \left (1+\frac {4 x^3}{4+x^4}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = x+\log (\log (x))+4 \int \frac {x^3}{4+x^4} \, dx \\ & = x+\log \left (4+x^4\right )+\log (\log (x)) \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {4+x^4+\left (4 x+4 x^4+x^5\right ) \log (x)}{\left (4 x+x^5\right ) \log (x)} \, dx=x+\log \left (4+x^4\right )+\log (\log (x)) \]
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Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
risch | \(x +\ln \left (x^{4}+4\right )+\ln \left (\ln \left (x \right )\right )\) | \(12\) |
default | \(\ln \left (\ln \left (x \right )\right )+x +\ln \left (x^{2}+2 x +2\right )+\ln \left (x^{2}-2 x +2\right )\) | \(24\) |
norman | \(\ln \left (\ln \left (x \right )\right )+x +\ln \left (x^{2}+2 x +2\right )+\ln \left (x^{2}-2 x +2\right )\) | \(24\) |
parallelrisch | \(\ln \left (\ln \left (x \right )\right )+x +\ln \left (x^{2}+2 x +2\right )+\ln \left (x^{2}-2 x +2\right )\) | \(24\) |
parts | \(\ln \left (\ln \left (x \right )\right )+x +\ln \left (x^{2}+2 x +2\right )+\ln \left (x^{2}-2 x +2\right )\) | \(24\) |
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Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {4+x^4+\left (4 x+4 x^4+x^5\right ) \log (x)}{\left (4 x+x^5\right ) \log (x)} \, dx=x + \log \left (x^{4} + 4\right ) + \log \left (\log \left (x\right )\right ) \]
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Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {4+x^4+\left (4 x+4 x^4+x^5\right ) \log (x)}{\left (4 x+x^5\right ) \log (x)} \, dx=x + \log {\left (x^{4} + 4 \right )} + \log {\left (\log {\left (x \right )} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (11) = 22\).
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.09 \[ \int \frac {4+x^4+\left (4 x+4 x^4+x^5\right ) \log (x)}{\left (4 x+x^5\right ) \log (x)} \, dx=x + \log \left (x^{2} + 2 \, x + 2\right ) + \log \left (x^{2} - 2 \, x + 2\right ) + \log \left (\log \left (x\right )\right ) \]
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Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {4+x^4+\left (4 x+4 x^4+x^5\right ) \log (x)}{\left (4 x+x^5\right ) \log (x)} \, dx=x + \log \left (x^{4} + 4\right ) + \log \left (\log \left (x\right )\right ) \]
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Time = 8.64 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {4+x^4+\left (4 x+4 x^4+x^5\right ) \log (x)}{\left (4 x+x^5\right ) \log (x)} \, dx=x+\ln \left (\ln \left (x\right )\,\left (x^4+4\right )\right ) \]
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