Integrand size = 24, antiderivative size = 16 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=-x^2+81 x^4+\log \left (\log ^2(x)\right ) \]
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Time = 0.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6873, 12, 6820, 2339, 29} \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=81 x^4-x^2+2 \log (\log (x)) \]
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Rule 12
Rule 29
Rule 2339
Rule 6820
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (1-x^2 \log (x)+162 x^4 \log (x)\right )}{x \log (x)} \, dx \\ & = 2 \int \frac {1-x^2 \log (x)+162 x^4 \log (x)}{x \log (x)} \, dx \\ & = 2 \int \left (-x+162 x^3+\frac {1}{x \log (x)}\right ) \, dx \\ & = -x^2+81 x^4+2 \int \frac {1}{x \log (x)} \, dx \\ & = -x^2+81 x^4+2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = -x^2+81 x^4+2 \log (\log (x)) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=-x^2+81 x^4+2 \log (\log (x)) \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06
method | result | size |
default | \(81 x^{4}-x^{2}+2 \ln \left (\ln \left (x \right )\right )\) | \(17\) |
norman | \(81 x^{4}-x^{2}+2 \ln \left (\ln \left (x \right )\right )\) | \(17\) |
parallelrisch | \(81 x^{4}-x^{2}+2 \ln \left (\ln \left (x \right )\right )\) | \(17\) |
risch | \(81 x^{4}-x^{2}+\frac {1}{324}+2 \ln \left (\ln \left (x \right )\right )\) | \(18\) |
parts | \(\frac {\left (162 x^{2}-1\right )^{2}}{324}+2 \ln \left (\ln \left (x \right )\right )\) | \(18\) |
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none
Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=81 \, x^{4} - x^{2} + 2 \, \log \left (\log \left (x\right )\right ) \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=81 x^{4} - x^{2} + 2 \log {\left (\log {\left (x \right )} \right )} \]
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none
Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=81 \, x^{4} - x^{2} + 2 \, \log \left (\log \left (x\right )\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=81 \, x^{4} - x^{2} + 2 \, \log \left (\log \left (x\right )\right ) \]
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Time = 12.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=2\,\ln \left (\ln \left (x\right )\right )-x^2+81\,x^4 \]
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