Integrand size = 9, antiderivative size = 20 \[ \int \frac {\log ^2(\log (x))}{x} \, dx=2 \log (x)-2 \log (x) \log (\log (x))+\log (x) \log ^2(\log (x)) \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2333, 2332} \[ \int \frac {\log ^2(\log (x))}{x} \, dx=\log (x) \log ^2(\log (x))-2 \log (x) \log (\log (x))+2 \log (x) \]
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Rule 2332
Rule 2333
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \log ^2(x) \, dx,x,\log (x)\right ) \\ & = \log (x) \log ^2(\log (x))-2 \text {Subst}(\int \log (x) \, dx,x,\log (x)) \\ & = 2 \log (x)-2 \log (x) \log (\log (x))+\log (x) \log ^2(\log (x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^2(\log (x))}{x} \, dx=2 \log (x)-2 \log (x) \log (\log (x))+\log (x) \log ^2(\log (x)) \]
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Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(2 \ln \left (x \right )-2 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}\) | \(21\) |
default | \(2 \ln \left (x \right )-2 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}\) | \(21\) |
norman | \(2 \ln \left (x \right )-2 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}\) | \(21\) |
risch | \(2 \ln \left (x \right )-2 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}\) | \(21\) |
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none
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^2(\log (x))}{x} \, dx=\log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 2 \, \log \left (x\right ) \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {\log ^2(\log (x))}{x} \, dx=\log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{2} - 2 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )} + 2 \log {\left (x \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {\log ^2(\log (x))}{x} \, dx={\left (\log \left (\log \left (x\right )\right )^{2} - 2 \, \log \left (\log \left (x\right )\right ) + 2\right )} \log \left (x\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^2(\log (x))}{x} \, dx=\log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 2 \, \log \left (x\right ) \]
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Time = 0.50 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {\log ^2(\log (x))}{x} \, dx=\ln \left (x\right )\,\left ({\ln \left (\ln \left (x\right )\right )}^2-2\,\ln \left (\ln \left (x\right )\right )+2\right ) \]
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