Integrand size = 10, antiderivative size = 5 \[ \int \frac {1}{x (3+\log (x))} \, dx=\log (3+\log (x)) \]
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Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2339, 29} \[ \int \frac {1}{x (3+\log (x))} \, dx=\log (\log (x)+3) \]
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Rule 29
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x} \, dx,x,3+\log (x)\right ) \\ & = \log (3+\log (x)) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (3+\log (x))} \, dx=\log (3+\log (x)) \]
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Time = 0.15 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\ln \left (3+\ln \left (x \right )\right )\) | \(6\) |
default | \(\ln \left (3+\ln \left (x \right )\right )\) | \(6\) |
norman | \(\ln \left (3+\ln \left (x \right )\right )\) | \(6\) |
risch | \(\ln \left (3+\ln \left (x \right )\right )\) | \(6\) |
parallelrisch | \(\ln \left (3+\ln \left (x \right )\right )\) | \(6\) |
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none
Time = 0.31 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (3+\log (x))} \, dx=\log \left (\log \left (x\right ) + 3\right ) \]
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Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (3+\log (x))} \, dx=\log {\left (\log {\left (x \right )} + 3 \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (3+\log (x))} \, dx=\log \left (\log \left (x\right ) + 3\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (5) = 10\).
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 4.40 \[ \int \frac {1}{x (3+\log (x))} \, dx=\frac {1}{2} \, \log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\left (x\right ) - 1\right )}^{2} + {\left (\log \left ({\left | x \right |}\right ) + 3\right )}^{2}\right ) \]
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Time = 1.79 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (3+\log (x))} \, dx=\ln \left (\ln \left (x\right )+3\right ) \]
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