Integrand size = 9, antiderivative size = 12 \[ \int \frac {-1+\log (x)}{\log ^2(x)} \, dx=\frac {x}{\log (x)}+\log (3 \log (5)) \]
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Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2407, 2334, 2335} \[ \int \frac {-1+\log (x)}{\log ^2(x)} \, dx=\frac {x}{\log (x)} \]
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Rule 2334
Rule 2335
Rule 2407
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{\log ^2(x)}+\frac {1}{\log (x)}\right ) \, dx \\ & = -\int \frac {1}{\log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx \\ & = \frac {x}{\log (x)}+\text {li}(x)-\int \frac {1}{\log (x)} \, dx \\ & = \frac {x}{\log (x)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int \frac {-1+\log (x)}{\log ^2(x)} \, dx=\frac {x}{\log (x)} \]
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Time = 0.22 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58
method | result | size |
default | \(\frac {x}{\ln \left (x \right )}\) | \(7\) |
norman | \(\frac {x}{\ln \left (x \right )}\) | \(7\) |
risch | \(\frac {x}{\ln \left (x \right )}\) | \(7\) |
parallelrisch | \(\frac {x}{\ln \left (x \right )}\) | \(7\) |
parts | \(\frac {x}{\ln \left (x \right )}\) | \(7\) |
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none
Time = 0.23 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int \frac {-1+\log (x)}{\log ^2(x)} \, dx=\frac {x}{\log \left (x\right )} \]
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Time = 0.04 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.25 \[ \int \frac {-1+\log (x)}{\log ^2(x)} \, dx=\frac {x}{\log {\left (x \right )}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {-1+\log (x)}{\log ^2(x)} \, dx={\rm Ei}\left (\log \left (x\right )\right ) - \Gamma \left (-1, -\log \left (x\right )\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int \frac {-1+\log (x)}{\log ^2(x)} \, dx=\frac {x}{\log \left (x\right )} \]
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Time = 8.91 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int \frac {-1+\log (x)}{\log ^2(x)} \, dx=\frac {x}{\ln \left (x\right )} \]
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