Integrand size = 17, antiderivative size = 13 \[ \int \frac {-6-3 x-6 \log (x)}{x^2 \log ^2(x)} \, dx=\frac {3 \left (1+\frac {2}{x}\right )}{\log (x)} \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.23, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {6873, 12, 6874, 2395, 2343, 2346, 2209, 2339, 30} \[ \int \frac {-6-3 x-6 \log (x)}{x^2 \log ^2(x)} \, dx=\frac {6}{x \log (x)}+\frac {3}{\log (x)} \]
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Rule 12
Rule 30
Rule 2209
Rule 2339
Rule 2343
Rule 2346
Rule 2395
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {3 (-2-x-2 \log (x))}{x^2 \log ^2(x)} \, dx \\ & = 3 \int \frac {-2-x-2 \log (x)}{x^2 \log ^2(x)} \, dx \\ & = 3 \int \left (\frac {-2-x}{x^2 \log ^2(x)}-\frac {2}{x^2 \log (x)}\right ) \, dx \\ & = 3 \int \frac {-2-x}{x^2 \log ^2(x)} \, dx-6 \int \frac {1}{x^2 \log (x)} \, dx \\ & = 3 \int \left (-\frac {2}{x^2 \log ^2(x)}-\frac {1}{x \log ^2(x)}\right ) \, dx-6 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right ) \\ & = -6 \operatorname {ExpIntegralEi}(-\log (x))-3 \int \frac {1}{x \log ^2(x)} \, dx-6 \int \frac {1}{x^2 \log ^2(x)} \, dx \\ & = -6 \operatorname {ExpIntegralEi}(-\log (x))+\frac {6}{x \log (x)}-3 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )+6 \int \frac {1}{x^2 \log (x)} \, dx \\ & = -6 \operatorname {ExpIntegralEi}(-\log (x))+\frac {3}{\log (x)}+\frac {6}{x \log (x)}+6 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right ) \\ & = \frac {3}{\log (x)}+\frac {6}{x \log (x)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.23 \[ \int \frac {-6-3 x-6 \log (x)}{x^2 \log ^2(x)} \, dx=\frac {3}{\log (x)}+\frac {6}{x \log (x)} \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {6+3 x}{x \ln \left (x \right )}\) | \(13\) |
norman | \(\frac {6+3 x}{x \ln \left (x \right )}\) | \(14\) |
parallelrisch | \(\frac {6+3 x}{x \ln \left (x \right )}\) | \(14\) |
default | \(\frac {3}{\ln \left (x \right )}+\frac {6}{x \ln \left (x \right )}\) | \(17\) |
parts | \(\frac {3}{\ln \left (x \right )}+\frac {6}{x \ln \left (x \right )}\) | \(17\) |
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none
Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {-6-3 x-6 \log (x)}{x^2 \log ^2(x)} \, dx=\frac {3 \, {\left (x + 2\right )}}{x \log \left (x\right )} \]
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Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {-6-3 x-6 \log (x)}{x^2 \log ^2(x)} \, dx=\frac {3 x + 6}{x \log {\left (x \right )}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54 \[ \int \frac {-6-3 x-6 \log (x)}{x^2 \log ^2(x)} \, dx=\frac {3}{\log \left (x\right )} - 6 \, {\rm Ei}\left (-\log \left (x\right )\right ) + 6 \, \Gamma \left (-1, \log \left (x\right )\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {-6-3 x-6 \log (x)}{x^2 \log ^2(x)} \, dx=\frac {3 \, {\left (x + 2\right )}}{x \log \left (x\right )} \]
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Time = 7.79 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {-6-3 x-6 \log (x)}{x^2 \log ^2(x)} \, dx=\frac {3\,\left (x+2\right )}{x\,\ln \left (x\right )} \]
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