Integrand size = 25, antiderivative size = 17 \[ \int \frac {4-5 x+4 \log (x)+x^2 \log ^2(x)}{x^2 \log ^2(x)} \, dx=x+\frac {5}{\log (x)}-\frac {4}{x \log (x)} \]
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Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6874, 2395, 2343, 2346, 2209, 2339, 30} \[ \int \frac {4-5 x+4 \log (x)+x^2 \log ^2(x)}{x^2 \log ^2(x)} \, dx=x+\frac {5}{\log (x)}-\frac {4}{x \log (x)} \]
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Rule 30
Rule 2209
Rule 2339
Rule 2343
Rule 2346
Rule 2395
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (1+\frac {4-5 x}{x^2 \log ^2(x)}+\frac {4}{x^2 \log (x)}\right ) \, dx \\ & = x+4 \int \frac {1}{x^2 \log (x)} \, dx+\int \frac {4-5 x}{x^2 \log ^2(x)} \, dx \\ & = x+4 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )+\int \left (\frac {4}{x^2 \log ^2(x)}-\frac {5}{x \log ^2(x)}\right ) \, dx \\ & = x+4 \operatorname {ExpIntegralEi}(-\log (x))+4 \int \frac {1}{x^2 \log ^2(x)} \, dx-5 \int \frac {1}{x \log ^2(x)} \, dx \\ & = x+4 \operatorname {ExpIntegralEi}(-\log (x))-\frac {4}{x \log (x)}-4 \int \frac {1}{x^2 \log (x)} \, dx-5 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right ) \\ & = x+4 \operatorname {ExpIntegralEi}(-\log (x))+\frac {5}{\log (x)}-\frac {4}{x \log (x)}-4 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right ) \\ & = x+\frac {5}{\log (x)}-\frac {4}{x \log (x)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {4-5 x+4 \log (x)+x^2 \log ^2(x)}{x^2 \log ^2(x)} \, dx=x+\frac {5}{\log (x)}-\frac {4}{x \log (x)} \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
risch | \(x +\frac {5 x -4}{x \ln \left (x \right )}\) | \(16\) |
default | \(x -\frac {4}{x \ln \left (x \right )}+\frac {5}{\ln \left (x \right )}\) | \(18\) |
parts | \(x -\frac {4}{x \ln \left (x \right )}+\frac {5}{\ln \left (x \right )}\) | \(18\) |
norman | \(\frac {-4+x^{2} \ln \left (x \right )+5 x}{x \ln \left (x \right )}\) | \(20\) |
parallelrisch | \(\frac {-4+x^{2} \ln \left (x \right )+5 x}{x \ln \left (x \right )}\) | \(20\) |
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none
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {4-5 x+4 \log (x)+x^2 \log ^2(x)}{x^2 \log ^2(x)} \, dx=\frac {x^{2} \log \left (x\right ) + 5 \, x - 4}{x \log \left (x\right )} \]
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Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \frac {4-5 x+4 \log (x)+x^2 \log ^2(x)}{x^2 \log ^2(x)} \, dx=x + \frac {5 x - 4}{x \log {\left (x \right )}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {4-5 x+4 \log (x)+x^2 \log ^2(x)}{x^2 \log ^2(x)} \, dx=x + \frac {5}{\log \left (x\right )} + 4 \, {\rm Ei}\left (-\log \left (x\right )\right ) - 4 \, \Gamma \left (-1, \log \left (x\right )\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {4-5 x+4 \log (x)+x^2 \log ^2(x)}{x^2 \log ^2(x)} \, dx=x + \frac {5 \, x - 4}{x \log \left (x\right )} \]
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Time = 9.16 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {4-5 x+4 \log (x)+x^2 \log ^2(x)}{x^2 \log ^2(x)} \, dx=x+\frac {5\,x-4}{x\,\ln \left (x\right )} \]
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