Integrand size = 23, antiderivative size = 22 \[ \int \frac {-5 e-5 x-5 e \log (x)}{512 x^2 \log ^2(x)} \, dx=4+\frac {5}{2} \left (-9+\frac {e+x}{256 x \log (x)}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {12, 6873, 6874, 2395, 2343, 2346, 2209, 2339, 30} \[ \int \frac {-5 e-5 x-5 e \log (x)}{512 x^2 \log ^2(x)} \, dx=\frac {5 e}{512 x \log (x)}+\frac {5}{512 \log (x)} \]
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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}& = \frac {1}{512} \int \frac {-5 e-5 x-5 e \log (x)}{x^2 \log ^2(x)} \, dx \\ & = \frac {1}{512} \int \frac {5 (-e-x-e \log (x))}{x^2 \log ^2(x)} \, dx \\ & = \frac {5}{512} \int \frac {-e-x-e \log (x)}{x^2 \log ^2(x)} \, dx \\ & = \frac {5}{512} \int \left (\frac {-e-x}{x^2 \log ^2(x)}-\frac {e}{x^2 \log (x)}\right ) \, dx \\ & = \frac {5}{512} \int \frac {-e-x}{x^2 \log ^2(x)} \, dx-\frac {1}{512} (5 e) \int \frac {1}{x^2 \log (x)} \, dx \\ & = \frac {5}{512} \int \left (-\frac {e}{x^2 \log ^2(x)}-\frac {1}{x \log ^2(x)}\right ) \, dx-\frac {1}{512} (5 e) \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {5}{512} e \text {Ei}(-\log (x))-\frac {5}{512} \int \frac {1}{x \log ^2(x)} \, dx-\frac {1}{512} (5 e) \int \frac {1}{x^2 \log ^2(x)} \, dx \\ & = -\frac {5}{512} e \text {Ei}(-\log (x))+\frac {5 e}{512 x \log (x)}-\frac {5}{512} \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )+\frac {1}{512} (5 e) \int \frac {1}{x^2 \log (x)} \, dx \\ & = -\frac {5}{512} e \text {Ei}(-\log (x))+\frac {5}{512 \log (x)}+\frac {5 e}{512 x \log (x)}+\frac {1}{512} (5 e) \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right ) \\ & = \frac {5}{512 \log (x)}+\frac {5 e}{512 x \log (x)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {-5 e-5 x-5 e \log (x)}{512 x^2 \log ^2(x)} \, dx=\frac {5 (e+x)}{512 x \log (x)} \]
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Time = 0.42 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64
method | result | size |
risch | \(\frac {\frac {5 x}{512}+\frac {5 \,{\mathrm e}}{512}}{x \ln \left (x \right )}\) | \(14\) |
norman | \(\frac {\frac {5 x}{512}+\frac {5 \,{\mathrm e}}{512}}{x \ln \left (x \right )}\) | \(17\) |
parallelrisch | \(\frac {5 \,{\mathrm e}+5 x}{512 x \ln \left (x \right )}\) | \(18\) |
default | \(\frac {5 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )}{512}-\frac {5 \,{\mathrm e} \left (-\frac {1}{x \ln \left (x \right )}+\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )\right )}{512}+\frac {5}{512 \ln \left (x \right )}\) | \(34\) |
parts | \(\frac {5 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )}{512}-\frac {5 \,{\mathrm e} \left (-\frac {1}{x \ln \left (x \right )}+\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )\right )}{512}+\frac {5}{512 \ln \left (x \right )}\) | \(34\) |
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Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int \frac {-5 e-5 x-5 e \log (x)}{512 x^2 \log ^2(x)} \, dx=\frac {5 \, {\left (x + e\right )}}{512 \, x \log \left (x\right )} \]
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Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {-5 e-5 x-5 e \log (x)}{512 x^2 \log ^2(x)} \, dx=\frac {5 x + 5 e}{512 x \log {\left (x \right )}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {-5 e-5 x-5 e \log (x)}{512 x^2 \log ^2(x)} \, dx=-\frac {5}{512} \, {\rm Ei}\left (-\log \left (x\right )\right ) e + \frac {5}{512} \, e \Gamma \left (-1, \log \left (x\right )\right ) + \frac {5}{512 \, \log \left (x\right )} \]
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Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int \frac {-5 e-5 x-5 e \log (x)}{512 x^2 \log ^2(x)} \, dx=\frac {5 \, {\left (x + e\right )}}{512 \, x \log \left (x\right )} \]
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Time = 10.59 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-5 e-5 x-5 e \log (x)}{512 x^2 \log ^2(x)} \, dx=\frac {5\,x^2+5\,\mathrm {e}\,x}{512\,x^2\,\ln \left (x\right )} \]
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