Integrand size = 32, antiderivative size = 20 \[ \int \frac {-e^2+\left (-2 e^2-e^{50} x\right ) \log (x)}{e^2 x \log (x)} \, dx=\log \left (\frac {3 e^{-e^{48} x}}{2 x^2 \log (x)}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {12, 6873, 6874, 45, 2339, 29} \[ \int \frac {-e^2+\left (-2 e^2-e^{50} x\right ) \log (x)}{e^2 x \log (x)} \, dx=-e^{48} x-2 \log (x)-\log (\log (x)) \]
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Rule 12
Rule 29
Rule 45
Rule 2339
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-e^2+\left (-2 e^2-e^{50} x\right ) \log (x)}{x \log (x)} \, dx}{e^2} \\ & = \frac {\int \frac {e^2 \left (-1-2 \log (x)-e^{48} x \log (x)\right )}{x \log (x)} \, dx}{e^2} \\ & = \int \frac {-1-2 \log (x)-e^{48} x \log (x)}{x \log (x)} \, dx \\ & = \int \left (\frac {-2-e^{48} x}{x}-\frac {1}{x \log (x)}\right ) \, dx \\ & = \int \frac {-2-e^{48} x}{x} \, dx-\int \frac {1}{x \log (x)} \, dx \\ & = \int \left (-e^{48}-\frac {2}{x}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = -e^{48} x-2 \log (x)-\log (\log (x)) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {-e^2+\left (-2 e^2-e^{50} x\right ) \log (x)}{e^2 x \log (x)} \, dx=-e^{48} x-2 \log (x)-\log (\log (x)) \]
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Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-{\mathrm e}^{48} x -2 \ln \left (x \right )-\ln \left (\ln \left (x \right )\right )\) | \(16\) |
parts | \(-\ln \left (\ln \left (x \right )\right )-2 \ln \left (x \right )-{\mathrm e}^{-2} x \,{\mathrm e}^{50}\) | \(20\) |
norman | \(-\ln \left (\ln \left (x \right )\right )-2 \ln \left (x \right )-{\mathrm e}^{-2} x \,{\mathrm e}^{50}\) | \(22\) |
default | \({\mathrm e}^{-2} \left (-x \,{\mathrm e}^{50}-2 \,{\mathrm e}^{2} \ln \left (x \right )-{\mathrm e}^{2} \ln \left (\ln \left (x \right )\right )\right )\) | \(27\) |
parallelrisch | \({\mathrm e}^{-2} \left (-x \,{\mathrm e}^{50}-2 \,{\mathrm e}^{2} \ln \left (x \right )-{\mathrm e}^{2} \ln \left (\ln \left (x \right )\right )\right )\) | \(27\) |
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Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {-e^2+\left (-2 e^2-e^{50} x\right ) \log (x)}{e^2 x \log (x)} \, dx=-x e^{48} - 2 \, \log \left (x\right ) - \log \left (\log \left (x\right )\right ) \]
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Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {-e^2+\left (-2 e^2-e^{50} x\right ) \log (x)}{e^2 x \log (x)} \, dx=- x e^{48} - 2 \log {\left (x \right )} - \log {\left (\log {\left (x \right )} \right )} \]
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Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {-e^2+\left (-2 e^2-e^{50} x\right ) \log (x)}{e^2 x \log (x)} \, dx=-{\left (x e^{50} + 2 \, e^{2} \log \left (x\right ) + e^{2} \log \left (\log \left (x\right )\right )\right )} e^{\left (-2\right )} \]
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {-e^2+\left (-2 e^2-e^{50} x\right ) \log (x)}{e^2 x \log (x)} \, dx=-{\left (x e^{50} + 2 \, e^{2} \log \left (x\right ) + e^{2} \log \left (\log \left (x\right )\right )\right )} e^{\left (-2\right )} \]
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Time = 14.71 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {-e^2+\left (-2 e^2-e^{50} x\right ) \log (x)}{e^2 x \log (x)} \, dx=-\ln \left (\ln \left (x\right )\right )-2\,\ln \left (x\right )-x\,{\mathrm {e}}^{48} \]
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