Integrand size = 35, antiderivative size = 17 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=2+\frac {1}{\log \left (-\frac {54 x^2}{(e-x)^2}\right )} \]
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Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 1607, 6818} \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\log \left (-\frac {54 x^2}{x^2-2 e x+e^2}\right )} \]
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Rule 12
Rule 1607
Rule 6818
Rubi steps \begin{align*} \text {integral}& = -\left ((2 e) \int \frac {1}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx\right ) \\ & = -\left ((2 e) \int \frac {1}{(e-x) x \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx\right ) \\ & = \frac {1}{\log \left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\log \left (-\frac {54 x^2}{(e-x)^2}\right )} \]
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Time = 0.78 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29
method | result | size |
risch | \(\frac {1}{\ln \left (-\frac {54 x^{2}}{{\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}}\right )}\) | \(22\) |
norman | \(\frac {1}{\ln \left (-\frac {54 x^{2}}{{\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}}\right )}\) | \(24\) |
parallelrisch | \(\frac {1}{\ln \left (-\frac {54 x^{2}}{{\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}}\right )}\) | \(24\) |
default | \(\frac {1}{\ln \left (54\right )+\ln \left (\frac {x^{2}}{-{\mathrm e}^{2}+2 x \,{\mathrm e}-x^{2}}\right )}\) | \(30\) |
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none
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\log \left (-\frac {54 \, x^{2}}{x^{2} - 2 \, x e + e^{2}}\right )} \]
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Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\log {\left (- \frac {54 x^{2}}{x^{2} - 2 e x + e^{2}} \right )}} \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.12 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {e}{{\left (3 i \, \pi + 3 \, \log \left (3\right ) + \log \left (2\right )\right )} e - 2 \, e \log \left (x - e\right ) + 2 \, e \log \left (x\right )} \]
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Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\log \left (-\frac {54 \, x^{2}}{x^{2} - 2 \, x e + e^{2}}\right )} \]
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Time = 13.12 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\ln \left (\frac {54\,x^2}{{\left (x-\mathrm {e}\right )}^2}\right )+\pi \,1{}\mathrm {i}} \]
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