Integrand size = 23, antiderivative size = 19 \[ \int \frac {-e^3+2 x}{e^3 x-x^2} \, dx=16+\log \left (\frac {1}{5 \left (e^3-x\right ) x}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {642} \[ \int \frac {-e^3+2 x}{e^3 x-x^2} \, dx=-\log \left (e^3 x-x^2\right ) \]
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Rule 642
Rubi steps \begin{align*} \text {integral}& = -\log \left (e^3 x-x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-e^3+2 x}{e^3 x-x^2} \, dx=-\log \left (e^3-x\right )-\log (x) \]
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Time = 1.15 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63
method | result | size |
default | \(-\ln \left (x \left (-x +{\mathrm e}^{3}\right )\right )\) | \(12\) |
risch | \(-\ln \left (-x \,{\mathrm e}^{3}+x^{2}\right )\) | \(13\) |
norman | \(-\ln \left (x \right )-\ln \left (-x +{\mathrm e}^{3}\right )\) | \(15\) |
parallelrisch | \(-\ln \left (x \right )-\ln \left (x -{\mathrm e}^{3}\right )\) | \(15\) |
meijerg | \(-\ln \left (x \right )+3-i \pi -\ln \left (1-{\mathrm e}^{-3} x \right )\) | \(21\) |
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none
Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {-e^3+2 x}{e^3 x-x^2} \, dx=-\log \left (x^{2} - x e^{3}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.53 \[ \int \frac {-e^3+2 x}{e^3 x-x^2} \, dx=- \log {\left (x^{2} - x e^{3} \right )} \]
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none
Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {-e^3+2 x}{e^3 x-x^2} \, dx=-\log \left (x^{2} - x e^{3}\right ) \]
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none
Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {-e^3+2 x}{e^3 x-x^2} \, dx=-\log \left ({\left | x^{2} - x e^{3} \right |}\right ) \]
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Time = 8.98 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int \frac {-e^3+2 x}{e^3 x-x^2} \, dx=-\ln \left (x\,\left (x-{\mathrm {e}}^3\right )\right ) \]
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