Integrand size = 18, antiderivative size = 21 \[ \int \frac {4 e^2 x}{9-6 x^2+x^4} \, dx=4-2 \left (-\frac {e^2}{3-x^2}+\log (4)\right ) \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 28, 267} \[ \int \frac {4 e^2 x}{9-6 x^2+x^4} \, dx=\frac {2 e^2}{3-x^2} \]
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Rule 12
Rule 28
Rule 267
Rubi steps \begin{align*} \text {integral}& = \left (4 e^2\right ) \int \frac {x}{9-6 x^2+x^4} \, dx \\ & = \left (4 e^2\right ) \int \frac {x}{\left (-3+x^2\right )^2} \, dx \\ & = \frac {2 e^2}{3-x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57 \[ \int \frac {4 e^2 x}{9-6 x^2+x^4} \, dx=-\frac {2 e^2}{-3+x^2} \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(-\frac {2 \,{\mathrm e}^{2}}{x^{2}-3}\) | \(12\) |
default | \(-\frac {2 \,{\mathrm e}^{2}}{x^{2}-3}\) | \(12\) |
norman | \(-\frac {2 \,{\mathrm e}^{2}}{x^{2}-3}\) | \(12\) |
risch | \(-\frac {2 \,{\mathrm e}^{2}}{x^{2}-3}\) | \(12\) |
parallelrisch | \(-\frac {2 \,{\mathrm e}^{2}}{x^{2}-3}\) | \(12\) |
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none
Time = 0.25 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.52 \[ \int \frac {4 e^2 x}{9-6 x^2+x^4} \, dx=-\frac {2 \, e^{2}}{x^{2} - 3} \]
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Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57 \[ \int \frac {4 e^2 x}{9-6 x^2+x^4} \, dx=- \frac {4 e^{2}}{2 x^{2} - 6} \]
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none
Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.52 \[ \int \frac {4 e^2 x}{9-6 x^2+x^4} \, dx=-\frac {2 \, e^{2}}{x^{2} - 3} \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.52 \[ \int \frac {4 e^2 x}{9-6 x^2+x^4} \, dx=-\frac {2 \, e^{2}}{x^{2} - 3} \]
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Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.52 \[ \int \frac {4 e^2 x}{9-6 x^2+x^4} \, dx=-\frac {2\,{\mathrm {e}}^2}{x^2-3} \]
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