Integrand size = 22, antiderivative size = 11 \[ \int \frac {1+2 x^2}{1-4 x^2+4 x^4} \, dx=\frac {x}{1-2 x^2} \]
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Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {28, 391} \[ \int \frac {1+2 x^2}{1-4 x^2+4 x^4} \, dx=\frac {x}{1-2 x^2} \]
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Rule 28
Rule 391
Rubi steps \begin{align*} \text {integral}& = 4 \int \frac {1+2 x^2}{\left (-2+4 x^2\right )^2} \, dx \\ & = \frac {x}{1-2 x^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {1+2 x^2}{1-4 x^2+4 x^4} \, dx=-\frac {x}{-1+2 x^2} \]
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Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {x}{2 \left (x^{2}-\frac {1}{2}\right )}\) | \(11\) |
risch | \(-\frac {x}{2 \left (x^{2}-\frac {1}{2}\right )}\) | \(11\) |
gosper | \(-\frac {x}{2 x^{2}-1}\) | \(13\) |
norman | \(-\frac {x}{2 x^{2}-1}\) | \(13\) |
parallelrisch | \(-\frac {x}{2 x^{2}-1}\) | \(13\) |
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none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {1+2 x^2}{1-4 x^2+4 x^4} \, dx=-\frac {x}{2 \, x^{2} - 1} \]
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Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {1+2 x^2}{1-4 x^2+4 x^4} \, dx=- \frac {x}{2 x^{2} - 1} \]
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none
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {1+2 x^2}{1-4 x^2+4 x^4} \, dx=-\frac {x}{2 \, x^{2} - 1} \]
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {1+2 x^2}{1-4 x^2+4 x^4} \, dx=-\frac {x}{2 \, x^{2} - 1} \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {1+2 x^2}{1-4 x^2+4 x^4} \, dx=-\frac {x}{2\,\left (x^2-\frac {1}{2}\right )} \]
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