Integrand size = 22, antiderivative size = 29 \[ \int \frac {1-2 x^2}{1+3 x^2+4 x^4} \, dx=-\frac {1}{2} \log \left (1-x+2 x^2\right )+\frac {1}{2} \log \left (1+x+2 x^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1178, 642} \[ \int \frac {1-2 x^2}{1+3 x^2+4 x^4} \, dx=\frac {1}{2} \log \left (2 x^2+x+1\right )-\frac {1}{2} \log \left (2 x^2-x+1\right ) \]
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Rule 642
Rule 1178
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int \frac {\frac {1}{2}+2 x}{-\frac {1}{2}-\frac {x}{2}-x^2} \, dx\right )-\frac {1}{2} \int \frac {\frac {1}{2}-2 x}{-\frac {1}{2}+\frac {x}{2}-x^2} \, dx \\ & = -\frac {1}{2} \log \left (1-x+2 x^2\right )+\frac {1}{2} \log \left (1+x+2 x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {1-2 x^2}{1+3 x^2+4 x^4} \, dx=-\frac {1}{2} \log \left (1-x+2 x^2\right )+\frac {1}{2} \log \left (1+x+2 x^2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(-\frac {\ln \left (x^{2}-\frac {1}{2} x +\frac {1}{2}\right )}{2}+\frac {\ln \left (x^{2}+\frac {1}{2} x +\frac {1}{2}\right )}{2}\) | \(24\) |
default | \(-\frac {\ln \left (2 x^{2}-x +1\right )}{2}+\frac {\ln \left (2 x^{2}+x +1\right )}{2}\) | \(26\) |
norman | \(-\frac {\ln \left (2 x^{2}-x +1\right )}{2}+\frac {\ln \left (2 x^{2}+x +1\right )}{2}\) | \(26\) |
risch | \(-\frac {\ln \left (2 x^{2}-x +1\right )}{2}+\frac {\ln \left (2 x^{2}+x +1\right )}{2}\) | \(26\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {1-2 x^2}{1+3 x^2+4 x^4} \, dx=\frac {1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac {1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \]
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Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {1-2 x^2}{1+3 x^2+4 x^4} \, dx=- \frac {\log {\left (x^{2} - \frac {x}{2} + \frac {1}{2} \right )}}{2} + \frac {\log {\left (x^{2} + \frac {x}{2} + \frac {1}{2} \right )}}{2} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {1-2 x^2}{1+3 x^2+4 x^4} \, dx=\frac {1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac {1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {1-2 x^2}{1+3 x^2+4 x^4} \, dx=\frac {1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac {1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \]
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Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.41 \[ \int \frac {1-2 x^2}{1+3 x^2+4 x^4} \, dx=\mathrm {atanh}\left (\frac {x}{2\,x^2+1}\right ) \]
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