Optimal. Leaf size=29 \[ \frac{1}{2} \log \left (2 x^2+x+1\right )-\frac{1}{2} \log \left (2 x^2-x+1\right ) \]
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Rubi [A] time = 0.0159712, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1164, 628} \[ \frac{1}{2} \log \left (2 x^2+x+1\right )-\frac{1}{2} \log \left (2 x^2-x+1\right ) \]
Antiderivative was successfully verified.
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Rule 1164
Rule 628
Rubi steps
\begin{align*} \int \frac{1-2 x^2}{1+3 x^2+4 x^4} \, dx &=-\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*}
Mathematica [A] time = 0.0059714, size = 29, normalized size = 1. \[ \frac{1}{2} \log \left (2 x^2+x+1\right )-\frac{1}{2} \log \left (2 x^2-x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 26, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( 2\,{x}^{2}-x+1 \right ) }{2}}+{\frac{\ln \left ( 2\,{x}^{2}+x+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963789, size = 34, normalized size = 1.17 \begin{align*} \frac{1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac{1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29054, size = 66, normalized size = 2.28 \begin{align*} \frac{1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac{1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.099563, size = 26, normalized size = 0.9 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12616, size = 34, normalized size = 1.17 \begin{align*} \frac{1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac{1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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