Optimal. Leaf size=31 \[ \frac{1}{4} \log \left (2 x^2+2 x+1\right )-\frac{1}{4} \log \left (2 x^2-2 x+1\right ) \]
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Rubi [A] time = 0.0149195, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1165, 628} \[ \frac{1}{4} \log \left (2 x^2+2 x+1\right )-\frac{1}{4} \log \left (2 x^2-2 x+1\right ) \]
Antiderivative was successfully verified.
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Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1-2 x^2}{1+4 x^4} \, dx &=-\left (\frac{1}{4} \int \frac{1+2 x}{-\frac{1}{2}-x-x^2} \, dx\right )-\frac{1}{4} \int \frac{1-2 x}{-\frac{1}{2}+x-x^2} \, dx\\ &=-\frac{1}{4} \log \left (1-2 x+2 x^2\right )+\frac{1}{4} \log \left (1+2 x+2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0046045, size = 31, normalized size = 1. \[ \frac{1}{4} \log \left (2 x^2+2 x+1\right )-\frac{1}{4} \log \left (2 x^2-2 x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 28, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( 2\,{x}^{2}-2\,x+1 \right ) }{4}}+{\frac{\ln \left ( 2\,{x}^{2}+2\,x+1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.978012, size = 36, normalized size = 1.16 \begin{align*} \frac{1}{4} \, \log \left (2 \, x^{2} + 2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x^{2} - 2 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39473, size = 72, normalized size = 2.32 \begin{align*} \frac{1}{4} \, \log \left (2 \, x^{2} + 2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x^{2} - 2 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.099135, size = 22, normalized size = 0.71 \begin{align*} - \frac{\log{\left (x^{2} - x + \frac{1}{2} \right )}}{4} + \frac{\log{\left (x^{2} + x + \frac{1}{2} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1064, size = 46, normalized size = 1.48 \begin{align*} \frac{1}{4} \, \log \left (x^{2} + \sqrt{2} \left (\frac{1}{4}\right )^{\frac{1}{4}} x + \frac{1}{2}\right ) - \frac{1}{4} \, \log \left (x^{2} - \sqrt{2} \left (\frac{1}{4}\right )^{\frac{1}{4}} x + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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