Optimal. Leaf size=50 \[ \frac{\log \left (2 x^2+\sqrt{2} x+1\right )}{2 \sqrt{2}}-\frac{\log \left (2 x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}} \]
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Rubi [A] time = 0.0230111, antiderivative size = 50, 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} \[ \frac{\log \left (2 x^2+\sqrt{2} x+1\right )}{2 \sqrt{2}}-\frac{\log \left (2 x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1164
Rule 628
Rubi steps
\begin{align*} \int \frac{1-2 x^2}{1+2 x^2+4 x^4} \, dx &=-\frac{\int \frac{\frac{1}{\sqrt{2}}+2 x}{-\frac{1}{2}-\frac{x}{\sqrt{2}}-x^2} \, dx}{2 \sqrt{2}}-\frac{\int \frac{\frac{1}{\sqrt{2}}-2 x}{-\frac{1}{2}+\frac{x}{\sqrt{2}}-x^2} \, dx}{2 \sqrt{2}}\\ &=-\frac{\log \left (1-\sqrt{2} x+2 x^2\right )}{2 \sqrt{2}}+\frac{\log \left (1+\sqrt{2} x+2 x^2\right )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0126572, size = 42, normalized size = 0.84 \[ \frac{\log \left (2 x^2+\sqrt{2} x+1\right )-\log \left (-2 x^2+\sqrt{2} x-1\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 39, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 1+2\,{x}^{2}-x\sqrt{2} \right ) \sqrt{2}}{4}}+{\frac{\ln \left ( 1+2\,{x}^{2}+x\sqrt{2} \right ) \sqrt{2}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{2 \, x^{2} - 1}{4 \, x^{4} + 2 \, x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35116, size = 111, normalized size = 2.22 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{4 \, x^{4} + 6 \, x^{2} + 2 \, \sqrt{2}{\left (2 \, x^{3} + x\right )} + 1}{4 \, x^{4} + 2 \, x^{2} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.101103, size = 46, normalized size = 0.92 \begin{align*} - \frac{\sqrt{2} \log{\left (x^{2} - \frac{\sqrt{2} x}{2} + \frac{1}{2} \right )}}{4} + \frac{\sqrt{2} \log{\left (x^{2} + \frac{\sqrt{2} x}{2} + \frac{1}{2} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{2 \, x^{2} - 1}{4 \, x^{4} + 2 \, x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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