Optimal. Leaf size=50 \[ \frac{\log \left (2 x^2+\sqrt{5} x+1\right )}{2 \sqrt{5}}-\frac{\log \left (2 x^2-\sqrt{5} x+1\right )}{2 \sqrt{5}} \]
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Rubi [A] time = 0.0228969, 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} \[ \frac{\log \left (2 x^2+\sqrt{5} x+1\right )}{2 \sqrt{5}}-\frac{\log \left (2 x^2-\sqrt{5} x+1\right )}{2 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 1164
Rule 628
Rubi steps
\begin{align*} \int \frac{1-2 x^2}{1-x^2+4 x^4} \, dx &=-\frac{\int \frac{\frac{\sqrt{5}}{2}+2 x}{-\frac{1}{2}-\frac{\sqrt{5} x}{2}-x^2} \, dx}{2 \sqrt{5}}-\frac{\int \frac{\frac{\sqrt{5}}{2}-2 x}{-\frac{1}{2}+\frac{\sqrt{5} x}{2}-x^2} \, dx}{2 \sqrt{5}}\\ &=-\frac{\log \left (1-\sqrt{5} x+2 x^2\right )}{2 \sqrt{5}}+\frac{\log \left (1+\sqrt{5} x+2 x^2\right )}{2 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.0137571, size = 42, normalized size = 0.84 \[ \frac{\log \left (2 x^2+\sqrt{5} x+1\right )-\log \left (-2 x^2+\sqrt{5} x-1\right )}{2 \sqrt{5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 39, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 1+2\,{x}^{2}-x\sqrt{5} \right ) \sqrt{5}}{10}}+{\frac{\ln \left ( 1+2\,{x}^{2}+x\sqrt{5} \right ) \sqrt{5}}{10}} \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} - x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42215, size = 109, normalized size = 2.18 \begin{align*} \frac{1}{10} \, \sqrt{5} \log \left (\frac{4 \, x^{4} + 9 \, x^{2} + 2 \, \sqrt{5}{\left (2 \, x^{3} + x\right )} + 1}{4 \, x^{4} - x^{2} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.102462, size = 46, normalized size = 0.92 \begin{align*} - \frac{\sqrt{5} \log{\left (x^{2} - \frac{\sqrt{5} x}{2} + \frac{1}{2} \right )}}{10} + \frac{\sqrt{5} \log{\left (x^{2} + \frac{\sqrt{5} x}{2} + \frac{1}{2} \right )}}{10} \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} - x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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