Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0312964, antiderivative size = 46, 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 = {1163, 203} \[ \frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1163
Rule 203
Rubi steps
\begin{align*} \int \frac{1-2 x^2}{1+6 x^2+4 x^4} \, dx &=\left (-1-\sqrt{5}\right ) \int \frac{1}{3+\sqrt{5}+4 x^2} \, dx+\left (-1+\sqrt{5}\right ) \int \frac{1}{3-\sqrt{5}+4 x^2} \, dx\\ &=\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0705627, size = 84, normalized size = 1.83 \[ \frac{-\left (\sqrt{5}-5\right ) \sqrt{3+\sqrt{5}} \tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )-\sqrt{3-\sqrt{5}} \left (5+\sqrt{5}\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{4 \sqrt{5}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 136, normalized size = 3. \begin{align*} -2\,{\frac{1}{2\,\sqrt{10}-2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }+2\,{\frac{\sqrt{5}}{2\,\sqrt{10}-2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }-2\,{\frac{\sqrt{5}}{2\,\sqrt{10}+2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) }-2\,{\frac{1}{2\,\sqrt{10}+2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) } \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} + 6 \, x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38737, size = 99, normalized size = 2.15 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (2 \, \sqrt{2}{\left (x^{3} + x\right )}\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.113693, size = 39, normalized size = 0.85 \begin{align*} - \frac{\sqrt{2} \left (2 \operatorname{atan}{\left (\sqrt{2} x \right )} - 2 \operatorname{atan}{\left (2 \sqrt{2} x^{3} + 2 \sqrt{2} x \right )}\right )}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16846, size = 53, normalized size = 1.15 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{4 \, x}{\sqrt{10} + \sqrt{2}}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{4 \, x}{\sqrt{10} - \sqrt{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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