Optimal. Leaf size=43 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{\sqrt{6}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{\sqrt{6}} \]
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Rubi [A] time = 0.049938, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1163, 203} \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{\sqrt{6}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{\sqrt{6}} \]
Antiderivative was successfully verified.
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Rule 1163
Rule 203
Rubi steps
\begin{align*} \int \frac{1+x^2}{1+4 x^2+x^4} \, dx &=\frac{1}{6} \left (3-\sqrt{3}\right ) \int \frac{1}{2-\sqrt{3}+x^2} \, dx+\frac{1}{6} \left (3+\sqrt{3}\right ) \int \frac{1}{2+\sqrt{3}+x^2} \, dx\\ &=\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{\sqrt{6}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{\sqrt{6}}\\ \end{align*}
Mathematica [A] time = 0.0673384, size = 81, normalized size = 1.88 \[ \frac{\left (\sqrt{3}-1\right ) \tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\left (1+\sqrt{3}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.065, size = 110, normalized size = 2.6 \begin{align*} -{\frac{\sqrt{3}}{3\,\sqrt{6}-3\,\sqrt{2}}\arctan \left ( 2\,{\frac{x}{\sqrt{6}-\sqrt{2}}} \right ) }+{\frac{1}{\sqrt{6}-\sqrt{2}}\arctan \left ( 2\,{\frac{x}{\sqrt{6}-\sqrt{2}}} \right ) }+{\frac{\sqrt{3}}{3\,\sqrt{2}+3\,\sqrt{6}}\arctan \left ( 2\,{\frac{x}{\sqrt{2}+\sqrt{6}}} \right ) }+{\frac{1}{\sqrt{2}+\sqrt{6}}\arctan \left ( 2\,{\frac{x}{\sqrt{2}+\sqrt{6}}} \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{x^{2} + 1}{x^{4} + 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 = 1.3479, size = 109, normalized size = 2.53 \begin{align*} \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{6} \, \sqrt{6}{\left (x^{3} + 5 \, x\right )}\right ) + \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{6} \, \sqrt{6} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.108176, size = 41, normalized size = 0.95 \begin{align*} \frac{\sqrt{6} \left (2 \operatorname{atan}{\left (\frac{\sqrt{6} x}{6} \right )} + 2 \operatorname{atan}{\left (\frac{\sqrt{6} x^{3}}{6} + \frac{5 \sqrt{6} x}{6} \right )}\right )}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11523, size = 35, normalized size = 0.81 \begin{align*} \frac{1}{12} \, \sqrt{6}{\left (\pi \mathrm{sgn}\left (x\right ) + 2 \, \arctan \left (\frac{\sqrt{6}{\left (x^{2} - 1\right )}}{6 \, x}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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