Optimal. Leaf size=50 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0398747, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1163, 203} \[ \frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1163
Rule 203
Rubi steps
\begin{align*} \int \frac{1-x^2}{1+5 x^2+x^4} \, dx &=\frac{1}{6} \left (-3+\sqrt{21}\right ) \int \frac{1}{\frac{5}{2}-\frac{\sqrt{21}}{2}+x^2} \, dx-\frac{1}{6} \left (3+\sqrt{21}\right ) \int \frac{1}{\frac{5}{2}+\frac{\sqrt{21}}{2}+x^2} \, dx\\ &=-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.12987, size = 87, normalized size = 1.74 \[ \frac{\left (7-\sqrt{21}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{5-\sqrt{21}}} x\right )}{\sqrt{42 \left (5-\sqrt{21}\right )}}+\frac{\left (-7-\sqrt{21}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{42 \left (5+\sqrt{21}\right )}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.058, size = 136, normalized size = 2.7 \begin{align*}{\frac{2\,\sqrt{21}}{6\,\sqrt{7}-6\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}-2\,\sqrt{3}}} \right ) }-2\,{\frac{1}{2\,\sqrt{7}-2\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}-2\,\sqrt{3}}} \right ) }-{\frac{2\,\sqrt{21}}{6\,\sqrt{7}+6\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}+2\,\sqrt{3}}} \right ) }-2\,{\frac{1}{2\,\sqrt{7}+2\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}+2\,\sqrt{3}}} \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} + 5 \, x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4109, size = 109, normalized size = 2.18 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x^{3} + 4 \, x\right )}\right ) - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.112025, size = 42, normalized size = 0.84 \begin{align*} - \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )} - 2 \operatorname{atan}{\left (\frac{\sqrt{3} x^{3}}{3} + \frac{4 \sqrt{3} x}{3} \right )}\right )}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1387, size = 35, normalized size = 0.7 \begin{align*} \frac{1}{6} \, \sqrt{3}{\left (\pi \mathrm{sgn}\left (x\right ) - 2 \, \arctan \left (\frac{\sqrt{3}{\left (x^{2} + 1\right )}}{3 \, x}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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