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 = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 203
Rule 1163
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.40, size = 31, normalized size = 0.72 \[ \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} {\left (x^{3} + 5 \, x\right )}\right ) + \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 26, normalized size = 0.60 \[ \frac {1}{12} \, \sqrt {6} {\left (\pi \mathrm {sgn}\relax (x) + 2 \, \arctan \left (\frac {\sqrt {6} {\left (x^{2} - 1\right )}}{6 \, x}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 110, normalized size = 2.56 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {2 x}{\sqrt {6}-\sqrt {2}}\right )}{3 \left (\sqrt {6}-\sqrt {2}\right )}+\frac {\arctan \left (\frac {2 x}{\sqrt {6}-\sqrt {2}}\right )}{\sqrt {6}-\sqrt {2}}+\frac {\sqrt {3}\, \arctan \left (\frac {2 x}{\sqrt {6}+\sqrt {2}}\right )}{3 \sqrt {6}+3 \sqrt {2}}+\frac {\arctan \left (\frac {2 x}{\sqrt {6}+\sqrt {2}}\right )}{\sqrt {6}+\sqrt {2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} + 1}{x^{4} + 4 \, x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 29, normalized size = 0.67 \[ \frac {\sqrt {6}\,\left (\mathrm {atan}\left (\frac {\sqrt {6}\,x^3}{6}+\frac {5\,\sqrt {6}\,x}{6}\right )+\mathrm {atan}\left (\frac {\sqrt {6}\,x}{6}\right )\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 41, normalized size = 0.95 \[ \frac {\sqrt {6} \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{6} \right )} + 2 \operatorname {atan}{\left (\frac {\sqrt {6} x^{3}}{6} + \frac {5 \sqrt {6} x}{6} \right )}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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