Optimal. Leaf size=45 \[ \frac {\tan ^{-1}\left (\frac {2 x}{\sqrt {3-\sqrt {5}}}\right )}{\sqrt {10}}+\frac {\tan ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {5}}}\right )}{\sqrt {10}} \]
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Rubi [A] time = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\tan ^{-1}\left (\frac {2 x}{\sqrt {3-\sqrt {5}}}\right )}{\sqrt {10}}+\frac {\tan ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {5}}}\right )}{\sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 1163
Rubi steps
\begin {align*} \int \frac {1+2 x^2}{1+6 x^2+4 x^4} \, dx &=\frac {1}{5} \left (5-\sqrt {5}\right ) \int \frac {1}{3-\sqrt {5}+4 x^2} \, dx+\frac {1}{5} \left (5+\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 {10}}+\frac {\tan ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {5}}}\right )}{\sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 83, normalized size = 1.84 \begin {gather*} \frac {\left (\sqrt {5}-1\right ) \tan ^{-1}\left (\frac {2 x}{\sqrt {3-\sqrt {5}}}\right )}{2 \sqrt {5 \left (3-\sqrt {5}\right )}}+\frac {\left (1+\sqrt {5}\right ) \tan ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {5}}}\right )}{2 \sqrt {5 \left (3+\sqrt {5}\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x^2}{1+6 x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.80, size = 31, normalized size = 0.69 \begin {gather*} \frac {1}{10} \, \sqrt {10} \arctan \left (\frac {2}{5} \, \sqrt {10} {\left (x^{3} + 2 \, x\right )}\right ) + \frac {1}{10} \, \sqrt {10} \arctan \left (\frac {1}{5} \, \sqrt {10} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 39, normalized size = 0.87 \begin {gather*} \frac {1}{10} \, \sqrt {10} \arctan \left (\frac {4 \, x}{\sqrt {10} + \sqrt {2}}\right ) + \frac {1}{10} \, \sqrt {10} \arctan \left (\frac {4 \, x}{\sqrt {10} - \sqrt {2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 136, normalized size = 3.02 \begin {gather*} -\frac {2 \sqrt {5}\, \arctan \left (\frac {8 x}{2 \sqrt {10}-2 \sqrt {2}}\right )}{5 \left (2 \sqrt {10}-2 \sqrt {2}\right )}+\frac {2 \arctan \left (\frac {8 x}{2 \sqrt {10}-2 \sqrt {2}}\right )}{2 \sqrt {10}-2 \sqrt {2}}+\frac {2 \sqrt {5}\, \arctan \left (\frac {8 x}{2 \sqrt {10}+2 \sqrt {2}}\right )}{5 \left (2 \sqrt {10}+2 \sqrt {2}\right )}+\frac {2 \arctan \left (\frac {8 x}{2 \sqrt {10}+2 \sqrt {2}}\right )}{2 \sqrt {10}+2 \sqrt {2}} \end {gather*}
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 \begin {gather*} \int \frac {2 \, x^{2} + 1}{4 \, x^{4} + 6 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 29, normalized size = 0.64 \begin {gather*} \frac {\sqrt {10}\,\left (\mathrm {atan}\left (\frac {2\,\sqrt {10}\,x^3}{5}+\frac {4\,\sqrt {10}\,x}{5}\right )+\mathrm {atan}\left (\frac {\sqrt {10}\,x}{5}\right )\right )}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 42, normalized size = 0.93 \begin {gather*} \frac {\sqrt {10} \left (2 \operatorname {atan}{\left (\frac {\sqrt {10} x}{5} \right )} + 2 \operatorname {atan}{\left (\frac {2 \sqrt {10} x^{3}}{5} + \frac {4 \sqrt {10} x}{5} \right )}\right )}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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