Optimal. Leaf size=49 \[ \frac {\tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt {5}}+\frac {\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt {5}} \]
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Rubi [A] time = 0.06, antiderivative size = 49, 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} \begin {gather*} \frac {\tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt {5}}+\frac {\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 1163
Rubi steps
\begin {align*} \int \frac {1+x^2}{1+3 x^2+x^4} \, dx &=\frac {1}{10} \left (5-\sqrt {5}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx+\frac {1}{10} \left (5+\sqrt {5}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx\\ &=\frac {\tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt {5}}+\frac {\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 83, normalized size = 1.69 \begin {gather*} \frac {\left (\sqrt {5}-1\right ) \tan ^{-1}\left (\sqrt {\frac {2}{3-\sqrt {5}}} x\right )}{\sqrt {10 \left (3-\sqrt {5}\right )}}+\frac {\left (1+\sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt {10 \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+x^2}{1+3 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.94, size = 31, normalized size = 0.63 \begin {gather*} \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (x^{3} + 4 \, x\right )}\right ) + \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 26, normalized size = 0.53 \begin {gather*} \frac {1}{10} \, \sqrt {5} {\left (\pi \mathrm {sgn}\relax (x) + 2 \, \arctan \left (\frac {\sqrt {5} {\left (x^{2} - 1\right )}}{5 \, x}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 104, normalized size = 2.12 \begin {gather*} -\frac {2 \sqrt {5}\, \arctan \left (\frac {4 x}{2 \sqrt {5}-2}\right )}{5 \left (2 \sqrt {5}-2\right )}+\frac {2 \arctan \left (\frac {4 x}{2 \sqrt {5}-2}\right )}{2 \sqrt {5}-2}+\frac {2 \sqrt {5}\, \arctan \left (\frac {4 x}{2 \sqrt {5}+2}\right )}{5 \left (2 \sqrt {5}+2\right )}+\frac {2 \arctan \left (\frac {4 x}{2 \sqrt {5}+2}\right )}{2 \sqrt {5}+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 {x^{2} + 1}{x^{4} + 3 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.39, size = 29, normalized size = 0.59 \begin {gather*} \frac {\sqrt {5}\,\left (\mathrm {atan}\left (\frac {\sqrt {5}\,x^3}{5}+\frac {4\,\sqrt {5}\,x}{5}\right )+\mathrm {atan}\left (\frac {\sqrt {5}\,x}{5}\right )\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 41, normalized size = 0.84 \begin {gather*} \frac {\sqrt {5} \left (2 \operatorname {atan}{\left (\frac {\sqrt {5} x}{5} \right )} + 2 \operatorname {atan}{\left (\frac {\sqrt {5} x^{3}}{5} + \frac {4 \sqrt {5} x}{5} \right )}\right )}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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