Optimal. Leaf size=74 \[ \frac {\left (3+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {10}}-\frac {1}{10} \sqrt {180-80 \sqrt {5}} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 74, 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 = {1166, 203} \begin {gather*} \frac {\left (3+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {10}}-\frac {1}{10} \sqrt {180-80 \sqrt {5}} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 1166
Rubi steps
\begin {align*} \int \frac {3+x^2}{1+3 x^2+x^4} \, dx &=\frac {1}{10} \left (5-3 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx+\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx\\ &=-\frac {1}{5} \sqrt {45-20 \sqrt {5}} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )+\frac {\left (3+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 73, normalized size = 0.99 \begin {gather*} \frac {\left (3+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )-\left (3-\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {10}} \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 {3+x^2}{1+3 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.05, size = 137, normalized size = 1.85 \begin {gather*} \frac {2}{5} \, \sqrt {5} \sqrt {-4 \, \sqrt {5} + 9} \arctan \left (\frac {1}{4} \, \sqrt {2 \, x^{2} + \sqrt {5} + 3} {\left (\sqrt {5} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {5} + 9} - \frac {1}{2} \, {\left (\sqrt {5} x + 3 \, x\right )} \sqrt {-4 \, \sqrt {5} + 9}\right ) + \frac {2}{5} \, \sqrt {5} \sqrt {4 \, \sqrt {5} + 9} \arctan \left (\frac {1}{4} \, {\left (\sqrt {2 \, x^{2} - \sqrt {5} + 3} {\left (\sqrt {5} \sqrt {2} - 3 \, \sqrt {2}\right )} - 2 \, \sqrt {5} x + 6 \, x\right )} \sqrt {4 \, \sqrt {5} + 9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 41, normalized size = 0.55 \begin {gather*} \frac {1}{5} \, {\left (2 \, \sqrt {5} - 5\right )} \arctan \left (\frac {2 \, x}{\sqrt {5} + 1}\right ) + \frac {1}{5} \, {\left (2 \, \sqrt {5} + 5\right )} \arctan \left (\frac {2 \, x}{\sqrt {5} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 104, normalized size = 1.41 \begin {gather*} \frac {2 \arctan \left (\frac {4 x}{2 \sqrt {5}-2}\right )}{2 \sqrt {5}-2}+\frac {6 \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 {6 \sqrt {5}\, \arctan \left (\frac {4 x}{2 \sqrt {5}+2}\right )}{5 \left (2 \sqrt {5}+2\right )} \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} + 3}{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 = 0.11, size = 117, normalized size = 1.58 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {80\,x\,\sqrt {\frac {\sqrt {5}}{5}-\frac {9}{20}}}{24\,\sqrt {5}-56}-\frac {48\,\sqrt {5}\,x\,\sqrt {\frac {\sqrt {5}}{5}-\frac {9}{20}}}{24\,\sqrt {5}-56}\right )\,\sqrt {\frac {\sqrt {5}}{5}-\frac {9}{20}}-2\,\mathrm {atanh}\left (\frac {80\,x\,\sqrt {-\frac {\sqrt {5}}{5}-\frac {9}{20}}}{24\,\sqrt {5}+56}+\frac {48\,\sqrt {5}\,x\,\sqrt {-\frac {\sqrt {5}}{5}-\frac {9}{20}}}{24\,\sqrt {5}+56}\right )\,\sqrt {-\frac {\sqrt {5}}{5}-\frac {9}{20}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 46, normalized size = 0.62 \begin {gather*} 2 \left (\frac {\sqrt {5}}{5} + \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {2 x}{-1 + \sqrt {5}} \right )} - 2 \left (\frac {1}{2} - \frac {\sqrt {5}}{5}\right ) \operatorname {atan}{\left (\frac {2 x}{1 + \sqrt {5}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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