Optimal. Leaf size=104 \[ \frac {\sqrt {\frac {3+\left (3-\sqrt {3}\right ) x^2}{3+\left (3+\sqrt {3}\right ) x^2}} \left (3+\left (3+\sqrt {3}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right )|-1+\sqrt {3}\right )}{\sqrt {3 \left (3+\sqrt {3}\right )} \sqrt {3+6 x^2+2 x^4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1113}
\begin {gather*} \frac {\sqrt {\frac {\left (3-\sqrt {3}\right ) x^2+3}{\left (3+\sqrt {3}\right ) x^2+3}} \left (\left (3+\sqrt {3}\right ) x^2+3\right ) F\left (\text {ArcTan}\left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right )|-1+\sqrt {3}\right )}{\sqrt {3 \left (3+\sqrt {3}\right )} \sqrt {2 x^4+6 x^2+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1113
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+6 x^2+2 x^4}} \, dx &=\frac {\sqrt {\frac {3+\left (3-\sqrt {3}\right ) x^2}{3+\left (3+\sqrt {3}\right ) x^2}} \left (3+\left (3+\sqrt {3}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right )|-1+\sqrt {3}\right )}{\sqrt {3 \left (3+\sqrt {3}\right )} \sqrt {3+6 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.04, size = 90, normalized size = 0.87 \begin {gather*} -\frac {i \sqrt {\frac {-3+\sqrt {3}-2 x^2}{-3+\sqrt {3}}} \sqrt {3+\sqrt {3}+2 x^2} F\left (i \sinh ^{-1}\left (\sqrt {1-\frac {1}{\sqrt {3}}} x\right )|2+\sqrt {3}\right )}{\sqrt {6+12 x^2+4 x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.06, size = 82, normalized size = 0.79
method | result | size |
default | \(\frac {3 \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}}\) | \(82\) |
elliptic | \(\frac {3 \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}}\) | \(82\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.08, size = 33, normalized size = 0.32 \begin {gather*} -\frac {1}{6} \, {\left (\sqrt {3} + 3\right )} \sqrt {\sqrt {3} - 3} {\rm ellipticF}\left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {3} - 3}, \sqrt {3} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 x^{4} + 6 x^{2} + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {2\,x^4+6\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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