Optimal. Leaf size=90 \[ \frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3-2 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{12} \left (6+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3-2 x^2+2 x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 90, 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 = {1117}
\begin {gather*} \frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4-2 x^2+3}{\left (\sqrt {6} x^2+3\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{12} \left (6+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {2 x^4-2 x^2+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1117
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-2 x^2+2 x^4}} \, dx &=\frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3-2 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{12} \left (6+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3-2 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.06, size = 144, normalized size = 1.60 \begin {gather*} -\frac {i \sqrt {1-\frac {2 x^2}{1-i \sqrt {5}}} \sqrt {1-\frac {2 x^2}{1+i \sqrt {5}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2}{1-i \sqrt {5}}} x\right )|\frac {1-i \sqrt {5}}{1+i \sqrt {5}}\right )}{\sqrt {2} \sqrt {-\frac {1}{1-i \sqrt {5}}} \sqrt {3-2 x^2+2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.05, size = 87, normalized size = 0.97
method | result | size |
default | \(\frac {3 \sqrt {1-\left (\frac {1}{3}+\frac {i \sqrt {5}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{3}-\frac {i \sqrt {5}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {3+3 i \sqrt {5}}}{3}, \frac {\sqrt {-6-3 i \sqrt {5}}}{3}\right )}{\sqrt {3+3 i \sqrt {5}}\, \sqrt {2 x^{4}-2 x^{2}+3}}\) | \(87\) |
elliptic | \(\frac {3 \sqrt {1-\left (\frac {1}{3}+\frac {i \sqrt {5}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{3}-\frac {i \sqrt {5}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {3+3 i \sqrt {5}}}{3}, \frac {\sqrt {-6-3 i \sqrt {5}}}{3}\right )}{\sqrt {3+3 i \sqrt {5}}\, \sqrt {2 x^{4}-2 x^{2}+3}}\) | \(87\) |
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 = 0.09, size = 35, normalized size = 0.39 \begin {gather*} -\frac {1}{6} \, \sqrt {\sqrt {-5} + 1} {\left (\sqrt {-5} - 1\right )} {\rm ellipticF}\left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {-5} + 1}, -\frac {1}{3} \, \sqrt {-5} - \frac {2}{3}\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} - 2 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^4-2\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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