Optimal. Leaf size=90 \[ \frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3+x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-3-x^2-2 x^4}} \]
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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+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}{24} \left (12-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-2 x^4-x^2-3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1117
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3-x^2-2 x^4}} \, dx &=\frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3+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}{24} \left (12-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-3-x^2-2 x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.05, size = 142, normalized size = 1.58 \begin {gather*} -\frac {i \sqrt {1-\frac {4 x^2}{-1-i \sqrt {23}}} \sqrt {1-\frac {4 x^2}{-1+i \sqrt {23}}} F\left (i \sinh ^{-1}\left (2 \sqrt {-\frac {1}{-1-i \sqrt {23}}} x\right )|\frac {-1-i \sqrt {23}}{-1+i \sqrt {23}}\right )}{2 \sqrt {-\frac {1}{-1-i \sqrt {23}}} \sqrt {-3-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 = 0.04, size = 87, normalized size = 0.97
method | result | size |
default | \(\frac {6 \sqrt {1-\left (-\frac {1}{6}-\frac {i \sqrt {23}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}+\frac {i \sqrt {23}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-6-6 i \sqrt {23}}\, x}{6}, \frac {\sqrt {-33-3 i \sqrt {23}}}{6}\right )}{\sqrt {-6-6 i \sqrt {23}}\, \sqrt {-2 x^{4}-x^{2}-3}}\) | \(87\) |
elliptic | \(\frac {6 \sqrt {1-\left (-\frac {1}{6}-\frac {i \sqrt {23}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}+\frac {i \sqrt {23}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-6-6 i \sqrt {23}}\, x}{6}, \frac {\sqrt {-33-3 i \sqrt {23}}}{6}\right )}{\sqrt {-6-6 i \sqrt {23}}\, \sqrt {-2 x^{4}-x^{2}-3}}\) | \(87\) |
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 = 0.07, size = 41, normalized size = 0.46 \begin {gather*} \frac {1}{72} \, \sqrt {6} \sqrt {-3} {\left (\sqrt {-23} + 1\right )} \sqrt {\sqrt {-23} - 1} {\rm ellipticF}\left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {-23} - 1}, \frac {1}{12} \, \sqrt {-23} - \frac {11}{12}\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} - 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-x^2-3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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