Optimal. Leaf size=112 \[ \frac {\sqrt {-3+\sqrt {6} x^2} \sqrt {\frac {3+\sqrt {6} x^2}{3-\sqrt {6} x^2}} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {6} x^2}}\right )|\frac {1}{2}\right )}{6^{3/4} \sqrt {\frac {1}{3-\sqrt {6} x^2}} \sqrt {-3+2 x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {229}
\begin {gather*} \frac {\sqrt {\sqrt {6} x^2-3} \sqrt {\frac {\sqrt {6} x^2+3}{3-\sqrt {6} x^2}} F\left (\text {ArcSin}\left (\frac {2^{3/4} \sqrt [4]{3} x}{\sqrt {\sqrt {6} x^2-3}}\right )|\frac {1}{2}\right )}{6^{3/4} \sqrt {\frac {1}{3-\sqrt {6} x^2}} \sqrt {2 x^4-3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 229
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3+2 x^4}} \, dx &=\frac {\sqrt {-3+\sqrt {6} x^2} \sqrt {\frac {3+\sqrt {6} x^2}{3-\sqrt {6} x^2}} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {6} x^2}}\right )|\frac {1}{2}\right )}{6^{3/4} \sqrt {\frac {1}{3-\sqrt {6} x^2}} \sqrt {-3+2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 10.04, size = 40, normalized size = 0.36 \begin {gather*} \frac {\sqrt {3-2 x^4} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )\right |-1\right )}{\sqrt [4]{6} \sqrt {-3+2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.12, size = 56, normalized size = 0.50
method | result | size |
meijerg | \(\frac {\sqrt {3}\, \sqrt {-\mathrm {signum}\left (-1+\frac {2 x^{4}}{3}\right )}\, x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], \frac {2 x^{4}}{3}\right )}{3 \sqrt {\mathrm {signum}\left (-1+\frac {2 x^{4}}{3}\right )}}\) | \(40\) |
default | \(\frac {\sqrt {9+3 x^{2} \sqrt {6}}\, \sqrt {9-3 x^{2} \sqrt {6}}\, \EllipticF \left (\frac {\sqrt {-3 \sqrt {6}}\, x}{3}, i\right )}{3 \sqrt {-3 \sqrt {6}}\, \sqrt {2 x^{4}-3}}\) | \(56\) |
elliptic | \(\frac {\sqrt {9+3 x^{2} \sqrt {6}}\, \sqrt {9-3 x^{2} \sqrt {6}}\, \EllipticF \left (\frac {\sqrt {-3 \sqrt {6}}\, x}{3}, i\right )}{3 \sqrt {-3 \sqrt {6}}\, \sqrt {2 x^{4}-3}}\) | \(56\) |
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 [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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Sympy [C] Result contains complex when optimal does not.
time = 0.34, size = 34, normalized size = 0.30 \begin {gather*} - \frac {\sqrt {3} i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {2 x^{4}}{3}} \right )}}{12 \Gamma \left (\frac {5}{4}\right )} \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 [B]
time = 0.08, size = 31, normalized size = 0.28 \begin {gather*} \frac {x\,\sqrt {9-6\,x^4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ \frac {2\,x^4}{3}\right )}{3\,\sqrt {2\,x^4-3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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