Optimal. Leaf size=72 \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {-3 x^4-2}} \]
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Rubi [A] time = 0.01, antiderivative size = 72, 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 = {220} \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {-3 x^4-2}} \]
Antiderivative was successfully verified.
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Rule 220
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-3 x^4}} \, dx &=\frac {\left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {-2-3 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 47, normalized size = 0.65 \[ -\frac {\sqrt [4]{-\frac {1}{6}} \sqrt {3 x^4+2} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {3}{2}} x\right )\right |-1\right )}{\sqrt {-3 x^4-2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-3 \, x^{4} - 2}}{3 \, x^{4} + 2}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-3 \, x^{4} - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 66, normalized size = 0.92 \[ \frac {\sqrt {2}\, \sqrt {2 i \sqrt {6}\, x^{2}+4}\, \sqrt {-2 i \sqrt {6}\, x^{2}+4}\, \EllipticF \left (\frac {\sqrt {2}\, \sqrt {-i \sqrt {6}}\, x}{2}, i\right )}{4 \sqrt {-i \sqrt {6}}\, \sqrt {-3 x^{4}-2}} \]
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 \[ \int \frac {1}{\sqrt {-3 \, x^{4} - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.18, size = 31, normalized size = 0.43 \[ \frac {x\,\sqrt {6\,x^4+4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ -\frac {3\,x^4}{2}\right )}{2\,\sqrt {-3\,x^4-2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.70, size = 39, normalized size = 0.54 \[ - \frac {\sqrt {2} i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {3 x^{4} e^{i \pi }}{2}} \right )}}{8 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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