Optimal. Leaf size=72 \[ \frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {3+2 x^4}} \]
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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 = {226}
\begin {gather*} \frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4+3}{\left (\sqrt {6} x^2+3\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {2 x^4+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+2 x^4}} \, dx &=\frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3+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}{2}\right )}{2 \sqrt [4]{6} \sqrt {3+2 x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.04, size = 25, normalized size = 0.35 \begin {gather*} -\sqrt [4]{-\frac {1}{6}} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {2}{3}} x\right )\right |-1\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.09, size = 66, normalized size = 0.92
method | result | size |
meijerg | \(\frac {\sqrt {3}\, x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], -\frac {2 x^{4}}{3}\right )}{3}\) | \(18\) |
default | \(\frac {\sqrt {3}\, \sqrt {9-3 i \sqrt {6}\, x^{2}}\, \sqrt {9+3 i \sqrt {6}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {3}\, \sqrt {i \sqrt {6}}}{3}, i\right )}{9 \sqrt {i \sqrt {6}}\, \sqrt {2 x^{4}+3}}\) | \(66\) |
elliptic | \(\frac {\sqrt {3}\, \sqrt {9-3 i \sqrt {6}\, x^{2}}\, \sqrt {9+3 i \sqrt {6}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {3}\, \sqrt {i \sqrt {6}}}{3}, i\right )}{9 \sqrt {i \sqrt {6}}\, \sqrt {2 x^{4}+3}}\) | \(66\) |
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 = 34, normalized size = 0.47 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \sqrt {-2} \sqrt {\sqrt {3} \sqrt {-2}} {\rm ellipticF}\left (\frac {1}{3} \, \sqrt {3} \sqrt {\sqrt {3} \sqrt {-2}} x, -1\right ) \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 = 0.33, size = 36, normalized size = 0.50 \begin {gather*} \frac {\sqrt {3} 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} e^{i \pi }}{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 = 0.09, size = 16, normalized size = 0.22 \begin {gather*} \frac {\sqrt {3}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ -\frac {2\,x^4}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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