Optimal. Leaf size=88 \[ \frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3+x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3+x^2+2 x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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 = 140, normalized size = 1.59 \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 = 0.05, size = 85, normalized size = 0.97
method | result | size |
default | \(\frac {6 \sqrt {1-\left (-\frac {1}{6}+\frac {i \sqrt {23}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}-\frac {i \sqrt {23}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-6+6 i \sqrt {23}}}{6}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )}{\sqrt {-6+6 i \sqrt {23}}\, \sqrt {2 x^{4}+x^{2}+3}}\) | \(85\) |
elliptic | \(\frac {6 \sqrt {1-\left (-\frac {1}{6}+\frac {i \sqrt {23}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}-\frac {i \sqrt {23}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-6+6 i \sqrt {23}}}{6}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )}{\sqrt {-6+6 i \sqrt {23}}\, \sqrt {2 x^{4}+x^{2}+3}}\) | \(85\) |
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 = 0.08, size = 41, normalized size = 0.47 \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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