Optimal. Leaf size=92 \[ \frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4+2 x^2+3}{\left (\sqrt {6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{12} \left (6-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {2 x^4+2 x^2+3}} \]
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Rubi [A] time = 0.01, antiderivative size = 92, 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 = {1103} \[ \frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4+2 x^2+3}{\left (\sqrt {6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{12} \left (6-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {2 x^4+2 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1103
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+2 x^2+2 x^4}} \, dx &=\frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3+2 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}{12} \left (6-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3+2 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 144, normalized size = 1.57 \[ -\frac {i \sqrt {1-\frac {2 x^2}{-1-i \sqrt {5}}} \sqrt {1-\frac {2 x^2}{-1+i \sqrt {5}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2}{-1-i \sqrt {5}}} x\right )|\frac {-1-i \sqrt {5}}{-1+i \sqrt {5}}\right )}{\sqrt {2} \sqrt {-\frac {1}{-1-i \sqrt {5}}} \sqrt {2 x^4+2 x^2+3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} + 2 \, x^{2} + 3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 2 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 87, normalized size = 0.95 \[ \frac {3 \sqrt {-\left (-\frac {1}{3}+\frac {i \sqrt {5}}{3}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {1}{3}-\frac {i \sqrt {5}}{3}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-3+3 i \sqrt {5}}\, x}{3}, \frac {\sqrt {-6+3 i \sqrt {5}}}{3}\right )}{\sqrt {-3+3 i \sqrt {5}}\, \sqrt {2 x^{4}+2 x^{2}+3}} \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 2 \, x^{2} + 3}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {2\,x^4+2\,x^2+3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {2 x^{4} + 2 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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