Optimal. Leaf size=48 \[ \sqrt {\frac {1}{6} \left (2+\sqrt {10}\right )} F\left (\sin ^{-1}\left (\sqrt {\frac {1}{2} \left (-2+\sqrt {10}\right )} x\right )|\frac {1}{3} \left (-7-2 \sqrt {10}\right )\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1095, 419} \[ \sqrt {\frac {1}{6} \left (2+\sqrt {10}\right )} F\left (\sin ^{-1}\left (\sqrt {\frac {1}{2} \left (-2+\sqrt {10}\right )} x\right )|\frac {1}{3} \left (-7-2 \sqrt {10}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+4 x^2-3 x^4}} \, dx &=\left (2 \sqrt {3}\right ) \int \frac {1}{\sqrt {4+2 \sqrt {10}-6 x^2} \sqrt {-4+2 \sqrt {10}+6 x^2}} \, dx\\ &=\sqrt {\frac {1}{6} \left (2+\sqrt {10}\right )} F\left (\sin ^{-1}\left (\sqrt {\frac {1}{2} \left (-2+\sqrt {10}\right )} x\right )|\frac {1}{3} \left (-7-2 \sqrt {10}\right )\right )\\ \end {align*}
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Mathematica [C] time = 0.06, size = 49, normalized size = 1.02 \[ -\frac {i F\left (i \sinh ^{-1}\left (\sqrt {1+\sqrt {\frac {5}{2}}} x\right )|\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{\sqrt {2+\sqrt {10}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-3 \, x^{4} + 4 \, x^{2} + 2}}{3 \, x^{4} - 4 \, x^{2} - 2}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-3 \, x^{4} + 4 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 84, normalized size = 1.75 \[ \frac {2 \sqrt {-\left (-1+\frac {\sqrt {10}}{2}\right ) x^{2}+1}\, \sqrt {-\left (-1-\frac {\sqrt {10}}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-4+2 \sqrt {10}}\, x}{2}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{\sqrt {-4+2 \sqrt {10}}\, \sqrt {-3 x^{4}+4 x^{2}+2}} \]
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 \[ \int \frac {1}{\sqrt {-3 \, x^{4} + 4 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {-3\,x^4+4\,x^2+2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 3 x^{4} + 4 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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