Optimal. Leaf size=44 \[ \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-2+\sqrt {10}}} x\right )|\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{\sqrt {2+\sqrt {10}}} \]
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Rubi [A] time = 0.07, antiderivative size = 44, 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} \[ \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-2+\sqrt {10}}} x\right )|\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{\sqrt {2+\sqrt {10}}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-4 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {-4+2 \sqrt {10}-4 x^2} \sqrt {4+2 \sqrt {10}+4 x^2}} \, dx\\ &=\frac {F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-2+\sqrt {10}}} x\right )|\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{\sqrt {2+\sqrt {10}}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 51, normalized size = 1.16 \[ -\frac {i F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{2+\sqrt {10}}} x\right )|-\frac {7}{3}-\frac {2 \sqrt {10}}{3}\right )}{\sqrt {\sqrt {10}-2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} - 4 \, x^{2} + 3}}{2 \, x^{4} + 4 \, x^{2} - 3}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} - 4 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 84, normalized size = 1.91 \[ \frac {3 \sqrt {-\left (\frac {\sqrt {10}}{3}+\frac {2}{3}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {\sqrt {10}}{3}+\frac {2}{3}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {6+3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {15}}{3}-\frac {i \sqrt {6}}{3}\right )}{\sqrt {6+3 \sqrt {10}}\, \sqrt {-2 x^{4}-4 x^{2}+3}} \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} - 4 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {-2\,x^4-4\,x^2+3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 2 x^{4} - 4 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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