Optimal. Leaf size=48 \[ \frac {1}{3} \sqrt {-x^4-2 x^2+3} x+\frac {4 F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}}-\frac {2 E\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1091, 1180, 524, 424, 419} \[ \frac {1}{3} \sqrt {-x^4-2 x^2+3} x+\frac {4 F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}}-\frac {2 E\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 1091
Rule 1180
Rubi steps
\begin {align*} \int \sqrt {3-2 x^2-x^4} \, dx &=\frac {1}{3} x \sqrt {3-2 x^2-x^4}+\frac {1}{3} \int \frac {6-2 x^2}{\sqrt {3-2 x^2-x^4}} \, dx\\ &=\frac {1}{3} x \sqrt {3-2 x^2-x^4}+\frac {2}{3} \int \frac {6-2 x^2}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx\\ &=\frac {1}{3} x \sqrt {3-2 x^2-x^4}-\frac {2}{3} \int \frac {\sqrt {6+2 x^2}}{\sqrt {2-2 x^2}} \, dx+8 \int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx\\ &=\frac {1}{3} x \sqrt {3-2 x^2-x^4}-\frac {2 E\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}}+\frac {4 F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 59, normalized size = 1.23 \[ \frac {1}{3} \left (\sqrt {-x^4-2 x^2+3} x-4 i F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )-2 i E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-x^{4} - 2 \, x^{2} + 3}, x\right ) \]
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 \[ \int \sqrt {-x^{4} - 2 \, x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 114, normalized size = 2.38 \[ \frac {\sqrt {-x^{4}-2 x^{2}+3}\, x}{3}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (-\EllipticE \left (x , \frac {i \sqrt {3}}{3}\right )+\EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}} \]
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 \[ \int \sqrt {-x^{4} - 2 \, x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {-x^4-2\,x^2+3} \,d x \]
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 \[ \int \sqrt {- x^{4} - 2 x^{2} + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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