Optimal. Leaf size=25 \[ 4 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )-E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1180, 21, 423, 424, 419} \[ 4 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )-E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right ) \]
Antiderivative was successfully verified.
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Rule 21
Rule 419
Rule 423
Rule 424
Rule 1180
Rubi steps
\begin {align*} \int \frac {3-x^2}{\sqrt {3+2 x^2-x^4}} \, dx &=2 \int \frac {3-x^2}{\sqrt {6-2 x^2} \sqrt {2+2 x^2}} \, dx\\ &=\int \frac {\sqrt {6-2 x^2}}{\sqrt {2+2 x^2}} \, dx\\ &=8 \int \frac {1}{\sqrt {6-2 x^2} \sqrt {2+2 x^2}} \, dx-\int \frac {\sqrt {2+2 x^2}}{\sqrt {6-2 x^2}} \, dx\\ &=-E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )+4 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )\\ \end {align*}
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Mathematica [C] time = 0.06, size = 19, normalized size = 0.76 \[ -i \sqrt {3} E\left (i \sinh ^{-1}(x)|-\frac {1}{3}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} + 2 \, x^{2} + 3}}{x^{2} + 1}, x\right ) \]
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 \[ \int -\frac {x^{2} - 3}{\sqrt {-x^{4} + 2 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 113, normalized size = 4.52 \[ \frac {\sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {\sqrt {3}\, x}{3}, i \sqrt {3}\right )}{\sqrt {-x^{4}+2 x^{2}+3}}+\frac {\sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {3}\, x}{3}, i \sqrt {3}\right )+\EllipticF \left (\frac {\sqrt {3}\, x}{3}, i \sqrt {3}\right )\right )}{3 \sqrt {-x^{4}+2 x^{2}+3}} \]
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 \[ -\int \frac {x^{2} - 3}{\sqrt {-x^{4} + 2 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int -\frac {x^2-3}{\sqrt {-x^4+2\,x^2+3}} \,d x \]
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 \[ - \int \frac {x^{2}}{\sqrt {- x^{4} + 2 x^{2} + 3}}\, dx - \int \left (- \frac {3}{\sqrt {- x^{4} + 2 x^{2} + 3}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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