Optimal. Leaf size=27 \[ 2 \sqrt{3} \text{EllipticF}\left (\sin ^{-1}(x),-\frac{1}{3}\right )-\sqrt{3} E\left (\sin ^{-1}(x)|-\frac{1}{3}\right ) \]
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Rubi [A] time = 0.0387074, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1180, 524, 424, 419} \[ 2 \sqrt{3} F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )-\sqrt{3} E\left (\sin ^{-1}(x)|-\frac{1}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 1180
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{3-x^2}{\sqrt{3-2 x^2-x^4}} \, dx &=2 \int \frac{3-x^2}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx\\ &=12 \int \frac{1}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx-\int \frac{\sqrt{6+2 x^2}}{\sqrt{2-2 x^2}} \, dx\\ &=-\sqrt{3} E\left (\sin ^{-1}(x)|-\frac{1}{3}\right )+2 \sqrt{3} F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )\\ \end{align*}
Mathematica [C] time = 0.0641494, size = 35, normalized size = 1.3 \[ -i \left (2 \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right ),-3\right )+E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 95, normalized size = 3.5 \begin{align*}{({\it EllipticF} \left ( x,{\frac{i}{3}}\sqrt{3} \right ) -{\it EllipticE} \left ( x,{\frac{i}{3}}\sqrt{3} \right ) )\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}}+{{\it EllipticF} \left ( x,{\frac{i}{3}}\sqrt{3} \right ) \sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2} - 3}{\sqrt{-x^{4} - 2 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-x^{4} - 2 \, x^{2} + 3}{\left (x^{2} - 3\right )}}{x^{4} + 2 \, x^{2} - 3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{2}}{\sqrt{- x^{4} - 2 x^{2} + 3}}\, dx - \int - \frac{3}{\sqrt{- x^{4} - 2 x^{2} + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{2} - 3}{\sqrt{-x^{4} - 2 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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