Optimal. Leaf size=96 \[ \sqrt{9+2 \sqrt{21}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right ),\frac{1}{2} \left (-5-\sqrt{21}\right )\right )-\sqrt{\frac{1}{2} \left (\sqrt{21}-3\right )} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5-\sqrt{21}\right )\right ) \]
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Rubi [A] time = 0.178087, antiderivative size = 96, 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} \[ \sqrt{9+2 \sqrt{21}} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5-\sqrt{21}\right )\right )-\sqrt{\frac{1}{2} \left (\sqrt{21}-3\right )} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5-\sqrt{21}\right )\right ) \]
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+3 x^2-x^4}} \, dx &=2 \int \frac{3-x^2}{\sqrt{3+\sqrt{21}-2 x^2} \sqrt{-3+\sqrt{21}+2 x^2}} \, dx\\ &=\left (3+\sqrt{21}\right ) \int \frac{1}{\sqrt{3+\sqrt{21}-2 x^2} \sqrt{-3+\sqrt{21}+2 x^2}} \, dx-\int \frac{\sqrt{-3+\sqrt{21}+2 x^2}}{\sqrt{3+\sqrt{21}-2 x^2}} \, dx\\ &=-\sqrt{\frac{1}{2} \left (-3+\sqrt{21}\right )} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5-\sqrt{21}\right )\right )+\frac{1}{2} \sqrt{36+8 \sqrt{21}} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5-\sqrt{21}\right )\right )\\ \end{align*}
Mathematica [C] time = 0.165314, size = 103, normalized size = 1.07 \[ -\frac{i \left (\left (3+\sqrt{21}\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{-3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5+\sqrt{21}\right )\right )-\left (\sqrt{21}-3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{\sqrt{21}-3}} x\right ),\frac{1}{2} \left (\sqrt{21}-5\right )\right )\right )}{\sqrt{2 \left (3+\sqrt{21}\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.296, size = 204, normalized size = 2.1 \begin{align*} 36\,{\frac{\sqrt{1- \left ( -1/2+1/6\,\sqrt{21} \right ){x}^{2}}\sqrt{1- \left ( -1/2-1/6\,\sqrt{21} \right ){x}^{2}} \left ({\it EllipticF} \left ( 1/6\,x\sqrt{-18+6\,\sqrt{21}},i/2\sqrt{3}+i/2\sqrt{7} \right ) -{\it EllipticE} \left ( 1/6\,x\sqrt{-18+6\,\sqrt{21}},i/2\sqrt{3}+i/2\sqrt{7} \right ) \right ) }{\sqrt{-18+6\,\sqrt{21}}\sqrt{-{x}^{4}+3\,{x}^{2}+3} \left ( 3+\sqrt{21} \right ) }}+18\,{\frac{\sqrt{1- \left ( -1/2+1/6\,\sqrt{21} \right ){x}^{2}}\sqrt{1- \left ( -1/2-1/6\,\sqrt{21} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-18+6\,\sqrt{21}},i/2\sqrt{3}+i/2\sqrt{7} \right ) }{\sqrt{-18+6\,\sqrt{21}}\sqrt{-{x}^{4}+3\,{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} + 3 \, 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} + 3 \, x^{2} + 3}{\left (x^{2} - 3\right )}}{x^{4} - 3 \, 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} + 3 x^{2} + 3}}\, dx - \int - \frac{3}{\sqrt{- x^{4} + 3 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} + 3 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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