Optimal. Leaf size=48 \[ \frac{1}{3} \sqrt{-x^4-2 x^2+3} x+\frac{4 F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}-\frac{2 E\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
[Out]
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Rubi [A] time = 0.139859, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{1}{3} \sqrt{-x^4-2 x^2+3} x+\frac{4 F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}-\frac{2 E\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[3 - 2*x^2 - x^4],x]
[Out]
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Rubi in Sympy [A] time = 22.9189, size = 49, normalized size = 1.02 \[ \frac{x \sqrt{- x^{4} - 2 x^{2} + 3}}{3} - \frac{2 \sqrt{3} E\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{3}\right )}{3} + \frac{4 \sqrt{3} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**4-2*x**2+3)**(1/2),x)
[Out]
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Mathematica [C] time = 0.10745, 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.
[In] Integrate[Sqrt[3 - 2*x^2 - x^4],x]
[Out]
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Maple [B] time = 0.019, size = 114, normalized size = 2.4 \[{\frac{x}{3}\sqrt{-{x}^{4}-2\,{x}^{2}+3}}+{\frac{2\,{\it EllipticF} \left ( x,i/3\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}}+{\frac{2\,{\it EllipticF} \left ( x,i/3\sqrt{3} \right ) -2\,{\it EllipticE} \left ( x,i/3\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^4-2*x^2+3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} - 2 \, x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 - 2*x^2 + 3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{-x^{4} - 2 \, x^{2} + 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 - 2*x^2 + 3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- x^{4} - 2 x^{2} + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**4-2*x**2+3)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} - 2 \, x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 - 2*x^2 + 3),x, algorithm="giac")
[Out]