3.122 \(\int \frac{3-x^2}{\sqrt{3+2 x^2-x^4}} \, dx\)

Optimal. Leaf size=25 \[ 4 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )-E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right ) \]

[Out]

-EllipticE[ArcSin[x/Sqrt[3]], -3] + 4*EllipticF[ArcSin[x/Sqrt[3]], -3]

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Rubi [A]  time = 0.0990293, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ 4 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )-E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right ) \]

Antiderivative was successfully verified.

[In]  Int[(3 - x^2)/Sqrt[3 + 2*x^2 - x^4],x]

[Out]

-EllipticE[ArcSin[x/Sqrt[3]], -3] + 4*EllipticF[ArcSin[x/Sqrt[3]], -3]

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Rubi in Sympy [A]  time = 18.821, size = 27, normalized size = 1.08 \[ - E\left (\operatorname{asin}{\left (\frac{\sqrt{3} x}{3} \right )}\middle | -3\right ) + 4 F\left (\operatorname{asin}{\left (\frac{\sqrt{3} x}{3} \right )}\middle | -3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+3)/(-x**4+2*x**2+3)**(1/2),x)

[Out]

-elliptic_e(asin(sqrt(3)*x/3), -3) + 4*elliptic_f(asin(sqrt(3)*x/3), -3)

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Mathematica [C]  time = 0.0434694, size = 19, normalized size = 0.76 \[ -i \sqrt{3} E\left (i \sinh ^{-1}(x)|-\frac{1}{3}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(3 - x^2)/Sqrt[3 + 2*x^2 - x^4],x]

[Out]

(-I)*Sqrt[3]*EllipticE[I*ArcSinh[x], -1/3]

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Maple [B]  time = 0.019, size = 113, normalized size = 4.5 \[{\frac{\sqrt{3}}{3}\sqrt{-3\,{x}^{2}+9}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}}{3}},i\sqrt{3} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{3}}{3}},i\sqrt{3} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+2\,{x}^{2}+3}}}}+{\sqrt{3}\sqrt{-3\,{x}^{2}+9}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{3}}{3}},i\sqrt{3} \right ){\frac{1}{\sqrt{-{x}^{4}+2\,{x}^{2}+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+3)/(-x^4+2*x^2+3)^(1/2),x)

[Out]

1/3*3^(1/2)*(-3*x^2+9)^(1/2)*(x^2+1)^(1/2)/(-x^4+2*x^2+3)^(1/2)*(EllipticF(1/3*x
*3^(1/2),I*3^(1/2))-EllipticE(1/3*x*3^(1/2),I*3^(1/2)))+3^(1/2)*(-3*x^2+9)^(1/2)
*(x^2+1)^(1/2)/(-x^4+2*x^2+3)^(1/2)*EllipticF(1/3*x*3^(1/2),I*3^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} - 3}{\sqrt{-x^{4} + 2 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 3)/sqrt(-x^4 + 2*x^2 + 3),x, algorithm="maxima")

[Out]

-integrate((x^2 - 3)/sqrt(-x^4 + 2*x^2 + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x^{2} - 3}{\sqrt{-x^{4} + 2 \, x^{2} + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 3)/sqrt(-x^4 + 2*x^2 + 3),x, algorithm="fricas")

[Out]

integral(-(x^2 - 3)/sqrt(-x^4 + 2*x^2 + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x^{2}}{\sqrt{- x^{4} + 2 x^{2} + 3}}\, dx - \int \left (- \frac{3}{\sqrt{- x^{4} + 2 x^{2} + 3}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+3)/(-x**4+2*x**2+3)**(1/2),x)

[Out]

-Integral(x**2/sqrt(-x**4 + 2*x**2 + 3), x) - Integral(-3/sqrt(-x**4 + 2*x**2 +
3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{2} - 3}{\sqrt{-x^{4} + 2 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 3)/sqrt(-x^4 + 2*x^2 + 3),x, algorithm="giac")

[Out]

integrate(-(x^2 - 3)/sqrt(-x^4 + 2*x^2 + 3), x)