3.33 \(\int \frac{1}{\sqrt{3+x^2-2 x^4}} \, dx\)

Optimal. Leaf size=20 \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right )|-\frac{3}{2}\right )}{\sqrt{2}} \]

[Out]

EllipticF[ArcSin[Sqrt[2/3]*x], -3/2]/Sqrt[2]

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Rubi [A]  time = 0.0399358, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right )|-\frac{3}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[3 + x^2 - 2*x^4],x]

[Out]

EllipticF[ArcSin[Sqrt[2/3]*x], -3/2]/Sqrt[2]

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Rubi in Sympy [A]  time = 7.2481, size = 20, normalized size = 1. \[ \frac{\sqrt{2} F\left (\operatorname{asin}{\left (\frac{\sqrt{6} x}{3} \right )}\middle | - \frac{3}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*elliptic_f(asin(sqrt(6)*x/3), -3/2)/2

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Mathematica [A]  time = 0.0455195, size = 20, normalized size = 1. \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right )|-\frac{3}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[3 + x^2 - 2*x^4],x]

[Out]

EllipticF[ArcSin[Sqrt[2/3]*x], -3/2]/Sqrt[2]

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Maple [B]  time = 0.034, size = 47, normalized size = 2.4 \[{\frac{\sqrt{6}}{6}\sqrt{-6\,{x}^{2}+9}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{6}}{3}},{\frac{i}{2}}\sqrt{6} \right ){\frac{1}{\sqrt{-2\,{x}^{4}+{x}^{2}+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-2*x^4+x^2+3)^(1/2),x)

[Out]

1/6*6^(1/2)*(-6*x^2+9)^(1/2)*(x^2+1)^(1/2)/(-2*x^4+x^2+3)^(1/2)*EllipticF(1/3*x*
6^(1/2),1/2*I*6^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-2 \, x^{4} + x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-2*x^4 + x^2 + 3),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-2*x^4 + x^2 + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-2 \, x^{4} + x^{2} + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-2*x^4 + x^2 + 3),x, algorithm="fricas")

[Out]

integral(1/sqrt(-2*x^4 + x^2 + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 2 x^{4} + x^{2} + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(-2*x**4 + x**2 + 3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-2 \, x^{4} + x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-2*x^4 + x^2 + 3),x, algorithm="giac")

[Out]

integrate(1/sqrt(-2*x^4 + x^2 + 3), x)