3.108 \(\int \frac{1}{\sqrt{-2+6 x^2-3 x^4}} \, dx\)

Optimal. Leaf size=42 \[ -\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{3}{3+\sqrt{3}}} x\right )|\frac{1}{2} \left (1+\sqrt{3}\right )\right )}{\sqrt{2} \sqrt [4]{3}} \]

[Out]

-(EllipticF[ArcCos[Sqrt[3/(3 + Sqrt[3])]*x], (1 + Sqrt[3])/2]/(Sqrt[2]*3^(1/4)))

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Rubi [A]  time = 0.13712, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{3}{3+\sqrt{3}}} x\right )|\frac{1}{2} \left (1+\sqrt{3}\right )\right )}{\sqrt{2} \sqrt [4]{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-2 + 6*x^2 - 3*x^4],x]

[Out]

-(EllipticF[ArcCos[Sqrt[3/(3 + Sqrt[3])]*x], (1 + Sqrt[3])/2]/(Sqrt[2]*3^(1/4)))

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Rubi in Sympy [A]  time = 27.4348, size = 63, normalized size = 1.5 \[ - \frac{\sqrt{2} \sqrt [4]{3} F\left (\operatorname{acos}{\left (\frac{\sqrt{2} x \sqrt{- \sqrt{3} + 3}}{2} \right )}\middle | \frac{1}{2} + \frac{\sqrt{3}}{2}\right )}{\sqrt{- \sqrt{3} + 3} \sqrt{2 \sqrt{3} + 6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-3*x**4+6*x**2-2)**(1/2),x)

[Out]

-sqrt(2)*3**(1/4)*elliptic_f(acos(sqrt(2)*x*sqrt(-sqrt(3) + 3)/2), 1/2 + sqrt(3)
/2)/(sqrt(-sqrt(3) + 3)*sqrt(2*sqrt(3) + 6))

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Mathematica [B]  time = 0.145924, size = 85, normalized size = 2.02 \[ \frac{\sqrt{-3 x^2-\sqrt{3}+3} \sqrt{\left (\sqrt{3}-3\right ) x^2+2} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{3}\right )} x\right )|2-\sqrt{3}\right )}{\sqrt{6} \sqrt{-3 x^4+6 x^2-2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[-2 + 6*x^2 - 3*x^4],x]

[Out]

(Sqrt[3 - Sqrt[3] - 3*x^2]*Sqrt[2 + (-3 + Sqrt[3])*x^2]*EllipticF[ArcSin[Sqrt[(3
+ Sqrt[3])/2]*x], 2 - Sqrt[3]])/(Sqrt[6]*Sqrt[-2 + 6*x^2 - 3*x^4])

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Maple [A]  time = 0.099, size = 82, normalized size = 2. \[ 2\,{\frac{\sqrt{1- \left ( -1/2\,\sqrt{3}+3/2 \right ){x}^{2}}\sqrt{1- \left ( 1/2\,\sqrt{3}+3/2 \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{6-2\,\sqrt{3}}x,1/2\,\sqrt{6}+1/2\,\sqrt{2} \right ) }{\sqrt{6-2\,\sqrt{3}}\sqrt{-3\,{x}^{4}+6\,{x}^{2}-2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^4+6*x^2-2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-3 \, x^{4} + 6 \, x^{2} - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-3*x^4 + 6*x^2 - 2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-3*x^4 + 6*x^2 - 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-3 \, x^{4} + 6 \, x^{2} - 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-3*x^4 + 6*x^2 - 2),x, algorithm="fricas")

[Out]

integral(1/sqrt(-3*x^4 + 6*x^2 - 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 3 x^{4} + 6 x^{2} - 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-3*x**4+6*x**2-2)**(1/2),x)

[Out]

Integral(1/sqrt(-3*x**4 + 6*x**2 - 2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-3 \, x^{4} + 6 \, x^{2} - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-3*x^4 + 6*x^2 - 2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-3*x^4 + 6*x^2 - 2), x)