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

Optimal. Leaf size=44 \[ -\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{1}{3} \left (3-\sqrt{3}\right )} x\right )|\frac{1}{2} \left (1+\sqrt{3}\right )\right )}{\sqrt{2} \sqrt [4]{3}} \]

[Out]

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

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Rubi [A]  time = 0.156819, antiderivative size = 44, 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 \[ -\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{1}{3} \left (3-\sqrt{3}\right )} x\right )|\frac{1}{2} \left (1+\sqrt{3}\right )\right )}{\sqrt{2} \sqrt [4]{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.6694, size = 63, normalized size = 1.43 \[ - \frac{\sqrt{2} \sqrt [4]{3} F\left (\operatorname{acos}{\left (\frac{\sqrt{3} x \sqrt{- \sqrt{3} + 3}}{3} \right )}\middle | \frac{1}{2} + \frac{\sqrt{3}}{2}\right )}{\sqrt{- \sqrt{3} + 3} \sqrt{2 \sqrt{3} + 6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0631317, size = 81, normalized size = 1.84 \[ \frac{\sqrt{-2 x^2-\sqrt{3}+3} \sqrt{\left (\sqrt{3}-3\right ) x^2+3} F\left (\sin ^{-1}\left (\sqrt{1+\frac{1}{\sqrt{3}}} x\right )|2-\sqrt{3}\right )}{\sqrt{6} \sqrt{-2 x^4+6 x^2-3}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 - Sqrt[3] - 2*x^2]*Sqrt[3 + (-3 + Sqrt[3])*x^2]*EllipticF[ArcSin[Sqrt[1
+ 1/Sqrt[3]]*x], 2 - Sqrt[3]])/(Sqrt[6]*Sqrt[-3 + 6*x^2 - 2*x^4])

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Maple [A]  time = 0.096, size = 82, normalized size = 1.9 \[ 3\,{\frac{\sqrt{1- \left ( 1-1/3\,\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( 1+1/3\,\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{9-3\,\sqrt{3}},1/2\,\sqrt{6}+1/2\,\sqrt{2} \right ) }{\sqrt{9-3\,\sqrt{3}}\sqrt{-2\,{x}^{4}+6\,{x}^{2}-3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-2*x^4+6*x^2-3)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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