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

Optimal. Leaf size=88 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4-4 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{2}+\frac{1}{\sqrt{6}}\right )}{2 \sqrt [4]{6} \sqrt{-3 x^4+4 x^2-2}} \]

[Out]

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

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Rubi [A]  time = 0.0573656, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4-4 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{2}+\frac{1}{\sqrt{6}}\right )}{2 \sqrt [4]{6} \sqrt{-3 x^4+4 x^2-2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.86046, size = 90, normalized size = 1.02 \[ \frac{6^{\frac{3}{4}} \sqrt{- \frac{- 3 x^{4} + 4 x^{2} - 2}{\left (\frac{\sqrt{6} x^{2}}{2} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{3} x}{2} \right )}\middle | \frac{\sqrt{6}}{6} + \frac{1}{2}\right )}{12 \sqrt{- 3 x^{4} + 4 x^{2} - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6**(3/4)*sqrt(-(-3*x**4 + 4*x**2 - 2)/(sqrt(6)*x**2/2 + 1)**2)*(sqrt(6)*x**2/2 +
1)*elliptic_f(2*atan(2**(3/4)*3**(1/4)*x/2), sqrt(6)/6 + 1/2)/(12*sqrt(-3*x**4
+ 4*x**2 - 2))

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Mathematica [C]  time = 0.14756, size = 144, normalized size = 1.64 \[ -\frac{i \sqrt{1-\frac{3 x^2}{2-i \sqrt{2}}} \sqrt{1-\frac{3 x^2}{2+i \sqrt{2}}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{3}{2-i \sqrt{2}}} x\right )|\frac{2-i \sqrt{2}}{2+i \sqrt{2}}\right )}{\sqrt{3} \sqrt{-\frac{1}{2-i \sqrt{2}}} \sqrt{-3 x^4+4 x^2-2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*Sqrt[1 - (3*x^2)/(2 - I*Sqrt[2])]*Sqrt[1 - (3*x^2)/(2 + I*Sqrt[2])]*Ellipt
icF[I*ArcSinh[Sqrt[-3/(2 - I*Sqrt[2])]*x], (2 - I*Sqrt[2])/(2 + I*Sqrt[2])])/(Sq
rt[3]*Sqrt[-(2 - I*Sqrt[2])^(-1)]*Sqrt[-2 + 4*x^2 - 3*x^4])

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Maple [C]  time = 0.073, size = 87, normalized size = 1. \[ 2\,{\frac{\sqrt{1- \left ( 1-i/2\sqrt{2} \right ){x}^{2}}\sqrt{1- \left ( 1+i/2\sqrt{2} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{4-2\,i\sqrt{2}}x,1/3\,\sqrt{3+6\,i\sqrt{2}} \right ) }{\sqrt{4-2\,i\sqrt{2}}\sqrt{-3\,{x}^{4}+4\,{x}^{2}-2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^4+4*x^2-2)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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