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

Optimal. Leaf size=88 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4-x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{24} \left (12+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{-3 x^4+x^2-2}} \]

[Out]

((2 + Sqrt[6]*x^2)*Sqrt[(2 - x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTa
n[(3/2)^(1/4)*x], (12 + Sqrt[6])/24])/(2*6^(1/4)*Sqrt[-2 + x^2 - 3*x^4])

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Rubi [A]  time = 0.0517739, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4-x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{24} \left (12+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{-3 x^4+x^2-2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-2 + x^2 - 3*x^4],x]

[Out]

((2 + Sqrt[6]*x^2)*Sqrt[(2 - x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTa
n[(3/2)^(1/4)*x], (12 + Sqrt[6])/24])/(2*6^(1/4)*Sqrt[-2 + x^2 - 3*x^4])

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Rubi in Sympy [A]  time = 3.17359, size = 87, normalized size = 0.99 \[ \frac{6^{\frac{3}{4}} \sqrt{- \frac{- 3 x^{4} + x^{2} - 2}{\left (\frac{\sqrt{6} x^{2}}{2} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{3} x}{2} \right )}\middle | \frac{\sqrt{6}}{24} + \frac{1}{2}\right )}{12 \sqrt{- 3 x^{4} + x^{2} - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6**(3/4)*sqrt(-(-3*x**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)/24 + 1/2)/(12*sqrt(-3*x**4 +
x**2 - 2))

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Mathematica [C]  time = 0.144128, size = 142, normalized size = 1.61 \[ -\frac{i \sqrt{1-\frac{6 x^2}{1-i \sqrt{23}}} \sqrt{1-\frac{6 x^2}{1+i \sqrt{23}}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{6}{1-i \sqrt{23}}} x\right )|\frac{1-i \sqrt{23}}{1+i \sqrt{23}}\right )}{\sqrt{6} \sqrt{-\frac{1}{1-i \sqrt{23}}} \sqrt{-3 x^4+x^2-2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*Sqrt[1 - (6*x^2)/(1 - I*Sqrt[23])]*Sqrt[1 - (6*x^2)/(1 + I*Sqrt[23])]*Elli
pticF[I*ArcSinh[Sqrt[-6/(1 - I*Sqrt[23])]*x], (1 - I*Sqrt[23])/(1 + I*Sqrt[23])]
)/(Sqrt[6]*Sqrt[-(1 - I*Sqrt[23])^(-1)]*Sqrt[-2 + x^2 - 3*x^4])

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Maple [C]  time = 0.06, size = 85, normalized size = 1. \[ 2\,{\frac{\sqrt{1- \left ( 1/4-i/4\sqrt{23} \right ){x}^{2}}\sqrt{1- \left ( 1/4+i/4\sqrt{23} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{1-i\sqrt{23}}x,1/6\,\sqrt{-33+3\,i\sqrt{23}} \right ) }{\sqrt{1-i\sqrt{23}}\sqrt{-3\,{x}^{4}+{x}^{2}-2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^4+x^2-2)^(1/2),x)

[Out]

2/(1-I*23^(1/2))^(1/2)*(1-(1/4-1/4*I*23^(1/2))*x^2)^(1/2)*(1-(1/4+1/4*I*23^(1/2)
)*x^2)^(1/2)/(-3*x^4+x^2-2)^(1/2)*EllipticF(1/2*(1-I*23^(1/2))^(1/2)*x,1/6*(-33+
3*I*23^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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