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

Optimal. Leaf size=90 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4+4 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{2}-\frac{1}{\sqrt{6}}\right )}{2 \sqrt [4]{6} \sqrt{-2 x^4-4 x^2-3}} \]

[Out]

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

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Rubi [A]  time = 0.0490624, antiderivative size = 90, 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 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4+4 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{2}-\frac{1}{\sqrt{6}}\right )}{2 \sqrt [4]{6} \sqrt{-2 x^4-4 x^2-3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.61516, size = 94, normalized size = 1.04 \[ \frac{6^{\frac{3}{4}} \sqrt{- \frac{- 2 x^{4} - 4 x^{2} - 3}{\left (\frac{\sqrt{6} x^{2}}{3} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{3} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} x}{3} \right )}\middle | - \frac{\sqrt{6}}{6} + \frac{1}{2}\right )}{12 \sqrt{- 2 x^{4} - 4 x^{2} - 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.157449, size = 144, normalized size = 1.6 \[ -\frac{i \sqrt{1-\frac{2 x^2}{-2-i \sqrt{2}}} \sqrt{1-\frac{2 x^2}{-2+i \sqrt{2}}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2}{-2-i \sqrt{2}}} x\right )|\frac{-2-i \sqrt{2}}{-2+i \sqrt{2}}\right )}{\sqrt{2} \sqrt{-\frac{1}{-2-i \sqrt{2}}} \sqrt{-2 x^4-4 x^2-3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.059, size = 87, normalized size = 1. \[ 3\,{\frac{\sqrt{1- \left ( -2/3-i/3\sqrt{2} \right ){x}^{2}}\sqrt{1- \left ( -2/3+i/3\sqrt{2} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,\sqrt{-6-3\,i\sqrt{2}}x,1/3\,\sqrt{3-6\,i\sqrt{2}} \right ) }{\sqrt{-6-3\,i\sqrt{2}}\sqrt{-2\,{x}^{4}-4\,{x}^{2}-3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-2*x^4-4*x^2-3)^(1/2),x)

[Out]

3/(-6-3*I*2^(1/2))^(1/2)*(1-(-2/3-1/3*I*2^(1/2))*x^2)^(1/2)*(1-(-2/3+1/3*I*2^(1/
2))*x^2)^(1/2)/(-2*x^4-4*x^2-3)^(1/2)*EllipticF(1/3*(-6-3*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. \[ \int \frac{1}{\sqrt{-2 \, x^{4} - 4 \, x^{2} - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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