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

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

[Out]

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

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Rubi [A]  time = 0.0547033, 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 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4-6 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{4} \left (2+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3 x^4-6 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.98896, size = 88, normalized size = 0.98 \[ \frac{6^{\frac{3}{4}} \sqrt{\frac{3 x^{4} - 6 x^{2} + 2}{\left (\frac{\sqrt{6} x^{2}}{2} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{3} x}{2} \right )}\middle | \frac{1}{2} + \frac{\sqrt{6}}{4}\right )}{12 \sqrt{3 x^{4} - 6 x^{2} + 2}} \]

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]

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

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Mathematica [A]  time = 0.144108, size = 85, normalized size = 0.94 \[ \frac{\sqrt{-3 x^2-\sqrt{3}+3} \sqrt{\left (\sqrt{3}-3\right ) x^2+2} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{3}\right )} x\right )|2-\sqrt{3}\right )}{\sqrt{6} \sqrt{3 x^4-6 x^2+2}} \]

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 = 0.223, size = 82, normalized size = 0.9 \[ 2\,{\frac{\sqrt{1- \left ( 1/2\,\sqrt{3}+3/2 \right ){x}^{2}}\sqrt{1- \left ( -1/2\,\sqrt{3}+3/2 \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{6+2\,\sqrt{3}},1/2\,\sqrt{6}-1/2\,\sqrt{2} \right ) }{\sqrt{6+2\,\sqrt{3}}\sqrt{3\,{x}^{4}-6\,{x}^{2}+2}}} \]

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

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)