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

Optimal. Leaf size=92 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4+2 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{12} \left (6-\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3 x^4+2 x^2+2}} \]

[Out]

((2 + Sqrt[6]*x^2)*Sqrt[(2 + 2*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*Arc
Tan[(3/2)^(1/4)*x], (6 - Sqrt[6])/12])/(2*6^(1/4)*Sqrt[2 + 2*x^2 + 3*x^4])

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Rubi [A]  time = 0.0568264, antiderivative size = 92, 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 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4+2 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{12} \left (6-\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3 x^4+2 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

((2 + Sqrt[6]*x^2)*Sqrt[(2 + 2*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*Arc
Tan[(3/2)^(1/4)*x], (6 - Sqrt[6])/12])/(2*6^(1/4)*Sqrt[2 + 2*x^2 + 3*x^4])

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Rubi in Sympy [A]  time = 3.65911, size = 88, normalized size = 0.96 \[ \frac{6^{\frac{3}{4}} \sqrt{\frac{3 x^{4} + 2 x^{2} + 2}{\left (\frac{\sqrt{6} x^{2}}{2} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{3} x}{2} \right )}\middle | - \frac{\sqrt{6}}{12} + \frac{1}{2}\right )}{12 \sqrt{3 x^{4} + 2 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.142799, size = 144, normalized size = 1.57 \[ -\frac{i \sqrt{1-\frac{3 x^2}{-1-i \sqrt{5}}} \sqrt{1-\frac{3 x^2}{-1+i \sqrt{5}}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{3}{-1-i \sqrt{5}}} x\right )|\frac{-1-i \sqrt{5}}{-1+i \sqrt{5}}\right )}{\sqrt{3} \sqrt{-\frac{1}{-1-i \sqrt{5}}} \sqrt{3 x^4+2 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*Sqrt[1 - (3*x^2)/(-1 - I*Sqrt[5])]*Sqrt[1 - (3*x^2)/(-1 + I*Sqrt[5])]*Elli
pticF[I*ArcSinh[Sqrt[-3/(-1 - I*Sqrt[5])]*x], (-1 - I*Sqrt[5])/(-1 + I*Sqrt[5])]
)/(Sqrt[3]*Sqrt[-(-1 - I*Sqrt[5])^(-1)]*Sqrt[2 + 2*x^2 + 3*x^4])

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Maple [C]  time = 0.177, size = 87, normalized size = 1. \[ 2\,{\frac{\sqrt{1- \left ( -1/2+i/2\sqrt{5} \right ){x}^{2}}\sqrt{1- \left ( -1/2-i/2\sqrt{5} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-2+2\,i\sqrt{5}},1/3\,\sqrt{-6+3\,i\sqrt{5}} \right ) }{\sqrt{-2+2\,i\sqrt{5}}\sqrt{3\,{x}^{4}+2\,{x}^{2}+2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^4+2*x^2+2)^(1/2),x)

[Out]

2/(-2+2*I*5^(1/2))^(1/2)*(1-(-1/2+1/2*I*5^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*I*5^(1/
2))*x^2)^(1/2)/(3*x^4+2*x^2+2)^(1/2)*EllipticF(1/2*x*(-2+2*I*5^(1/2))^(1/2),1/3*
(-6+3*I*5^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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