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

Optimal. Leaf size=42 \[ -\frac{\text{EllipticF}\left (\cos ^{-1}\left (\sqrt{\frac{3}{3+\sqrt{3}}} x\right ),\frac{1}{2} \left (1+\sqrt{3}\right )\right )}{\sqrt{2} \sqrt [4]{3}} \]

[Out]

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

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Rubi [A]  time = 0.0520854, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1095, 420} \[ -\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{3}{3+\sqrt{3}}} x\right )|\frac{1}{2} \left (1+\sqrt{3}\right )\right )}{\sqrt{2} \sqrt [4]{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 + 6*x^2 - 3*x^4],x]

[Out]

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

Rule 1095

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], I
nt[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] &&
LtQ[c, 0]

Rule 420

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> -Simp[EllipticF[ArcCos[Rt[-(d/c), 2]
*x], (b*c)/(b*c - a*d)]/(Sqrt[c]*Rt[-(d/c), 2]*Sqrt[a - (b*c)/d]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &
& GtQ[c, 0] && GtQ[a - (b*c)/d, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-2+6 x^2-3 x^4}} \, dx &=\left (2 \sqrt{3}\right ) \int \frac{1}{\sqrt{6+2 \sqrt{3}-6 x^2} \sqrt{-6+2 \sqrt{3}+6 x^2}} \, dx\\ &=-\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{3}{3+\sqrt{3}}} x\right )|\frac{1}{2} \left (1+\sqrt{3}\right )\right )}{\sqrt{2} \sqrt [4]{3}}\\ \end{align*}

Mathematica [B]  time = 0.082354, size = 85, normalized size = 2.02 \[ \frac{\sqrt{-3 x^2-\sqrt{3}+3} \sqrt{\left (\sqrt{3}-3\right ) x^2+2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{3}\right )} x\right ),2-\sqrt{3}\right )}{\sqrt{6} \sqrt{-3 x^4+6 x^2-2}} \]

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 = 0.175, size = 82, normalized size = 2. \begin{align*} 2\,{\frac{\sqrt{1- \left ( 3/2-1/2\,\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( 1/2\,\sqrt{3}+3/2 \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{6-2\,\sqrt{3}}x,1/2\,\sqrt{6}+1/2\,\sqrt{2} \right ) }{\sqrt{6-2\,\sqrt{3}}\sqrt{-3\,{x}^{4}+6\,{x}^{2}-2}}} \end{align*}

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-(3/2-1/2*3^(1/2))*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)*E
llipticF(1/2*(6-2*3^(1/2))^(1/2)*x,1/2*6^(1/2)+1/2*2^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-3 \, x^{4} + 6 \, x^{2} - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^4+6*x^2-2)^(1/2),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. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-3 \, x^{4} + 6 \, x^{2} - 2}}{3 \, x^{4} - 6 \, x^{2} + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^4+6*x^2-2)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-3*x^4 + 6*x^2 - 2)/(3*x^4 - 6*x^2 + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 3 x^{4} + 6 x^{2} - 2}}\, dx \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-3 \, x^{4} + 6 \, x^{2} - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^4+6*x^2-2)^(1/2),x, algorithm="giac")

[Out]

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