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]

1/12*(cos(2*arctan(1/2*3^(1/4)*2^(3/4)*x))^2)^(1/2)/cos(2*arctan(1/2*3^(1/4)*2^(3/4)*x))*EllipticF(sin(2*arcta
n(1/2*3^(1/4)*2^(3/4)*x)),1/2*(2+6^(1/2))^(1/2))*(2+x^2*6^(1/2))*((3*x^4-6*x^2+2)/(2+x^2*6^(1/2))^2)^(1/2)*6^(
3/4)/(3*x^4-6*x^2+2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1096} \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-6 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{4} \left (2+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3 x^4-6 x^2+2}} \]

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

Rule 1096

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

Rubi steps

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

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Mathematica [A]  time = 0.08, 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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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {3 \, x^{4} - 6 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 \, x^{4} - 6 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 82, normalized size = 0.91 \[ \frac {2 \sqrt {-\left (\frac {\sqrt {3}}{2}+\frac {3}{2}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {\sqrt {3}}{2}+\frac {3}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {6+2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {6+2 \sqrt {3}}\, \sqrt {3 x^{4}-6 x^{2}+2}} \]

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)*E
llipticF(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 \, x^{4} - 6 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {3\,x^4-6\,x^2+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^4 - 6*x^2 + 2)^(1/2),x)

[Out]

int(1/(3*x^4 - 6*x^2 + 2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 x^{4} - 6 x^{2} + 2}}\, dx \]

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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