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

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

[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)),1/6*(18-6*6^(1/2))^(1/2))*(2+x^2*6^(1/2))*((3*x^4+4*x^2+2)/(2+x^2*6^(1/2))^2)^(1/2)*
6^(3/4)/(-3*x^4-4*x^2-2)^(1/2)

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Rubi [A]  time = 0.01, 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 = {1103} \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+4 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{2}-\frac {1}{\sqrt {6}}\right )}{2 \sqrt [4]{6} \sqrt {-3 x^4-4 x^2-2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 - 4*x^2 - 3*x^4],x]

[Out]

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

Rule 1103

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] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps

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

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Mathematica [C]  time = 0.08, size = 144, normalized size = 1.60 \[ -\frac {i \sqrt {1-\frac {3 x^2}{-2-i \sqrt {2}}} \sqrt {1-\frac {3 x^2}{-2+i \sqrt {2}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {3}{-2-i \sqrt {2}}} x\right )|\frac {-2-i \sqrt {2}}{-2+i \sqrt {2}}\right )}{\sqrt {3} \sqrt {-\frac {1}{-2-i \sqrt {2}}} \sqrt {-3 x^4-4 x^2-2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-3 \, x^{4} - 4 \, x^{2} - 2}}{3 \, x^{4} + 4 \, x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.05, size = 87, normalized size = 0.97 \[ \frac {2 \sqrt {-\left (-1-\frac {i \sqrt {2}}{2}\right ) x^{2}+1}\, \sqrt {-\left (-1+\frac {i \sqrt {2}}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-4-2 i \sqrt {2}}\, x}{2}, \frac {\sqrt {3-6 i \sqrt {2}}}{3}\right )}{\sqrt {-4-2 i \sqrt {2}}\, \sqrt {-3 x^{4}-4 x^{2}-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(- 4*x^2 - 3*x^4 - 2)^(1/2),x)

[Out]

int(1/(- 4*x^2 - 3*x^4 - 2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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