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

Optimal. Leaf size=72 \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {-3 x^4-2}} \]

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

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Rubi [A]  time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {220} \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {-3 x^4-2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 - 3*x^4],x]

[Out]

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

Rule 220

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

Rubi steps

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

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Mathematica [C]  time = 0.03, size = 47, normalized size = 0.65 \[ -\frac {\sqrt [4]{-\frac {1}{6}} \sqrt {3 x^4+2} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {3}{2}} x\right )\right |-1\right )}{\sqrt {-3 x^4-2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-2 - 3*x^4],x]

[Out]

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

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-3 \, x^{4} - 2}}{3 \, x^{4} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-3*x^4 - 2), x)

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maple [C]  time = 0.02, size = 66, normalized size = 0.92 \[ \frac {\sqrt {2}\, \sqrt {2 i \sqrt {6}\, x^{2}+4}\, \sqrt {-2 i \sqrt {6}\, x^{2}+4}\, \EllipticF \left (\frac {\sqrt {2}\, \sqrt {-i \sqrt {6}}\, x}{2}, i\right )}{4 \sqrt {-i \sqrt {6}}\, \sqrt {-3 x^{4}-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-3*x^4 - 2), x)

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mupad [B]  time = 4.18, size = 31, normalized size = 0.43 \[ \frac {x\,\sqrt {6\,x^4+4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ -\frac {3\,x^4}{2}\right )}{2\,\sqrt {-3\,x^4-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(- 3*x^4 - 2)^(1/2),x)

[Out]

(x*(6*x^4 + 4)^(1/2)*hypergeom([1/4, 1/2], 5/4, -(3*x^4)/2))/(2*(- 3*x^4 - 2)^(1/2))

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sympy [C]  time = 0.70, size = 39, normalized size = 0.54 \[ - \frac {\sqrt {2} i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {3 x^{4} e^{i \pi }}{2}} \right )}}{8 \Gamma \left (\frac {5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*I*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), 3*x**4*exp_polar(I*pi)/2)/(8*gamma(5/4))

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