3.1.71 \(\int \frac {1}{\sqrt {2+3 x^4}} \, dx\) [71]

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

[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 = {226} \begin {gather*} \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 \text {ArcTan}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {3 x^4+2}} \end {gather*}

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 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], 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] Result contains complex when optimal does not.
time = 10.03, size = 25, normalized size = 0.35 \begin {gather*} -\sqrt [4]{-\frac {1}{6}} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {3}{2}} x\right )\right |-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[2 + 3*x^4],x]

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 0.09, size = 66, normalized size = 0.92

method	result	size
meijerg	\(\frac {\sqrt {2}\, x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], -\frac {3 x^{4}}{2}\right )}{2}\)	\(18\)
default	\(\frac {\sqrt {2}\, \sqrt {4-2 i \sqrt {6}\, x^{2}}\, \sqrt {4+2 i \sqrt {6}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {i \sqrt {6}}}{2}, i\right )}{4 \sqrt {i \sqrt {6}}\, \sqrt {3 x^{4}+2}}\)	\(66\)
elliptic	\(\frac {\sqrt {2}\, \sqrt {4-2 i \sqrt {6}\, x^{2}}\, \sqrt {4+2 i \sqrt {6}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {i \sqrt {6}}}{2}, i\right )}{4 \sqrt {i \sqrt {6}}\, \sqrt {3 x^{4}+2}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^4+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.08, size = 16, normalized size = 0.22 \begin {gather*} -\frac {1}{6} \, \left (-6\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {2} \left (-6\right )^{\frac {1}{4}} x, -1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.33, size = 36, normalized size = 0.50 \begin {gather*} \frac {\sqrt {2} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [B]
time = 0.09, size = 16, normalized size = 0.22 \begin {gather*} \frac {\sqrt {2}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ -\frac {3\,x^4}{2}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^4 + 2)^(1/2),x)

[Out]

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

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