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

Optimal. Leaf size=115 \[ \frac {\sqrt {\sqrt {6} x^2-2} \sqrt {\frac {\sqrt {6} x^2+2}{2-\sqrt {6} x^2}} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{3} x}{\sqrt {\sqrt {6} x^2-2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {\frac {1}{2-\sqrt {6} x^2}} \sqrt {3 x^4-2}} \]

[Out]

1/12*EllipticF(2^(3/4)*3^(1/4)*x/(-2+x^2*6^(1/2))^(1/2),1/2*2^(1/2))*(-2+x^2*6^(1/2))^(1/2)*((2+x^2*6^(1/2))/(
2-x^2*6^(1/2)))^(1/2)*6^(3/4)/(3*x^4-2)^(1/2)/(1/(2-x^2*6^(1/2)))^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 115, 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 = {223} \[ \frac {\sqrt {\sqrt {6} x^2-2} \sqrt {\frac {\sqrt {6} x^2+2}{2-\sqrt {6} x^2}} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{3} x}{\sqrt {\sqrt {6} x^2-2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {\frac {1}{2-\sqrt {6} x^2}} \sqrt {3 x^4-2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 + 3*x^4],x]

[Out]

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

Rule 223

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 40, normalized size = 0.35 \[ \frac {\sqrt {2-3 x^4} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )\right |-1\right )}{\sqrt [4]{6} \sqrt {3 x^4-2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-2 + 3*x^4],x]

[Out]

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

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fricas [F]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {3 \, x^{4} - 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(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 = 56, normalized size = 0.49 \[ \frac {\sqrt {2 \sqrt {6}\, x^{2}+4}\, \sqrt {-2 \sqrt {6}\, x^{2}+4}\, \EllipticF \left (\frac {\sqrt {-2 \sqrt {6}}\, x}{2}, i\right )}{2 \sqrt {-2 \sqrt {6}}\, \sqrt {3 x^{4}-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(-2*6^(1/2))^(1/2)*(2*6^(1/2)*x^2+4)^(1/2)*(-2*6^(1/2)*x^2+4)^(1/2)/(3*x^4-2)^(1/2)*EllipticF(1/2*(-2*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 = 0.08, size = 31, normalized size = 0.27 \[ \frac {x\,\sqrt {4-6\,x^4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ \frac {3\,x^4}{2}\right )}{2\,\sqrt {3\,x^4-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^4 - 2)^(1/2),x)

[Out]

(x*(4 - 6*x^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 = 0.73, size = 34, normalized size = 0.30 \[ - \frac {\sqrt {2} i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {3 x^{4}}{2}} \right )}}{8 \Gamma \left (\frac {5}{4}\right )} \]

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**4/2)/(8*gamma(5/4))

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