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

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

[Out]

(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])

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Rubi [A]  time = 0.016283, antiderivative size = 115, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0236983, size = 40, normalized size = 0.35 \[ \frac{\sqrt{2-3 x^4} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right ),-1\right )}{\sqrt [4]{6} \sqrt{3 x^4-2}} \]

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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Maple [C]  time = 0.166, size = 56, normalized size = 0.5 \begin{align*}{\frac{1}{2\,\sqrt{-2\,\sqrt{6}}}\sqrt{4+2\,{x}^{2}\sqrt{6}}\sqrt{4-2\,{x}^{2}\sqrt{6}}{\it EllipticF} \left ({\frac{\sqrt{-2\,\sqrt{6}}x}{2}},i \right ){\frac{1}{\sqrt{3\,{x}^{4}-2}}}} \end{align*}

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)*(4+2*x^2*6^(1/2))^(1/2)*(4-2*x^2*6^(1/2))^(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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \, x^{4} - 2}}\,{d x} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{3 \, x^{4} - 2}}, x\right ) \end{align*}

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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Sympy [C]  time = 0.667135, size = 34, normalized size = 0.3 \begin{align*} - \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 )} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \, x^{4} - 2}}\,{d x} \end{align*}

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)