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

Optimal. Leaf size=112 \[ \frac{\sqrt{\sqrt{6} x^2-3} \sqrt{\frac{\sqrt{6} x^2+3}{3-\sqrt{6} x^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{3} x}{\sqrt{\sqrt{6} x^2-3}}\right ),\frac{1}{2}\right )}{6^{3/4} \sqrt{\frac{1}{3-\sqrt{6} x^2}} \sqrt{2 x^4-3}} \]

[Out]

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

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Rubi [A]  time = 0.0154549, antiderivative size = 112, 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} \[ \frac{\sqrt{\sqrt{6} x^2-3} \sqrt{\frac{\sqrt{6} x^2+3}{3-\sqrt{6} x^2}} F\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{3} x}{\sqrt{\sqrt{6} x^2-3}}\right )|\frac{1}{2}\right )}{6^{3/4} \sqrt{\frac{1}{3-\sqrt{6} x^2}} \sqrt{2 x^4-3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-3 + 2*x^4],x]

[Out]

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

Mathematica [A]  time = 0.0244002, size = 40, normalized size = 0.36 \[ \frac{\sqrt{3-2 x^4} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right ),-1\right )}{\sqrt [4]{6} \sqrt{2 x^4-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-3 + 2*x^4],x]

[Out]

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

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Maple [C]  time = 0.173, size = 56, normalized size = 0.5 \begin{align*}{\frac{1}{3\,\sqrt{-3\,\sqrt{6}}}\sqrt{9+3\,{x}^{2}\sqrt{6}}\sqrt{9-3\,{x}^{2}\sqrt{6}}{\it EllipticF} \left ({\frac{x\sqrt{-3\,\sqrt{6}}}{3}},i \right ){\frac{1}{\sqrt{2\,{x}^{4}-3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/(-3*6^(1/2))^(1/2)*(9+3*x^2*6^(1/2))^(1/2)*(9-3*x^2*6^(1/2))^(1/2)/(2*x^4-3)^(1/2)*EllipticF(1/3*x*(-3*6^(
1/2))^(1/2),I)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(2*x^4 - 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(2*x^4 - 3), x)

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Sympy [C]  time = 0.62534, size = 34, normalized size = 0.3 \begin{align*} - \frac{\sqrt{3} 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{2 x^{4}}{3}} \right )}}{12 \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*I*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), 2*x**4/3)/(12*gamma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(2*x^4 - 3), x)