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

Optimal. Leaf size=46 \[ \sqrt{\frac{2}{3+\sqrt{33}}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{6}{\sqrt{33}-3}} x\right ),\frac{1}{4} \left (\sqrt{33}-7\right )\right ) \]

[Out]

Sqrt[2/(3 + Sqrt[33])]*EllipticF[ArcSin[Sqrt[6/(-3 + Sqrt[33])]*x], (-7 + Sqrt[33])/4]

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Rubi [A]  time = 0.0889233, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1095, 419} \[ \sqrt{\frac{2}{3+\sqrt{33}}} F\left (\sin ^{-1}\left (\sqrt{\frac{6}{-3+\sqrt{33}}} x\right )|\frac{1}{4} \left (-7+\sqrt{33}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 - 3*x^2 - 3*x^4],x]

[Out]

Sqrt[2/(3 + Sqrt[33])]*EllipticF[ArcSin[Sqrt[6/(-3 + Sqrt[33])]*x], (-7 + Sqrt[33])/4]

Rule 1095

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], I
nt[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] &&
LtQ[c, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{2-3 x^2-3 x^4}} \, dx &=\left (2 \sqrt{3}\right ) \int \frac{1}{\sqrt{-3+\sqrt{33}-6 x^2} \sqrt{3+\sqrt{33}+6 x^2}} \, dx\\ &=\sqrt{\frac{2}{3+\sqrt{33}}} F\left (\sin ^{-1}\left (\sqrt{\frac{6}{-3+\sqrt{33}}} x\right )|\frac{1}{4} \left (-7+\sqrt{33}\right )\right )\\ \end{align*}

Mathematica [C]  time = 0.0643541, size = 55, normalized size = 1.2 \[ -i \sqrt{\frac{2}{\sqrt{33}-3}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{6}{3+\sqrt{33}}} x\right ),-\frac{7}{4}-\frac{\sqrt{33}}{4}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[2 - 3*x^2 - 3*x^4],x]

[Out]

(-I)*Sqrt[2/(-3 + Sqrt[33])]*EllipticF[I*ArcSinh[Sqrt[6/(3 + Sqrt[33])]*x], -7/4 - Sqrt[33]/4]

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Maple [B]  time = 0.256, size = 80, normalized size = 1.7 \begin{align*} 2\,{\frac{\sqrt{1- \left ( 1/4\,\sqrt{33}+3/4 \right ){x}^{2}}\sqrt{1- \left ( -1/4\,\sqrt{33}+3/4 \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{3+\sqrt{33}},i/4\sqrt{22}-i/4\sqrt{6} \right ) }{\sqrt{3+\sqrt{33}}\sqrt{-3\,{x}^{4}-3\,{x}^{2}+2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(3+33^(1/2))^(1/2)*(1-(1/4*33^(1/2)+3/4)*x^2)^(1/2)*(1-(-1/4*33^(1/2)+3/4)*x^2)^(1/2)/(-3*x^4-3*x^2+2)^(1/2)
*EllipticF(1/2*x*(3+33^(1/2))^(1/2),1/4*I*22^(1/2)-1/4*I*6^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-3*x**4 - 3*x**2 + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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