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

Optimal. Leaf size=12 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-\frac{3}{2}\right )}{\sqrt{2}} \]

[Out]

EllipticF[ArcSin[x], -3/2]/Sqrt[2]

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Rubi [A]  time = 0.0119981, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1095, 419} \[ \frac{F\left (\sin ^{-1}(x)|-\frac{3}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + x^2 - 3*x^4],x]

[Out]

EllipticF[ArcSin[x], -3/2]/Sqrt[2]

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

Mathematica [C]  time = 0.0227184, size = 63, normalized size = 5.25 \[ -\frac{i \sqrt{1-x^2} \sqrt{3 x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),-\frac{2}{3}\right )}{\sqrt{3} \sqrt{-3 x^4+x^2+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[2 + x^2 - 3*x^4],x]

[Out]

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

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Maple [B]  time = 0.049, size = 41, normalized size = 3.4 \begin{align*}{\frac{{\it EllipticF} \left ( x,{\frac{i}{2}}\sqrt{6} \right ) }{2}\sqrt{-{x}^{2}+1}\sqrt{6\,{x}^{2}+4}{\frac{1}{\sqrt{-3\,{x}^{4}+{x}^{2}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(-3*x^4+x^2+2)^(1/2)*EllipticF(x,1/2*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} + x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-3*x^4 + 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} + x^{2} + 2}}{3 \, x^{4} - x^{2} - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-3*x^4 + x^2 + 2)/(3*x^4 - 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} + x^{2} + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-3*x**4 + 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} + x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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