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

Optimal. Leaf size=10 \[ \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right ) \]

[Out]

EllipticF[ArcSin[x/Sqrt[2]], -2]

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Rubi [A]  time = 0.0105507, antiderivative size = 10, 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} \[ F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + x^2 - x^4],x]

[Out]

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

Mathematica [C]  time = 0.0152104, size = 19, normalized size = 1.9 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[2 + x^2 - x^4],x]

[Out]

((-I)*EllipticF[I*ArcSinh[x], -1/2])/Sqrt[2]

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Maple [B]  time = 0.004, size = 47, normalized size = 4.7 \begin{align*}{\frac{\sqrt{2}}{2}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-x^4 + x^2 + 2), x)