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

Optimal. Leaf size=12 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]

[Out]

EllipticF[ArcSin[x], -1/3]/Sqrt[3]

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Rubi [A]  time = 0.0131315, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1988, 1095, 419} \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(1 - x^2)*(3 + x^2)],x]

[Out]

EllipticF[ArcSin[x], -1/3]/Sqrt[3]

Rule 1988

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && TrinomialQ[u, x] &&  !TrinomialMatch
Q[u, x]

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

Mathematica [C]  time = 0.0157029, size = 18, normalized size = 1.5 \[ -i F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(1 - x^2)*(3 + x^2)],x]

[Out]

(-I)*EllipticF[I*ArcSinh[x/Sqrt[3]], -3]

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Maple [B]  time = 0.007, size = 43, normalized size = 3.6 \begin{align*}{\frac{{\it EllipticF} \left ( x,{\frac{i}{3}}\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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