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

Optimal. Leaf size=25 \[ 4 \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ),-3\right )-E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right ) \]

[Out]

-EllipticE[ArcSin[x/Sqrt[3]], -3] + 4*EllipticF[ArcSin[x/Sqrt[3]], -3]

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Rubi [A]  time = 0.032492, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1180, 21, 423, 424, 419} \[ 4 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )-E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right ) \]

Antiderivative was successfully verified.

[In]

Int[(3 - x^2)/Sqrt[3 + 2*x^2 - x^4],x]

[Out]

-EllipticE[ArcSin[x/Sqrt[3]], -3] + 4*EllipticF[ArcSin[x/Sqrt[3]], -3]

Rule 1180

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 423

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[b/d, Int[Sqrt[c + d*x^2]/Sqrt[a + b
*x^2], x], x] - Dist[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x]
&& PosQ[d/c] && NegQ[b/a]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 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{3-x^2}{\sqrt{3+2 x^2-x^4}} \, dx &=2 \int \frac{3-x^2}{\sqrt{6-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=\int \frac{\sqrt{6-2 x^2}}{\sqrt{2+2 x^2}} \, dx\\ &=8 \int \frac{1}{\sqrt{6-2 x^2} \sqrt{2+2 x^2}} \, dx-\int \frac{\sqrt{2+2 x^2}}{\sqrt{6-2 x^2}} \, dx\\ &=-E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )+4 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )\\ \end{align*}

Mathematica [C]  time = 0.0532473, size = 19, normalized size = 0.76 \[ -i \sqrt{3} E\left (i \sinh ^{-1}(x)|-\frac{1}{3}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(3 - x^2)/Sqrt[3 + 2*x^2 - x^4],x]

[Out]

(-I)*Sqrt[3]*EllipticE[I*ArcSinh[x], -1/3]

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Maple [B]  time = 0.054, size = 113, normalized size = 4.5 \begin{align*}{\frac{\sqrt{3}}{3}\sqrt{-3\,{x}^{2}+9}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}}{3}},i\sqrt{3} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{3}}{3}},i\sqrt{3} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+2\,{x}^{2}+3}}}}+{\sqrt{3}\sqrt{-3\,{x}^{2}+9}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{3}}{3}},i\sqrt{3} \right ){\frac{1}{\sqrt{-{x}^{4}+2\,{x}^{2}+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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