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

Optimal. Leaf size=27 \[ 2 \sqrt {3} F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )-\sqrt {3} E\left (\sin ^{-1}(x)|-\frac {1}{3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1180, 524, 424, 419} \[ 2 \sqrt {3} F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )-\sqrt {3} E\left (\sin ^{-1}(x)|-\frac {1}{3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[3]*EllipticE[ArcSin[x], -1/3]) + 2*Sqrt[3]*EllipticF[ArcSin[x], -1/3]

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)])

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 524

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-(b/a), -(d/c)]))))))

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]

Rubi steps

\begin {align*} \int \frac {3-x^2}{\sqrt {3-2 x^2-x^4}} \, dx &=2 \int \frac {3-x^2}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx\\ &=12 \int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx-\int \frac {\sqrt {6+2 x^2}}{\sqrt {2-2 x^2}} \, dx\\ &=-\sqrt {3} E\left (\sin ^{-1}(x)|-\frac {1}{3}\right )+2 \sqrt {3} F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.06, size = 35, normalized size = 1.30 \[ -i \left (2 F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )+E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} - 2 \, x^{2} + 3} {\left (x^{2} - 3\right )}}{x^{4} + 2 \, x^{2} - 3}, x\right ) \]

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 - 3)/(x^4 + 2*x^2 - 3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x^{2} - 3}{\sqrt {-x^{4} - 2 \, x^{2} + 3}}\,{d x} \]

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)

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maple [B]  time = 0.01, size = 95, normalized size = 3.52 \[ \frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )}{\sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (-\EllipticE \left (x , \frac {i \sqrt {3}}{3}\right )+\EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )\right )}{\sqrt {-x^{4}-2 x^{2}+3}} \]

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]

(-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))-EllipticE(x,1/3*I*3^(1/2)))+(-
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 = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x^{2} - 3}{\sqrt {-x^{4} - 2 \, x^{2} + 3}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int -\frac {x^2-3}{\sqrt {-x^4-2\,x^2+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x^{2}}{\sqrt {- x^{4} - 2 x^{2} + 3}}\, dx - \int \left (- \frac {3}{\sqrt {- x^{4} - 2 x^{2} + 3}}\right )\, dx \]

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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