3.10.54 \(\int \sqrt {3-2 x^2-x^4} \, dx\) [954]

Optimal. Leaf size=48 \[ \frac {1}{3} x \sqrt {3-2 x^2-x^4}-\frac {2 E\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}}+\frac {4 F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1105, 1194, 538, 435, 430} \begin {gather*} \frac {4 F\left (\text {ArcSin}(x)\left |-\frac {1}{3}\right .\right )}{\sqrt {3}}-\frac {2 E\left (\text {ArcSin}(x)\left |-\frac {1}{3}\right .\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {-x^4-2 x^2+3} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 - 2*x^2 - x^4],x]

[Out]

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

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 538

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 1105

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b*x^2 + c*x^4)^p/(4*p + 1)), x] + Dis
t[2*(p/(4*p + 1)), Int[(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4
*a*c, 0] && GtQ[p, 0] && IntegerQ[2*p]

Rule 1194

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 \sqrt {3-2 x^2-x^4} \, dx &=\frac {1}{3} x \sqrt {3-2 x^2-x^4}+\frac {1}{3} \int \frac {6-2 x^2}{\sqrt {3-2 x^2-x^4}} \, dx\\ &=\frac {1}{3} x \sqrt {3-2 x^2-x^4}+\frac {2}{3} \int \frac {6-2 x^2}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx\\ &=\frac {1}{3} x \sqrt {3-2 x^2-x^4}-\frac {2}{3} \int \frac {\sqrt {6+2 x^2}}{\sqrt {2-2 x^2}} \, dx+8 \int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx\\ &=\frac {1}{3} x \sqrt {3-2 x^2-x^4}-\frac {2 E\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}}+\frac {4 F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 4.04, size = 59, normalized size = 1.23 \begin {gather*} \frac {1}{3} \left (x \sqrt {3-2 x^2-x^4}-2 i E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )-4 i F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 - 2*x^2 - x^4],x]

[Out]

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

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (44 ) = 88\).
time = 0.04, size = 114, normalized size = 2.38

method	result	size
default	\(\frac {x \sqrt {-x^{4}-2 x^{2}+3}}{3}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )-\EllipticE \left (x , \frac {i \sqrt {3}}{3}\right )\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(114\)
elliptic	\(\frac {x \sqrt {-x^{4}-2 x^{2}+3}}{3}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )-\EllipticE \left (x , \frac {i \sqrt {3}}{3}\right )\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(114\)
risch	\(-\frac {x \left (x^{4}+2 x^{2}-3\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )-\EllipticE \left (x , \frac {i \sqrt {3}}{3}\right )\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x^4 - 2*x^2 + 3), x)

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Fricas [A]
time = 0.08, size = 24, normalized size = 0.50 \begin {gather*} \frac {\sqrt {-x^{4} - 2 \, x^{2} + 3} {\left (x^{2} + 2\right )}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- x^{4} - 2 x^{2} + 3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-x**4 - 2*x**2 + 3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x^4 - 2*x^2 + 3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {-x^4-2\,x^2+3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3 - x^4 - 2*x^2)^(1/2),x)

[Out]

int((3 - x^4 - 2*x^2)^(1/2), x)

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