3.1.35 \(\int \frac {1}{\sqrt {3-x^2-2 x^4}} \, dx\) [35]

Optimal. Leaf size=12 \[ \frac {F\left (\sin ^{-1}(x)|-\frac {2}{3}\right )}{\sqrt {3}} \]

[Out]

1/3*EllipticF(x,1/3*I*6^(1/2))*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1109, 430} \begin {gather*} \frac {F\left (\text {ArcSin}(x)\left |-\frac {2}{3}\right .\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 - x^2 - 2*x^4],x]

[Out]

EllipticF[ArcSin[x], -2/3]/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 1109

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]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 10.02, size = 65, normalized size = 5.42 \begin {gather*} -\frac {i \sqrt {1-x^2} \sqrt {3+2 x^2} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{3}} x\right )|-\frac {3}{2}\right )}{\sqrt {2} \sqrt {3-x^2-2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[3 - x^2 - 2*x^4],x]

[Out]

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

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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. 42 vs. \(2 (13 ) = 26\).
time = 0.03, size = 43, normalized size = 3.58

method	result	size
default	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {6 x^{2}+9}\, \EllipticF \left (x , \frac {i \sqrt {6}}{3}\right )}{3 \sqrt {-2 x^{4}-x^{2}+3}}\)	\(43\)
elliptic	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {6 x^{2}+9}\, \EllipticF \left (x , \frac {i \sqrt {6}}{3}\right )}{3 \sqrt {-2 x^{4}-x^{2}+3}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-x^2+1)^(1/2)*(6*x^2+9)^(1/2)/(-2*x^4-x^2+3)^(1/2)*EllipticF(x,1/3*I*6^(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(1/(-2*x^4-x^2+3)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.08, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{3} \, \sqrt {3} {\rm ellipticF}\left (x, -\frac {2}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(3)*ellipticF(x, -2/3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- 2 x^{4} - x^{2} + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int \frac {1}{\sqrt {-2\,x^4-x^2+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3 - 2*x^4 - x^2)^(1/2),x)

[Out]

int(1/(3 - 2*x^4 - x^2)^(1/2), x)

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