3.3.62 \(\int \frac {1}{\sqrt {(1-x^2) (3+x^2)}} \, dx\) [262]

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

[Out]

1/3*EllipticF(x,1/3*I*3^(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 = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1976, 1109, 430} \begin {gather*} \frac {F\left (\text {ArcSin}(x)\left |-\frac {1}{3}\right .\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(1 - x^2)*(3 + x^2)],x]

[Out]

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 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]

Rule 1976

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*(a*c*e + (b*c
+ a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.02, size = 18, normalized size = 1.50 \begin {gather*} -i F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right ) \end {gather*}

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] 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 = 0.02, size = 43, normalized size = 3.58

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, 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/((-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 [A]
time = 0.09, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{3} \, \sqrt {3} {\rm ellipticF}\left (x, -\frac {1}{3}\right ) \end {gather*}

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]

1/3*sqrt(3)*ellipticF(x, -1/3)

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

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 = 0.00, 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/((-x^2+1)*(x^2+3))^(1/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-(x^2 - 1)*(x^2 + 3))^(1/2),x)

[Out]

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

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