\(\int \frac {3-x^2}{\sqrt {3-2 x^2-x^4}} \, dx\) [117]

Optimal result

Integrand size = 24, antiderivative size = 27 \[ \int \frac {3-x^2}{\sqrt {3-2 x^2-x^4}} \, dx=-\sqrt {3} E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right )+2 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1194, 538, 435, 430} \[ \int \frac {3-x^2}{\sqrt {3-2 x^2-x^4}} \, dx=2 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )-\sqrt {3} E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right ) \]

[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 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 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*} \text {integral}& = 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*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {3-x^2}{\sqrt {3-2 x^2-x^4}} \, dx=-i \left (E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )+2 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right ),-3\right )\right ) \]

[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])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (27 ) = 54\).

Time = 1.56 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.52

method	result	size
default	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, F\left (x , \frac {i \sqrt {3}}{3}\right )}{\sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (F\left (x , \frac {i \sqrt {3}}{3}\right )-E\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{\sqrt {-x^{4}-2 x^{2}+3}}\)	\(95\)
elliptic	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, F\left (x , \frac {i \sqrt {3}}{3}\right )}{\sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (F\left (x , \frac {i \sqrt {3}}{3}\right )-E\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{\sqrt {-x^{4}-2 x^{2}+3}}\)	\(95\)

[In]

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

[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))+(-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)))

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {3-x^2}{\sqrt {3-2 x^2-x^4}} \, dx=\frac {i \, x E(\arcsin \left (\frac {1}{x}\right )\,|\,-3) + 2 i \, x F(\arcsin \left (\frac {1}{x}\right )\,|\,-3) + \sqrt {-x^{4} - 2 \, x^{2} + 3}}{x} \]

[In]

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

[Out]

(I*x*elliptic_e(arcsin(1/x), -3) + 2*I*x*elliptic_f(arcsin(1/x), -3) + sqrt(-x^4 - 2*x^2 + 3))/x

Sympy [F]

\[ \int \frac {3-x^2}{\sqrt {3-2 x^2-x^4}} \, dx=- \int \frac {x^{2}}{\sqrt {- x^{4} - 2 x^{2} + 3}}\, dx - \int \left (- \frac {3}{\sqrt {- x^{4} - 2 x^{2} + 3}}\right )\, dx \]

[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)

Maxima [F]

\[ \int \frac {3-x^2}{\sqrt {3-2 x^2-x^4}} \, dx=\int { -\frac {x^{2} - 3}{\sqrt {-x^{4} - 2 \, x^{2} + 3}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {3-x^2}{\sqrt {3-2 x^2-x^4}} \, dx=\int { -\frac {x^{2} - 3}{\sqrt {-x^{4} - 2 \, x^{2} + 3}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {3-x^2}{\sqrt {3-2 x^2-x^4}} \, dx=\int -\frac {x^2-3}{\sqrt {-x^4-2\,x^2+3}} \,d x \]

[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)