\(\int \frac {1}{\sqrt {-2+6 x^2-3 x^4}} \, dx\) [108]

Optimal result

Integrand size = 16, antiderivative size = 42 \[ \int \frac {1}{\sqrt {-2+6 x^2-3 x^4}} \, dx=-\frac {\operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {3}{3+\sqrt {3}}} x\right ),\frac {1}{2} \left (1+\sqrt {3}\right )\right )}{\sqrt {2} \sqrt [4]{3}} \]

[Out]

-1/6*3^(3/4)*(x^2/(3^(1/2)+3))^(1/2)/x*(3^(1/2)+3)^(1/2)*EllipticF((1-3*x^2/(3^(1/2)+3))^(1/2),1/2*(2+2*3^(1/2
))^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1109, 431} \[ \int \frac {1}{\sqrt {-2+6 x^2-3 x^4}} \, dx=-\frac {\operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {3}{3+\sqrt {3}}} x\right ),\frac {1}{2} \left (1+\sqrt {3}\right )\right )}{\sqrt {2} \sqrt [4]{3}} \]

[In]

Int[1/Sqrt[-2 + 6*x^2 - 3*x^4],x]

[Out]

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

Rule 431

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(42)=84\).

Time = 10.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.02 \[ \int \frac {1}{\sqrt {-2+6 x^2-3 x^4}} \, dx=\frac {\sqrt {3-\sqrt {3}-3 x^2} \sqrt {2+\left (-3+\sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} x\right ),2-\sqrt {3}\right )}{\sqrt {6} \sqrt {-2+6 x^2-3 x^4}} \]

[In]

Integrate[1/Sqrt[-2 + 6*x^2 - 3*x^4],x]

[Out]

(Sqrt[3 - Sqrt[3] - 3*x^2]*Sqrt[2 + (-3 + Sqrt[3])*x^2]*EllipticF[ArcSin[Sqrt[(3 + Sqrt[3])/2]*x], 2 - Sqrt[3]
])/(Sqrt[6]*Sqrt[-2 + 6*x^2 - 3*x^4])

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.95

method	result	size
default	\(\frac {2 \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {3}}{2}+\frac {3}{2}\right ) x^{2}}\, F\left (\frac {\sqrt {6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {6-2 \sqrt {3}}\, \sqrt {-3 x^{4}+6 x^{2}-2}}\)	\(82\)
elliptic	\(\frac {2 \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {3}}{2}+\frac {3}{2}\right ) x^{2}}\, F\left (\frac {\sqrt {6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {6-2 \sqrt {3}}\, \sqrt {-3 x^{4}+6 x^{2}-2}}\)	\(82\)

[In]

int(1/(-3*x^4+6*x^2-2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/(6-2*3^(1/2))^(1/2)*(1-(3/2-1/2*3^(1/2))*x^2)^(1/2)*(1-(1/2*3^(1/2)+3/2)*x^2)^(1/2)/(-3*x^4+6*x^2-2)^(1/2)*E
llipticF(1/2*(6-2*3^(1/2))^(1/2)*x,1/2*6^(1/2)+1/2*2^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\sqrt {-2+6 x^2-3 x^4}} \, dx=\frac {1}{12} \, {\left (\sqrt {3} \sqrt {2} \sqrt {-2} - 3 \, \sqrt {2} \sqrt {-2}\right )} \sqrt {\sqrt {3} + 3} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {3} + 3}\right )\,|\,-\sqrt {3} + 2) \]

[In]

integrate(1/(-3*x^4+6*x^2-2)^(1/2),x, algorithm="fricas")

[Out]

1/12*(sqrt(3)*sqrt(2)*sqrt(-2) - 3*sqrt(2)*sqrt(-2))*sqrt(sqrt(3) + 3)*elliptic_f(arcsin(1/2*sqrt(2)*x*sqrt(sq
rt(3) + 3)), -sqrt(3) + 2)

Sympy [F]

\[ \int \frac {1}{\sqrt {-2+6 x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {- 3 x^{4} + 6 x^{2} - 2}}\, dx \]

[In]

integrate(1/(-3*x**4+6*x**2-2)**(1/2),x)

[Out]

Integral(1/sqrt(-3*x**4 + 6*x**2 - 2), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {-2+6 x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} + 6 \, x^{2} - 2}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(-3*x^4 + 6*x^2 - 2), x)

Giac [F]

\[ \int \frac {1}{\sqrt {-2+6 x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} + 6 \, x^{2} - 2}} \,d x } \]

[In]

integrate(1/(-3*x^4+6*x^2-2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-3*x^4 + 6*x^2 - 2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-2+6 x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {-3\,x^4+6\,x^2-2}} \,d x \]

[In]

int(1/(6*x^2 - 3*x^4 - 2)^(1/2),x)

[Out]

int(1/(6*x^2 - 3*x^4 - 2)^(1/2), x)