\(\int \frac {1}{\sqrt {3-6 x^2-2 x^4}} \, dx\) [40]

Optimal result

Integrand size = 16, antiderivative size = 42 \[ \int \frac {1}{\sqrt {3-6 x^2-2 x^4}} \, dx=\sqrt {\frac {1}{6} \left (-3+\sqrt {15}\right )} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{3} \left (3+\sqrt {15}\right )} x\right ),-4+\sqrt {15}\right ) \]

[Out]

1/6*EllipticF(1/3*x*(9+3*15^(1/2))^(1/2),1/2*I*10^(1/2)-1/2*I*6^(1/2))*(-18+6*15^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.06 (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, 430} \[ \int \frac {1}{\sqrt {3-6 x^2-2 x^4}} \, dx=\sqrt {\frac {1}{6} \left (\sqrt {15}-3\right )} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{3} \left (3+\sqrt {15}\right )} x\right ),-4+\sqrt {15}\right ) \]

[In]

Int[1/Sqrt[3 - 6*x^2 - 2*x^4],x]

[Out]

Sqrt[(-3 + Sqrt[15])/6]*EllipticF[ArcSin[Sqrt[(3 + Sqrt[15])/3]*x], -4 + Sqrt[15]]

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*} \text {integral}& = \left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {-6+2 \sqrt {15}-4 x^2} \sqrt {6+2 \sqrt {15}+4 x^2}} \, dx \\ & = \sqrt {\frac {1}{6} \left (-3+\sqrt {15}\right )} F\left (\sin ^{-1}\left (\sqrt {\frac {1}{3} \left (3+\sqrt {15}\right )} x\right )|-4+\sqrt {15}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {3-6 x^2-2 x^4}} \, dx=-\frac {i \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-1+\sqrt {\frac {5}{3}}} x\right ),-4-\sqrt {15}\right )}{\sqrt {-3+\sqrt {15}}} \]

[In]

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

[Out]

((-I)*EllipticF[I*ArcSinh[Sqrt[-1 + Sqrt[5/3]]*x], -4 - Sqrt[15]])/Sqrt[-3 + Sqrt[15]]

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (37 ) = 74\).

Time = 0.60 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.00

method	result	size
default	\(\frac {3 \sqrt {1-\left (1+\frac {\sqrt {15}}{3}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {15}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {9+3 \sqrt {15}}}{3}, \frac {i \sqrt {10}}{2}-\frac {i \sqrt {6}}{2}\right )}{\sqrt {9+3 \sqrt {15}}\, \sqrt {-2 x^{4}-6 x^{2}+3}}\)	\(84\)
elliptic	\(\frac {3 \sqrt {1-\left (1+\frac {\sqrt {15}}{3}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {15}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {9+3 \sqrt {15}}}{3}, \frac {i \sqrt {10}}{2}-\frac {i \sqrt {6}}{2}\right )}{\sqrt {9+3 \sqrt {15}}\, \sqrt {-2 x^{4}-6 x^{2}+3}}\)	\(84\)

[In]

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

[Out]

3/(9+3*15^(1/2))^(1/2)*(1-(1+1/3*15^(1/2))*x^2)^(1/2)*(1-(1-1/3*15^(1/2))*x^2)^(1/2)/(-2*x^4-6*x^2+3)^(1/2)*El
lipticF(1/3*x*(9+3*15^(1/2))^(1/2),1/2*I*10^(1/2)-1/2*I*6^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\sqrt {3-6 x^2-2 x^4}} \, dx=\frac {1}{6} \, \sqrt {\sqrt {5} \sqrt {3} + 3} {\left (\sqrt {5} \sqrt {3} - 3\right )} F(\arcsin \left (\frac {1}{3} \, \sqrt {3} \sqrt {\sqrt {5} \sqrt {3} + 3} x\right )\,|\,\sqrt {5} \sqrt {3} - 4) \]

[In]

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

[Out]

1/6*sqrt(sqrt(5)*sqrt(3) + 3)*(sqrt(5)*sqrt(3) - 3)*elliptic_f(arcsin(1/3*sqrt(3)*sqrt(sqrt(5)*sqrt(3) + 3)*x)
, sqrt(5)*sqrt(3) - 4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/(3 - 2*x^4 - 6*x^2)^(1/2),x)

[Out]

int(1/(3 - 2*x^4 - 6*x^2)^(1/2), x)