\(\int \frac {1}{\sqrt {2-2 x^2-3 x^4}} \, dx\) [23]

Optimal result

Integrand size = 16, antiderivative size = 42 \[ \int \frac {1}{\sqrt {2-2 x^2-3 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}} \]

[Out]

EllipticF(x*3^(1/2)/(-1+7^(1/2))^(1/2),1/6*I*42^(1/2)-1/6*I*6^(1/2))/(1+7^(1/2))^(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, 430} \[ \int \frac {1}{\sqrt {2-2 x^2-3 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}} \]

[In]

Int[1/Sqrt[2 - 2*x^2 - 3*x^4],x]

[Out]

EllipticF[ArcSin[Sqrt[3/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3]/Sqrt[1 + Sqrt[7]]

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 {3}\right ) \int \frac {1}{\sqrt {-2+2 \sqrt {7}-6 x^2} \sqrt {2+2 \sqrt {7}+6 x^2}} \, dx \\ & = \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\sqrt {2-2 x^2-3 x^4}} \, dx=-\frac {i \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right ),-\frac {4}{3}-\frac {\sqrt {7}}{3}\right )}{\sqrt {-1+\sqrt {7}}} \]

[In]

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

[Out]

((-I)*EllipticF[I*ArcSinh[Sqrt[3/(1 + Sqrt[7])]*x], -4/3 - Sqrt[7]/3])/Sqrt[-1 + Sqrt[7]]

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. \(2 (34 ) = 68\).

Time = 0.36 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.00

method	result	size
default	\(\frac {2 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {2+2 \sqrt {7}}}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{\sqrt {2+2 \sqrt {7}}\, \sqrt {-3 x^{4}-2 x^{2}+2}}\)	\(84\)
elliptic	\(\frac {2 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {2+2 \sqrt {7}}}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{\sqrt {2+2 \sqrt {7}}\, \sqrt {-3 x^{4}-2 x^{2}+2}}\)	\(84\)

[In]

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

[Out]

2/(2+2*7^(1/2))^(1/2)*(1-(1/2+1/2*7^(1/2))*x^2)^(1/2)*(1-(1/2-1/2*7^(1/2))*x^2)^(1/2)/(-3*x^4-2*x^2+2)^(1/2)*E
llipticF(1/2*x*(2+2*7^(1/2))^(1/2),1/6*I*42^(1/2)-1/6*I*6^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {2-2 x^2-3 x^4}} \, dx=\frac {1}{6} \, \sqrt {\sqrt {7} + 1} {\left (\sqrt {7} - 1\right )} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {7} + 1}\right )\,|\,\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) \]

[In]

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

[Out]

1/6*sqrt(sqrt(7) + 1)*(sqrt(7) - 1)*elliptic_f(arcsin(1/2*sqrt(2)*x*sqrt(sqrt(7) + 1)), 1/3*sqrt(7) - 4/3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/(2 - 3*x^4 - 2*x^2)^(1/2),x)

[Out]

int(1/(2 - 3*x^4 - 2*x^2)^(1/2), x)