\(\int \frac {1}{\sqrt {-2+x^2-3 x^4}} \, dx\) [113]

Optimal result

Integrand size = 14, antiderivative size = 88 \[ \int \frac {1}{\sqrt {-2+x^2-3 x^4}} \, dx=\frac {\left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2-x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{24} \left (12+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-2+x^2-3 x^4}} \]

[Out]

1/12*(cos(2*arctan(1/2*3^(1/4)*2^(3/4)*x))^2)^(1/2)/cos(2*arctan(1/2*3^(1/4)*2^(3/4)*x))*EllipticF(sin(2*arcta
n(1/2*3^(1/4)*2^(3/4)*x)),1/12*(72+6*6^(1/2))^(1/2))*(2+x^2*6^(1/2))*((3*x^4-x^2+2)/(2+x^2*6^(1/2))^2)^(1/2)*6
^(3/4)/(-3*x^4+x^2-2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1117} \[ \int \frac {1}{\sqrt {-2+x^2-3 x^4}} \, dx=\frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{24} \left (12+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-3 x^4+x^2-2}} \]

[In]

Int[1/Sqrt[-2 + x^2 - 3*x^4],x]

[Out]

((2 + Sqrt[6]*x^2)*Sqrt[(2 - x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTan[(3/2)^(1/4)*x], (12 + Sqrt[6
])/24])/(2*6^(1/4)*Sqrt[-2 + x^2 - 3*x^4])

Rule 1117

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2-x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{24} \left (12+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-2+x^2-3 x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.07 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\sqrt {-2+x^2-3 x^4}} \, dx=-\frac {i \sqrt {1-\frac {6 x^2}{1-i \sqrt {23}}} \sqrt {1-\frac {6 x^2}{1+i \sqrt {23}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {6}{1-i \sqrt {23}}} x\right ),\frac {1-i \sqrt {23}}{1+i \sqrt {23}}\right )}{\sqrt {6} \sqrt {-\frac {1}{1-i \sqrt {23}}} \sqrt {-2+x^2-3 x^4}} \]

[In]

Integrate[1/Sqrt[-2 + x^2 - 3*x^4],x]

[Out]

((-I)*Sqrt[1 - (6*x^2)/(1 - I*Sqrt[23])]*Sqrt[1 - (6*x^2)/(1 + I*Sqrt[23])]*EllipticF[I*ArcSinh[Sqrt[-6/(1 - I
*Sqrt[23])]*x], (1 - I*Sqrt[23])/(1 + I*Sqrt[23])])/(Sqrt[6]*Sqrt[-(1 - I*Sqrt[23])^(-1)]*Sqrt[-2 + x^2 - 3*x^
4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.61 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {2 \sqrt {1-\left (\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, F\left (\frac {\sqrt {1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )}{\sqrt {1-i \sqrt {23}}\, \sqrt {-3 x^{4}+x^{2}-2}}\)	\(85\)
elliptic	\(\frac {2 \sqrt {1-\left (\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, F\left (\frac {\sqrt {1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )}{\sqrt {1-i \sqrt {23}}\, \sqrt {-3 x^{4}+x^{2}-2}}\)	\(85\)

[In]

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

[Out]

2/(1-I*23^(1/2))^(1/2)*(1-(1/4-1/4*I*23^(1/2))*x^2)^(1/2)*(1-(1/4+1/4*I*23^(1/2))*x^2)^(1/2)/(-3*x^4+x^2-2)^(1
/2)*EllipticF(1/2*(1-I*23^(1/2))^(1/2)*x,1/6*(-33+3*I*23^(1/2))^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\sqrt {-2+x^2-3 x^4}} \, dx=\frac {1}{24} \, \sqrt {-2} \sqrt {\sqrt {-23} + 1} {\left (\sqrt {-23} - 1\right )} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {-23} + 1}\right )\,|\,-\frac {1}{12} \, \sqrt {-23} - \frac {11}{12}) \]

[In]

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

[Out]

1/24*sqrt(-2)*sqrt(sqrt(-23) + 1)*(sqrt(-23) - 1)*elliptic_f(arcsin(1/2*x*sqrt(sqrt(-23) + 1)), -1/12*sqrt(-23
) - 11/12)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/(x^2 - 3*x^4 - 2)^(1/2),x)

[Out]

int(1/(x^2 - 3*x^4 - 2)^(1/2), x)