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

Optimal. Leaf size=88 \[ \frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4-x^2+3}{\left (\sqrt {6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{24} \left (12+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-2 x^4+x^2-3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1103} \[ \frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4-x^2+3}{\left (\sqrt {6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{24} \left (12+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-2 x^4+x^2-3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-3 + x^2 - 2*x^4],x]

[Out]

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

Rule 1103

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)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.06, size = 140, normalized size = 1.59 \[ -\frac {i \sqrt {1-\frac {4 x^2}{1-i \sqrt {23}}} \sqrt {1-\frac {4 x^2}{1+i \sqrt {23}}} F\left (i \sinh ^{-1}\left (2 \sqrt {-\frac {1}{1-i \sqrt {23}}} x\right )|\frac {1-i \sqrt {23}}{1+i \sqrt {23}}\right )}{2 \sqrt {-\frac {1}{1-i \sqrt {23}}} \sqrt {-2 x^4+x^2-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-3 + x^2 - 2*x^4],x]

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} + x^{2} - 3}}{2 \, x^{4} - x^{2} + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + x^{2} - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.05, size = 85, normalized size = 0.97 \[ \frac {6 \sqrt {-\left (\frac {1}{6}-\frac {i \sqrt {23}}{6}\right ) x^{2}+1}\, \sqrt {-\left (\frac {1}{6}+\frac {i \sqrt {23}}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {6-6 i \sqrt {23}}\, x}{6}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )}{\sqrt {6-6 i \sqrt {23}}\, \sqrt {-2 x^{4}+x^{2}-3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + x^{2} - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {-2\,x^4+x^2-3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2 - 2*x^4 - 3)^(1/2),x)

[Out]

int(1/(x^2 - 2*x^4 - 3)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- 2 x^{4} + x^{2} - 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________