3.2.16 \(\int \frac {1}{\sqrt {-2-2 x^2-3 x^4}} \, dx\) [116]

Optimal. Leaf size=92 \[ \frac {\left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2+2 x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{12} \left (6-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-2-2 x^2-3 x^4}} \]

[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)),1/6*(18-3*6^(1/2))^(1/2))*(2+x^2*6^(1/2))*((3*x^4+2*x^2+2)/(2+x^2*6^(1/2))^2)^(1/2)*
6^(3/4)/(-3*x^4-2*x^2-2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1117} \begin {gather*} \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+2 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{12} \left (6-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-3 x^4-2 x^2-2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 - 2*x^2 - 3*x^4],x]

[Out]

((2 + Sqrt[6]*x^2)*Sqrt[(2 + 2*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTan[(3/2)^(1/4)*x], (6 - Sqrt[
6])/12])/(2*6^(1/4)*Sqrt[-2 - 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*} \int \frac {1}{\sqrt {-2-2 x^2-3 x^4}} \, dx &=\frac {\left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2+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}{12} \left (6-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-2-2 x^2-3 x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.05, size = 144, normalized size = 1.57 \begin {gather*} -\frac {i \sqrt {1-\frac {3 x^2}{-1-i \sqrt {5}}} \sqrt {1-\frac {3 x^2}{-1+i \sqrt {5}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {3}{-1-i \sqrt {5}}} x\right )|\frac {-1-i \sqrt {5}}{-1+i \sqrt {5}}\right )}{\sqrt {3} \sqrt {-\frac {1}{-1-i \sqrt {5}}} \sqrt {-2-2 x^2-3 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[1 - (3*x^2)/(-1 - I*Sqrt[5])]*Sqrt[1 - (3*x^2)/(-1 + I*Sqrt[5])]*EllipticF[I*ArcSinh[Sqrt[-3/(-1 -
I*Sqrt[5])]*x], (-1 - I*Sqrt[5])/(-1 + I*Sqrt[5])])/(Sqrt[3]*Sqrt[-(-1 - I*Sqrt[5])^(-1)]*Sqrt[-2 - 2*x^2 - 3*
x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.04, size = 87, normalized size = 0.95

method	result	size
default	\(\frac {2 \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-2-2 i \sqrt {5}}\, x}{2}, \frac {\sqrt {-6-3 i \sqrt {5}}}{3}\right )}{\sqrt {-2-2 i \sqrt {5}}\, \sqrt {-3 x^{4}-2 x^{2}-2}}\)	\(87\)
elliptic	\(\frac {2 \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-2-2 i \sqrt {5}}\, x}{2}, \frac {\sqrt {-6-3 i \sqrt {5}}}{3}\right )}{\sqrt {-2-2 i \sqrt {5}}\, \sqrt {-3 x^{4}-2 x^{2}-2}}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-2-2*I*5^(1/2))^(1/2)*(1-(-1/2-1/2*I*5^(1/2))*x^2)^(1/2)*(1-(-1/2+1/2*I*5^(1/2))*x^2)^(1/2)/(-3*x^4-2*x^2-2
)^(1/2)*EllipticF(1/2*(-2-2*I*5^(1/2))^(1/2)*x,1/3*(-6-3*I*5^(1/2))^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 0.07, size = 41, normalized size = 0.45 \begin {gather*} \frac {1}{12} \, \sqrt {2} \sqrt {-2} {\left (\sqrt {-5} + 1\right )} \sqrt {\sqrt {-5} - 1} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-5} - 1}, \frac {1}{3} \, \sqrt {-5} - \frac {2}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(2)*sqrt(-2)*(sqrt(-5) + 1)*sqrt(sqrt(-5) - 1)*ellipticF(1/2*sqrt(2)*x*sqrt(sqrt(-5) - 1), 1/3*sqrt(-
5) - 2/3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- 3 x^{4} - 2 x^{2} - 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {-3\,x^4-2\,x^2-2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(- 2*x^2 - 3*x^4 - 2)^(1/2),x)

[Out]

int(1/(- 2*x^2 - 3*x^4 - 2)^(1/2), x)

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