3.2.17 \(\int \frac {1}{\sqrt {-2-3 x^2-3 x^4}} \, dx\) [117]

Optimal. Leaf size=92 \[ \frac {\left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2+3 x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{8} \left (4-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-2-3 x^2-3 x^4}} \]

[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)),1/4*(8-2*6^(1/2))^(1/2))*(2+x^2*6^(1/2))*((3*x^4+3*x^2+2)/(2+x^2*6^(1/2))^2)^(1/2)*6
^(3/4)/(-3*x^4-3*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+3 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}{8} \left (4-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-3 x^4-3 x^2-2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 - 3*x^2 - 3*x^4],x]

[Out]

((2 + Sqrt[6]*x^2)*Sqrt[(2 + 3*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTan[(3/2)^(1/4)*x], (4 - Sqrt[
6])/8])/(2*6^(1/4)*Sqrt[-2 - 3*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 {-2-3 x^2-3 x^4}} \, dx &=\frac {\left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2+3 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}{8} \left (4-\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-2-3 x^2-3 x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.11, size = 144, normalized size = 1.57 \begin {gather*} -\frac {i \sqrt {1-\frac {6 x^2}{-3-i \sqrt {15}}} \sqrt {1-\frac {6 x^2}{-3+i \sqrt {15}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {6}{-3-i \sqrt {15}}} x\right )|\frac {-3-i \sqrt {15}}{-3+i \sqrt {15}}\right )}{\sqrt {6} \sqrt {-\frac {1}{-3-i \sqrt {15}}} \sqrt {-2-3 x^2-3 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[1 - (6*x^2)/(-3 - I*Sqrt[15])]*Sqrt[1 - (6*x^2)/(-3 + I*Sqrt[15])]*EllipticF[I*ArcSinh[Sqrt[-6/(-3
- I*Sqrt[15])]*x], (-3 - I*Sqrt[15])/(-3 + I*Sqrt[15])])/(Sqrt[6]*Sqrt[-(-3 - I*Sqrt[15])^(-1)]*Sqrt[-2 - 3*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	\(\frac {2 \sqrt {1-\left (-\frac {3}{4}-\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}+\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-3-i \sqrt {15}}\, x}{2}, \frac {\sqrt {-1-i \sqrt {15}}}{2}\right )}{\sqrt {-3-i \sqrt {15}}\, \sqrt {-3 x^{4}-3 x^{2}-2}}\)	\(87\)
elliptic	\(\frac {2 \sqrt {1-\left (-\frac {3}{4}-\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}+\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-3-i \sqrt {15}}\, x}{2}, \frac {\sqrt {-1-i \sqrt {15}}}{2}\right )}{\sqrt {-3-i \sqrt {15}}\, \sqrt {-3 x^{4}-3 x^{2}-2}}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-3-I*15^(1/2))^(1/2)*(1-(-3/4-1/4*I*15^(1/2))*x^2)^(1/2)*(1-(-3/4+1/4*I*15^(1/2))*x^2)^(1/2)/(-3*x^4-3*x^2-
2)^(1/2)*EllipticF(1/2*(-3-I*15^(1/2))^(1/2)*x,1/2*(-1-I*15^(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^4-3*x^2-2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.07, size = 35, normalized size = 0.38 \begin {gather*} \frac {1}{24} \, \sqrt {-2} {\left (\sqrt {-15} + 3\right )} \sqrt {\sqrt {-15} - 3} {\rm ellipticF}\left (\frac {1}{2} \, x \sqrt {\sqrt {-15} - 3}, \frac {1}{4} \, \sqrt {-15} - \frac {1}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt(-2)*(sqrt(-15) + 3)*sqrt(sqrt(-15) - 3)*ellipticF(1/2*x*sqrt(sqrt(-15) - 3), 1/4*sqrt(-15) - 1/4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/sqrt(-3*x^4 - 3*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^4-3\,x^2-2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(- 3*x^2 - 3*x^4 - 2)^(1/2),x)

[Out]

int(1/(- 3*x^2 - 3*x^4 - 2)^(1/2), x)

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