3.1.96 \(\int \frac {1}{\sqrt {-3+6 x^2-2 x^4}} \, dx\) [96]

Optimal. Leaf size=44 \[ -\frac {F\left (\cos ^{-1}\left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right )|\frac {1}{2} \left (1+\sqrt {3}\right )\right )}{\sqrt {2} \sqrt [4]{3}} \]

[Out]

-1/6*(x^2*(9-3*3^(1/2)))^(1/2)/x/(9-3*3^(1/2))^(1/2)*EllipticF(1/3*(9-x^2*(9-3*3^(1/2)))^(1/2),1/2*(2+2*3^(1/2
))^(1/2))*3^(3/4)*2^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1109, 431} \begin {gather*} -\frac {F\left (\text {ArcCos}\left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right )|\frac {1}{2} \left (1+\sqrt {3}\right )\right )}{\sqrt {2} \sqrt [4]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-3 + 6*x^2 - 2*x^4],x]

[Out]

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

Rule 431

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(-(Sqrt[c]*Rt[-d/c, 2]*Sqrt[a -
b*(c/d)])^(-1))*EllipticF[ArcCos[Rt[-d/c, 2]*x], b*(c/(b*c - a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c
] && GtQ[c, 0] && GtQ[a - b*(c/d), 0]

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

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Mathematica [A]
time = 10.04, size = 81, normalized size = 1.84 \begin {gather*} \frac {\sqrt {3-\sqrt {3}-2 x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2} F\left (\sin ^{-1}\left (\sqrt {1+\frac {1}{\sqrt {3}}} x\right )|2-\sqrt {3}\right )}{\sqrt {6} \sqrt {-3+6 x^2-2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-3 + 6*x^2 - 2*x^4],x]

[Out]

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

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Maple [A]
time = 0.04, size = 82, normalized size = 1.86

method	result	size
default	\(\frac {3 \sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {9-3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {9-3 \sqrt {3}}\, \sqrt {-2 x^{4}+6 x^{2}-3}}\)	\(82\)
elliptic	\(\frac {3 \sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {9-3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {9-3 \sqrt {3}}\, \sqrt {-2 x^{4}+6 x^{2}-3}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-2*x^4+6*x^2-3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

3/(9-3*3^(1/2))^(1/2)*(1-(1-1/3*3^(1/2))*x^2)^(1/2)*(1-(1+1/3*3^(1/2))*x^2)^(1/2)/(-2*x^4+6*x^2-3)^(1/2)*Ellip
ticF(1/3*x*(9-3*3^(1/2))^(1/2),1/2*6^(1/2)+1/2*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/(-2*x^4+6*x^2-3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-2*x^4 + 6*x^2 - 3), x)

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Fricas [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/(-2*x^4+6*x^2-3)^(1/2),x, algorithm="fricas")

[Out]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- 2 x^{4} + 6 x^{2} - 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x**4+6*x**2-3)**(1/2),x)

[Out]

Integral(1/sqrt(-2*x**4 + 6*x**2 - 3), 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/(-2*x^4+6*x^2-3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-2*x^4 + 6*x^2 - 3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {-2\,x^4+6\,x^2-3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(6*x^2 - 2*x^4 - 3)^(1/2),x)

[Out]

int(1/(6*x^2 - 2*x^4 - 3)^(1/2), x)

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