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

Optimal. Leaf size=153 \[ \frac{\sqrt{-\left (3-\sqrt{33}\right ) x^2-4} \sqrt{\frac{\left (3+\sqrt{33}\right ) x^2+4}{\left (3-\sqrt{33}\right ) x^2+4}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{-\left (3-\sqrt{33}\right ) x^2-4}}\right ),\frac{1}{22} \left (11-\sqrt{33}\right )\right )}{2 \sqrt{2} \sqrt [4]{33} \sqrt{\frac{1}{\left (3-\sqrt{33}\right ) x^2+4}} \sqrt{3 x^4-3 x^2-2}} \]

[Out]

(Sqrt[-4 - (3 - Sqrt[33])*x^2]*Sqrt[(4 + (3 + Sqrt[33])*x^2)/(4 + (3 - Sqrt[33])*x^2)]*EllipticF[ArcSin[(Sqrt[
2]*33^(1/4)*x)/Sqrt[-4 - (3 - Sqrt[33])*x^2]], (11 - Sqrt[33])/22])/(2*Sqrt[2]*33^(1/4)*Sqrt[(4 + (3 - Sqrt[33
])*x^2)^(-1)]*Sqrt[-2 - 3*x^2 + 3*x^4])

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Rubi [A]  time = 0.030337, antiderivative size = 153, normalized size of antiderivative = 1., 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 = {1098} \[ \frac{\sqrt{-\left (3-\sqrt{33}\right ) x^2-4} \sqrt{\frac{\left (3+\sqrt{33}\right ) x^2+4}{\left (3-\sqrt{33}\right ) x^2+4}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{-\left (3-\sqrt{33}\right ) x^2-4}}\right )|\frac{1}{22} \left (11-\sqrt{33}\right )\right )}{2 \sqrt{2} \sqrt [4]{33} \sqrt{\frac{1}{\left (3-\sqrt{33}\right ) x^2+4}} \sqrt{3 x^4-3 x^2-2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 - 3*x^2 + 3*x^4],x]

[Out]

(Sqrt[-4 - (3 - Sqrt[33])*x^2]*Sqrt[(4 + (3 + Sqrt[33])*x^2)/(4 + (3 - Sqrt[33])*x^2)]*EllipticF[ArcSin[(Sqrt[
2]*33^(1/4)*x)/Sqrt[-4 - (3 - Sqrt[33])*x^2]], (11 - Sqrt[33])/22])/(2*Sqrt[2]*33^(1/4)*Sqrt[(4 + (3 - Sqrt[33
])*x^2)^(-1)]*Sqrt[-2 - 3*x^2 + 3*x^4])

Rule 1098

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-2-3 x^2+3 x^4}} \, dx &=\frac{\sqrt{-4-\left (3-\sqrt{33}\right ) x^2} \sqrt{\frac{4+\left (3+\sqrt{33}\right ) x^2}{4+\left (3-\sqrt{33}\right ) x^2}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{-4-\left (3-\sqrt{33}\right ) x^2}}\right )|\frac{1}{22} \left (11-\sqrt{33}\right )\right )}{2 \sqrt{2} \sqrt [4]{33} \sqrt{\frac{1}{4+\left (3-\sqrt{33}\right ) x^2}} \sqrt{-2-3 x^2+3 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0672875, size = 81, normalized size = 0.53 \[ -\frac{i \sqrt{-6 x^4+6 x^2+4} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{6}{\sqrt{33}-3}} x\right ),\frac{1}{4} \left (\sqrt{33}-7\right )\right )}{\sqrt{3+\sqrt{33}} \sqrt{3 x^4-3 x^2-2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*Sqrt[4 + 6*x^2 - 6*x^4]*EllipticF[I*ArcSinh[Sqrt[6/(-3 + Sqrt[33])]*x], (-7 + Sqrt[33])/4])/(Sqrt[3 + Sq
rt[33]]*Sqrt[-2 - 3*x^2 + 3*x^4])

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Maple [C]  time = 0.176, size = 84, normalized size = 0.6 \begin{align*} 2\,{\frac{\sqrt{1- \left ( -3/4-1/4\,\sqrt{33} \right ){x}^{2}}\sqrt{1- \left ( -3/4+1/4\,\sqrt{33} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{-\sqrt{33}-3}x,i/4\sqrt{22}-i/4\sqrt{6} \right ) }{\sqrt{-\sqrt{33}-3}\sqrt{3\,{x}^{4}-3\,{x}^{2}-2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-33^(1/2)-3)^(1/2)*(1-(-3/4-1/4*33^(1/2))*x^2)^(1/2)*(1-(-3/4+1/4*33^(1/2))*x^2)^(1/2)/(3*x^4-3*x^2-2)^(1/2
)*EllipticF(1/2*(-33^(1/2)-3)^(1/2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \, x^{4} - 3 \, x^{2} - 2}}\,{d x} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{3 \, x^{4} - 3 \, x^{2} - 2}}, x\right ) \end{align*}

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]

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

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

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \, x^{4} - 3 \, x^{2} - 2}}\,{d x} \end{align*}

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)