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

Optimal. Leaf size=48 \[ \sqrt{\frac{1}{6} \left (2+\sqrt{10}\right )} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{1}{2} \left (\sqrt{10}-2\right )} x\right ),\frac{1}{3} \left (-7-2 \sqrt{10}\right )\right ) \]

[Out]

Sqrt[(2 + Sqrt[10])/6]*EllipticF[ArcSin[Sqrt[(-2 + Sqrt[10])/2]*x], (-7 - 2*Sqrt[10])/3]

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Rubi [A]  time = 0.133588, antiderivative size = 48, normalized size of antiderivative = 1., 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 = {1095, 419} \[ \sqrt{\frac{1}{6} \left (2+\sqrt{10}\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{2} \left (-2+\sqrt{10}\right )} x\right )|\frac{1}{3} \left (-7-2 \sqrt{10}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 4*x^2 - 3*x^4],x]

[Out]

Sqrt[(2 + Sqrt[10])/6]*EllipticF[ArcSin[Sqrt[(-2 + Sqrt[10])/2]*x], (-7 - 2*Sqrt[10])/3]

Rule 1095

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]

Rule 419

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{2+4 x^2-3 x^4}} \, dx &=\left (2 \sqrt{3}\right ) \int \frac{1}{\sqrt{4+2 \sqrt{10}-6 x^2} \sqrt{-4+2 \sqrt{10}+6 x^2}} \, dx\\ &=\sqrt{\frac{1}{6} \left (2+\sqrt{10}\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{2} \left (-2+\sqrt{10}\right )} x\right )|\frac{1}{3} \left (-7-2 \sqrt{10}\right )\right )\\ \end{align*}

Mathematica [C]  time = 0.0562925, size = 49, normalized size = 1.02 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{1+\sqrt{\frac{5}{2}}} x\right ),\frac{1}{3} \left (2 \sqrt{10}-7\right )\right )}{\sqrt{2+\sqrt{10}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*EllipticF[I*ArcSinh[Sqrt[1 + Sqrt[5/2]]*x], (-7 + 2*Sqrt[10])/3])/Sqrt[2 + Sqrt[10]]

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Maple [B]  time = 0.24, size = 84, normalized size = 1.8 \begin{align*} 2\,{\frac{\sqrt{1- \left ( -1+1/2\,\sqrt{10} \right ){x}^{2}}\sqrt{1- \left ( -1-1/2\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-4+2\,\sqrt{10}},i/3\sqrt{6}+i/3\sqrt{15} \right ) }{\sqrt{-4+2\,\sqrt{10}}\sqrt{-3\,{x}^{4}+4\,{x}^{2}+2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-4+2*10^(1/2))^(1/2)*(1-(-1+1/2*10^(1/2))*x^2)^(1/2)*(1-(-1-1/2*10^(1/2))*x^2)^(1/2)/(-3*x^4+4*x^2+2)^(1/2)
*EllipticF(1/2*x*(-4+2*10^(1/2))^(1/2),1/3*I*6^(1/2)+1/3*I*15^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-3*x^4 + 4*x^2 + 2)/(3*x^4 - 4*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} + 4 x^{2} + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-3*x**4 + 4*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} + 4 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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