3.28 \(\int \frac{1}{\sqrt{3+6 x^2-2 x^4}} \, dx\)

Optimal. Leaf size=44 \[ \sqrt{\frac{1}{6} \left (3+\sqrt{15}\right )} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{1}{3} \left (\sqrt{15}-3\right )} x\right ),-4-\sqrt{15}\right ) \]

[Out]

Sqrt[(3 + Sqrt[15])/6]*EllipticF[ArcSin[Sqrt[(-3 + Sqrt[15])/3]*x], -4 - Sqrt[15]]

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Rubi [A]  time = 0.132986, antiderivative size = 44, 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} \[ \sqrt{\frac{1}{6} \left (3+\sqrt{15}\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{3} \left (-3+\sqrt{15}\right )} x\right )|-4-\sqrt{15}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 + 6*x^2 - 2*x^4],x]

[Out]

Sqrt[(3 + Sqrt[15])/6]*EllipticF[ArcSin[Sqrt[(-3 + Sqrt[15])/3]*x], -4 - Sqrt[15]]

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{3+6 x^2-2 x^4}} \, dx &=\left (2 \sqrt{2}\right ) \int \frac{1}{\sqrt{6+2 \sqrt{15}-4 x^2} \sqrt{-6+2 \sqrt{15}+4 x^2}} \, dx\\ &=\sqrt{\frac{1}{6} \left (3+\sqrt{15}\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{3} \left (-3+\sqrt{15}\right )} x\right )|-4-\sqrt{15}\right )\\ \end{align*}

Mathematica [C]  time = 0.0520907, size = 43, normalized size = 0.98 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{1+\sqrt{\frac{5}{3}}} x\right ),\sqrt{15}-4\right )}{\sqrt{3+\sqrt{15}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*EllipticF[I*ArcSinh[Sqrt[1 + Sqrt[5/3]]*x], -4 + Sqrt[15]])/Sqrt[3 + Sqrt[15]]

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Maple [B]  time = 0.24, size = 84, normalized size = 1.9 \begin{align*} 3\,{\frac{\sqrt{1- \left ( -1+1/3\,\sqrt{15} \right ){x}^{2}}\sqrt{1- \left ( -1-1/3\,\sqrt{15} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{-9+3\,\sqrt{15}},i/2\sqrt{6}+i/2\sqrt{10} \right ) }{\sqrt{-9+3\,\sqrt{15}}\sqrt{-2\,{x}^{4}+6\,{x}^{2}+3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/(-9+3*15^(1/2))^(1/2)*(1-(-1+1/3*15^(1/2))*x^2)^(1/2)*(1-(-1-1/3*15^(1/2))*x^2)^(1/2)/(-2*x^4+6*x^2+3)^(1/2)
*EllipticF(1/3*x*(-9+3*15^(1/2))^(1/2),1/2*I*6^(1/2)+1/2*I*10^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} + 6 \, x^{2} + 3}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-2 \, x^{4} + 6 \, x^{2} + 3}}{2 \, x^{4} - 6 \, x^{2} - 3}, x\right ) \end{align*}

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]

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

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

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} + 6 \, x^{2} + 3}}\,{d x} \end{align*}

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)