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

Optimal. Leaf size=42 \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-1}} x\right ),\frac{1}{3} \left (\sqrt{7}-4\right )\right )}{\sqrt{1+\sqrt{7}}} \]

[Out]

EllipticF[ArcSin[Sqrt[2/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3]/Sqrt[1 + Sqrt[7]]

________________________________________________________________________________________

Rubi [A]  time = 0.0473294, antiderivative size = 42, 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} \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{7}}} x\right )|\frac{1}{3} \left (-4+\sqrt{7}\right )\right )}{\sqrt{1+\sqrt{7}}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 - 2*x^2 - 2*x^4],x]

[Out]

EllipticF[ArcSin[Sqrt[2/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3]/Sqrt[1 + Sqrt[7]]

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

Mathematica [C]  time = 0.0440687, size = 51, normalized size = 1.21 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{7}}} x\right ),-\frac{4}{3}-\frac{\sqrt{7}}{3}\right )}{\sqrt{\sqrt{7}-1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*EllipticF[I*ArcSinh[Sqrt[2/(1 + Sqrt[7])]*x], -4/3 - Sqrt[7]/3])/Sqrt[-1 + Sqrt[7]]

________________________________________________________________________________________

Maple [B]  time = 0.208, size = 84, normalized size = 2. \begin{align*} 3\,{\frac{\sqrt{1- \left ( 1/3\,\sqrt{7}+1/3 \right ){x}^{2}}\sqrt{1- \left ( -1/3\,\sqrt{7}+1/3 \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{3+3\,\sqrt{7}},i/6\sqrt{42}-i/6\sqrt{6} \right ) }{\sqrt{3+3\,\sqrt{7}}\sqrt{-2\,{x}^{4}-2\,{x}^{2}+3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/(3+3*7^(1/2))^(1/2)*(1-(1/3*7^(1/2)+1/3)*x^2)^(1/2)*(1-(-1/3*7^(1/2)+1/3)*x^2)^(1/2)/(-2*x^4-2*x^2+3)^(1/2)*
EllipticF(1/3*x*(3+3*7^(1/2))^(1/2),1/6*I*42^(1/2)-1/6*I*6^(1/2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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