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

Optimal. Leaf size=44 \[ \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-2+\sqrt {10}}} x\right )|\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{\sqrt {2+\sqrt {10}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, 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 = {1095, 419} \[ \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-2+\sqrt {10}}} x\right )|\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{\sqrt {2+\sqrt {10}}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 - 4*x^2 - 2*x^4],x]

[Out]

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

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)])

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.05, size = 51, normalized size = 1.16 \[ -\frac {i F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{2+\sqrt {10}}} x\right )|-\frac {7}{3}-\frac {2 \sqrt {10}}{3}\right )}{\sqrt {\sqrt {10}-2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} - 4 \, x^{2} + 3}}{2 \, x^{4} + 4 \, x^{2} - 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-2 \, x^{4} - 4 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.08, size = 84, normalized size = 1.91 \[ \frac {3 \sqrt {-\left (\frac {\sqrt {10}}{3}+\frac {2}{3}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {\sqrt {10}}{3}+\frac {2}{3}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {6+3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {15}}{3}-\frac {i \sqrt {6}}{3}\right )}{\sqrt {6+3 \sqrt {10}}\, \sqrt {-2 x^{4}-4 x^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-2 \, x^{4} - 4 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {-2\,x^4-4\,x^2+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3 - 2*x^4 - 4*x^2)^(1/2),x)

[Out]

int(1/(3 - 2*x^4 - 4*x^2)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- 2 x^{4} - 4 x^{2} + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________