3.1.17 \(\int \frac {1}{\sqrt {2+4 x^2-3 x^4}} \, dx\) [17]

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

[Out]

1/6*EllipticF(1/2*x*(-4+2*10^(1/2))^(1/2),1/3*I*6^(1/2)+1/3*I*15^(1/2))*(12+6*10^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 48, 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 = {1109, 430} \begin {gather*} \sqrt {\frac {1}{6} \left (2+\sqrt {10}\right )} F\left (\text {ArcSin}\left (\sqrt {\frac {1}{2} \left (-2+\sqrt {10}\right )} x\right )|\frac {1}{3} \left (-7-2 \sqrt {10}\right )\right ) \end {gather*}

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 430

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

Rule 1109

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 {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] Result contains complex when optimal does not.
time = 10.05, size = 49, normalized size = 1.02 \begin {gather*} -\frac {i F\left (i \sinh ^{-1}\left (\sqrt {1+\sqrt {\frac {5}{2}}} x\right )|\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{\sqrt {2+\sqrt {10}}} \end {gather*}

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]]

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (37 ) = 74\).
time = 0.07, size = 84, normalized size = 1.75

method	result	size
default	\(\frac {2 \sqrt {1-\left (-1+\frac {\sqrt {10}}{2}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {10}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-4+2 \sqrt {10}}}{2}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{\sqrt {-4+2 \sqrt {10}}\, \sqrt {-3 x^{4}+4 x^{2}+2}}\)	\(84\)
elliptic	\(\frac {2 \sqrt {1-\left (-1+\frac {\sqrt {10}}{2}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {10}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-4+2 \sqrt {10}}}{2}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{\sqrt {-4+2 \sqrt {10}}\, \sqrt {-3 x^{4}+4 x^{2}+2}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.09, size = 35, normalized size = 0.73 \begin {gather*} \frac {1}{6} \, {\left (\sqrt {10} + 2\right )} \sqrt {\sqrt {10} - 2} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {10} - 2}, -\frac {2}{3} \, \sqrt {10} - \frac {7}{3}\right ) \end {gather*}

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]

1/6*(sqrt(10) + 2)*sqrt(sqrt(10) - 2)*ellipticF(1/2*sqrt(2)*x*sqrt(sqrt(10) - 2), -2/3*sqrt(10) - 7/3)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x^2 - 3*x^4 + 2)^(1/2),x)

[Out]

int(1/(4*x^2 - 3*x^4 + 2)^(1/2), x)

________________________________________________________________________________________