3.1.28 \(\int \frac {1}{\sqrt {3+6 x^2-2 x^4}} \, dx\) [28]

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

[Out]

1/6*EllipticF(1/3*x*(-9+3*15^(1/2))^(1/2),1/2*I*6^(1/2)+1/2*I*10^(1/2))*(18+6*15^(1/2))^(1/2)

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Rubi [A]
time = 0.09, 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 = {1109, 430} \begin {gather*} \sqrt {\frac {1}{6} \left (3+\sqrt {15}\right )} F\left (\text {ArcSin}\left (\sqrt {\frac {1}{3} \left (-3+\sqrt {15}\right )} x\right )|-4-\sqrt {15}\right ) \end {gather*}

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 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 {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.06, size = 43, normalized size = 0.98 \begin {gather*} -\frac {i F\left (i \sinh ^{-1}\left (\sqrt {1+\sqrt {\frac {5}{3}}} x\right )|-4+\sqrt {15}\right )}{\sqrt {3+\sqrt {15}}} \end {gather*}

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] 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 = 1.91

method	result	size
default	\(\frac {3 \sqrt {1-\left (-1+\frac {\sqrt {15}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {15}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-9+3 \sqrt {15}}}{3}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right )}{\sqrt {-9+3 \sqrt {15}}\, \sqrt {-2 x^{4}+6 x^{2}+3}}\)	\(84\)
elliptic	\(\frac {3 \sqrt {1-\left (-1+\frac {\sqrt {15}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {15}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-9+3 \sqrt {15}}}{3}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right )}{\sqrt {-9+3 \sqrt {15}}\, \sqrt {-2 x^{4}+6 x^{2}+3}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, 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/(-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 [A]
time = 0.08, size = 50, normalized size = 1.14 \begin {gather*} \frac {1}{6} \, {\left (\sqrt {5} \sqrt {3} + 3\right )} \sqrt {\sqrt {5} \sqrt {3} - 3} {\rm ellipticF}\left (\frac {1}{3} \, \sqrt {3} \sqrt {\sqrt {5} \sqrt {3} - 3} x, -\sqrt {5} \sqrt {3} - 4\right ) \end {gather*}

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]

1/6*(sqrt(5)*sqrt(3) + 3)*sqrt(sqrt(5)*sqrt(3) - 3)*ellipticF(1/3*sqrt(3)*sqrt(sqrt(5)*sqrt(3) - 3)*x, -sqrt(5
)*sqrt(3) - 4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- 2 x^{4} + 6 x^{2} + 3}}\, dx \end {gather*}

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 = 0.00, 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/(-2*x^4+6*x^2+3)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {-2\,x^4+6\,x^2+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(6*x^2 - 2*x^4 + 3)^(1/2),x)

[Out]

int(1/(6*x^2 - 2*x^4 + 3)^(1/2), x)

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