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

Optimal. Leaf size=104 \[ \frac {\sqrt {\frac {\left (3-\sqrt {3}\right ) x^2+3}{\left (3+\sqrt {3}\right ) x^2+3}} \left (\left (3+\sqrt {3}\right ) x^2+3\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right )|-1+\sqrt {3}\right )}{\sqrt {3 \left (3+\sqrt {3}\right )} \sqrt {2 x^4+6 x^2+3}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1099} \[ \frac {\sqrt {\frac {\left (3-\sqrt {3}\right ) x^2+3}{\left (3+\sqrt {3}\right ) x^2+3}} \left (\left (3+\sqrt {3}\right ) x^2+3\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right )|-1+\sqrt {3}\right )}{\sqrt {3 \left (3+\sqrt {3}\right )} \sqrt {2 x^4+6 x^2+3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 + 6*x^2 + 2*x^4],x]

[Out]

(Sqrt[(3 + (3 - Sqrt[3])*x^2)/(3 + (3 + Sqrt[3])*x^2)]*(3 + (3 + Sqrt[3])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt
[3])/3]*x], -1 + Sqrt[3]])/(Sqrt[3*(3 + Sqrt[3])]*Sqrt[3 + 6*x^2 + 2*x^4])

Rule 1099

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[((2*a + (b +
q)*x^2)*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*EllipticF[ArcTan[Rt[(b + q)/(2*a), 2]*x], (2*q)/(b + q)]
)/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]), x] /; PosQ[(b + q)/a] &&  !(PosQ[(b - q)/a] && SimplerSq
rtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [C]  time = 0.06, size = 90, normalized size = 0.87 \[ -\frac {i \sqrt {\frac {-2 x^2+\sqrt {3}-3}{\sqrt {3}-3}} \sqrt {2 x^2+\sqrt {3}+3} F\left (i \sinh ^{-1}\left (\sqrt {1-\frac {1}{\sqrt {3}}} x\right )|2+\sqrt {3}\right )}{\sqrt {4 x^4+12 x^2+6}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*Sqrt[(-3 + Sqrt[3] - 2*x^2)/(-3 + Sqrt[3])]*Sqrt[3 + Sqrt[3] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - 1/Sqr
t[3]]*x], 2 + Sqrt[3]])/Sqrt[6 + 12*x^2 + 4*x^4]

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fricas [F]  time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} + 6 \, x^{2} + 3}}, x\right ) \]

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(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 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 6 \, x^{2} + 3}}\,{d x} \]

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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maple [A]  time = 0.08, size = 82, normalized size = 0.79 \[ \frac {3 \sqrt {-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}+1}\, \sqrt {-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-9+3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}} \]

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*3^(1/2))^(1/2)*(1-(-1+1/3*3^(1/2))*x^2)^(1/2)*(1-(-1-1/3*3^(1/2))*x^2)^(1/2)/(2*x^4+6*x^2+3)^(1/2)*Ell
ipticF(1/3*x*(-9+3*3^(1/2))^(1/2),1/2*6^(1/2)+1/2*2^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 6 \, x^{2} + 3}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {2\,x^4+6\,x^2+3}} \,d x \]

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 x^{4} + 6 x^{2} + 3}}\, dx \]

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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