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]

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

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Rubi [A]  time = 0.139387, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\sqrt{\frac{\left (3-\sqrt{3}\right ) x^2+3}{\left (3+\sqrt{3}\right ) x^2+3}} \left (\left (3+\sqrt{3}\right ) x^2+3\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{3} \left (3+\sqrt{3}\right )} x\right )|-1+\sqrt{3}\right )}{\sqrt{3 \left (3+\sqrt{3}\right )} \sqrt{2 x^4+6 x^2+3}} \]

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

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Rubi in Sympy [A]  time = 4.96566, size = 102, normalized size = 0.98 \[ \frac{\sqrt{3} \sqrt{\frac{x^{2} \left (- 2 \sqrt{3} + 6\right ) + 6}{x^{2} \left (2 \sqrt{3} + 6\right ) + 6}} \left (x^{2} \left (2 \sqrt{3} + 6\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{3} x \sqrt{\sqrt{3} + 3}}{3} \right )}\middle | -1 + \sqrt{3}\right )}{6 \sqrt{\sqrt{3} + 3} \sqrt{2 x^{4} + 6 x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*x**4+6*x**2+3)**(1/2),x)

[Out]

sqrt(3)*sqrt((x**2*(-2*sqrt(3) + 6) + 6)/(x**2*(2*sqrt(3) + 6) + 6))*(x**2*(2*sq
rt(3) + 6) + 6)*elliptic_f(atan(sqrt(3)*x*sqrt(sqrt(3) + 3)/3), -1 + sqrt(3))/(6
*sqrt(sqrt(3) + 3)*sqrt(2*x**4 + 6*x**2 + 3))

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Mathematica [C]  time = 0.0977932, 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]*Elli
pticF[I*ArcSinh[Sqrt[1 - 1/Sqrt[3]]*x], 2 + Sqrt[3]])/Sqrt[6 + 12*x^2 + 4*x^4]

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Maple [A]  time = 0.221, size = 82, normalized size = 0.8 \[ 3\,{\frac{\sqrt{1- \left ( -1+1/3\,\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -1-1/3\,\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{-9+3\,\sqrt{3}},1/2\,\sqrt{6}+1/2\,\sqrt{2} \right ) }{\sqrt{-9+3\,\sqrt{3}}\sqrt{2\,{x}^{4}+6\,{x}^{2}+3}}} \]

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)*EllipticF(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., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 6 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 + 6*x^2 + 3),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 + 6*x^2 + 3),x, algorithm="fricas")

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 + 6*x^2 + 3),x, algorithm="giac")

[Out]

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