3.79 \(\int \frac{1}{\sqrt{3+8 x^2+2 x^4}} \, dx\)

Optimal. Leaf size=110 \[ \frac{\sqrt{\frac{\left (4-\sqrt{10}\right ) x^2+3}{\left (4+\sqrt{10}\right ) x^2+3}} \left (\left (4+\sqrt{10}\right ) x^2+3\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{3} \left (4+\sqrt{10}\right )} x\right )|-\frac{2}{3} \left (5-2 \sqrt{10}\right )\right )}{\sqrt{3 \left (4+\sqrt{10}\right )} \sqrt{2 x^4+8 x^2+3}} \]

[Out]

(Sqrt[(3 + (4 - Sqrt[10])*x^2)/(3 + (4 + Sqrt[10])*x^2)]*(3 + (4 + Sqrt[10])*x^2
)*EllipticF[ArcTan[Sqrt[(4 + Sqrt[10])/3]*x], (-2*(5 - 2*Sqrt[10]))/3])/(Sqrt[3*
(4 + Sqrt[10])]*Sqrt[3 + 8*x^2 + 2*x^4])

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Rubi [A]  time = 0.184747, antiderivative size = 110, 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 \[ \frac{\sqrt{\frac{\left (4-\sqrt{10}\right ) x^2+3}{\left (4+\sqrt{10}\right ) x^2+3}} \left (\left (4+\sqrt{10}\right ) x^2+3\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{3} \left (4+\sqrt{10}\right )} x\right )|-\frac{2}{3} \left (5-2 \sqrt{10}\right )\right )}{\sqrt{3 \left (4+\sqrt{10}\right )} \sqrt{2 x^4+8 x^2+3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[3 + 8*x^2 + 2*x^4],x]

[Out]

(Sqrt[(3 + (4 - Sqrt[10])*x^2)/(3 + (4 + Sqrt[10])*x^2)]*(3 + (4 + Sqrt[10])*x^2
)*EllipticF[ArcTan[Sqrt[(4 + Sqrt[10])/3]*x], (-2*(5 - 2*Sqrt[10]))/3])/(Sqrt[3*
(4 + Sqrt[10])]*Sqrt[3 + 8*x^2 + 2*x^4])

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Rubi in Sympy [A]  time = 5.06709, size = 107, normalized size = 0.97 \[ \frac{\sqrt{3} \sqrt{\frac{x^{2} \left (- 2 \sqrt{10} + 8\right ) + 6}{x^{2} \left (2 \sqrt{10} + 8\right ) + 6}} \left (x^{2} \left (2 \sqrt{10} + 8\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{3} x \sqrt{\sqrt{10} + 4}}{3} \right )}\middle | - \frac{10}{3} + \frac{4 \sqrt{10}}{3}\right )}{6 \sqrt{\sqrt{10} + 4} \sqrt{2 x^{4} + 8 x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*x**4+8*x**2+3)**(1/2),x)

[Out]

sqrt(3)*sqrt((x**2*(-2*sqrt(10) + 8) + 6)/(x**2*(2*sqrt(10) + 8) + 6))*(x**2*(2*
sqrt(10) + 8) + 6)*elliptic_f(atan(sqrt(3)*x*sqrt(sqrt(10) + 4)/3), -10/3 + 4*sq
rt(10)/3)/(6*sqrt(sqrt(10) + 4)*sqrt(2*x**4 + 8*x**2 + 3))

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Mathematica [C]  time = 0.140025, size = 98, normalized size = 0.89 \[ -\frac{i \sqrt{\frac{-2 x^2+\sqrt{10}-4}{\sqrt{10}-4}} \sqrt{2 x^2+\sqrt{10}+4} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{4+\sqrt{10}}} x\right )|\frac{13}{3}+\frac{4 \sqrt{10}}{3}\right )}{\sqrt{4 x^4+16 x^2+6}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/Sqrt[3 + 8*x^2 + 2*x^4],x]

[Out]

((-I)*Sqrt[(-4 + Sqrt[10] - 2*x^2)/(-4 + Sqrt[10])]*Sqrt[4 + Sqrt[10] + 2*x^2]*E
llipticF[I*ArcSinh[Sqrt[2/(4 + Sqrt[10])]*x], 13/3 + (4*Sqrt[10])/3])/Sqrt[6 + 1
6*x^2 + 4*x^4]

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Maple [A]  time = 0.234, size = 82, normalized size = 0.8 \[ 3\,{\frac{\sqrt{1- \left ( -4/3+1/3\,\sqrt{10} \right ){x}^{2}}\sqrt{1- \left ( -4/3-1/3\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{-12+3\,\sqrt{10}},2/3\,\sqrt{6}+1/3\,\sqrt{15} \right ) }{\sqrt{-12+3\,\sqrt{10}}\sqrt{2\,{x}^{4}+8\,{x}^{2}+3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*x^4+8*x^2+3)^(1/2),x)

[Out]

3/(-12+3*10^(1/2))^(1/2)*(1-(-4/3+1/3*10^(1/2))*x^2)^(1/2)*(1-(-4/3-1/3*10^(1/2)
)*x^2)^(1/2)/(2*x^4+8*x^2+3)^(1/2)*EllipticF(1/3*x*(-12+3*10^(1/2))^(1/2),2/3*6^
(1/2)+1/3*15^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 8 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 + 8*x^2 + 3),x, algorithm="maxima")

[Out]

integrate(1/sqrt(2*x^4 + 8*x^2 + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{2 \, x^{4} + 8 \, x^{2} + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 + 8*x^2 + 3),x, algorithm="fricas")

[Out]

integral(1/sqrt(2*x^4 + 8*x^2 + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 x^{4} + 8 x^{2} + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*x**4+8*x**2+3)**(1/2),x)

[Out]

Integral(1/sqrt(2*x**4 + 8*x**2 + 3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 8 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 + 8*x^2 + 3),x, algorithm="giac")

[Out]

integrate(1/sqrt(2*x^4 + 8*x^2 + 3), x)