3.38 \(\int \frac{1}{\sqrt{3-4 x^2-2 x^4}} \, dx\)

Optimal. Leaf size=44 \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{-2+\sqrt{10}}} x\right )|\frac{1}{3} \left (-7+2 \sqrt{10}\right )\right )}{\sqrt{2+\sqrt{10}}} \]

[Out]

EllipticF[ArcSin[Sqrt[2/(-2 + Sqrt[10])]*x], (-7 + 2*Sqrt[10])/3]/Sqrt[2 + Sqrt[
10]]

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Rubi [A]  time = 0.219771, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{-2+\sqrt{10}}} x\right )|\frac{1}{3} \left (-7+2 \sqrt{10}\right )\right )}{\sqrt{2+\sqrt{10}}} \]

Antiderivative was successfully verified.

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

[Out]

EllipticF[ArcSin[Sqrt[2/(-2 + Sqrt[10])]*x], (-7 + 2*Sqrt[10])/3]/Sqrt[2 + Sqrt[
10]]

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Rubi in Sympy [A]  time = 19.6938, size = 71, normalized size = 1.61 \[ \frac{2 \sqrt{6} F\left (\operatorname{asin}{\left (\frac{\sqrt{3} x \sqrt{2 + \sqrt{10}}}{3} \right )}\middle | - \frac{7}{3} + \frac{2 \sqrt{10}}{3}\right )}{\sqrt{-4 + 2 \sqrt{10}} \sqrt{2 + \sqrt{10}} \sqrt{4 + 2 \sqrt{10}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(6)*elliptic_f(asin(sqrt(3)*x*sqrt(2 + sqrt(10))/3), -7/3 + 2*sqrt(10)/3)/
(sqrt(-4 + 2*sqrt(10))*sqrt(2 + sqrt(10))*sqrt(4 + 2*sqrt(10)))

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Mathematica [C]  time = 0.101334, size = 51, normalized size = 1.16 \[ -\frac{i F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{2+\sqrt{10}}} x\right )|-\frac{7}{3}-\frac{2 \sqrt{10}}{3}\right )}{\sqrt{\sqrt{10}-2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-I)*EllipticF[I*ArcSinh[Sqrt[2/(2 + Sqrt[10])]*x], -7/3 - (2*Sqrt[10])/3])/Sqr
t[-2 + Sqrt[10]]

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Maple [B]  time = 0.091, size = 84, normalized size = 1.9 \[ 3\,{\frac{\sqrt{1- \left ( 2/3+1/3\,\sqrt{10} \right ){x}^{2}}\sqrt{1- \left ( 2/3-1/3\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{6+3\,\sqrt{10}},i/3\sqrt{15}-i/3\sqrt{6} \right ) }{\sqrt{6+3\,\sqrt{10}}\sqrt{-2\,{x}^{4}-4\,{x}^{2}+3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-2*x^4-4*x^2+3)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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