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

Optimal. Leaf size=42 \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{-1+\sqrt{7}}} x\right )|\frac{1}{3} \left (-4+\sqrt{7}\right )\right )}{\sqrt{1+\sqrt{7}}} \]

[Out]

EllipticF[ArcSin[Sqrt[3/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3]/Sqrt[1 + Sqrt[7]]

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Rubi [A]  time = 0.13774, antiderivative size = 42, 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 \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{-1+\sqrt{7}}} x\right )|\frac{1}{3} \left (-4+\sqrt{7}\right )\right )}{\sqrt{1+\sqrt{7}}} \]

Antiderivative was successfully verified.

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

[Out]

EllipticF[ArcSin[Sqrt[3/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3]/Sqrt[1 + Sqrt[7]]

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Rubi in Sympy [A]  time = 20.848, size = 70, normalized size = 1.67 \[ \frac{2 \sqrt{6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x \sqrt{1 + \sqrt{7}}}{2} \right )}\middle | - \frac{4}{3} + \frac{\sqrt{7}}{3}\right )}{\sqrt{-2 + 2 \sqrt{7}} \sqrt{1 + \sqrt{7}} \sqrt{2 + 2 \sqrt{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(6)*elliptic_f(asin(sqrt(2)*x*sqrt(1 + sqrt(7))/2), -4/3 + sqrt(7)/3)/(sqr
t(-2 + 2*sqrt(7))*sqrt(1 + sqrt(7))*sqrt(2 + 2*sqrt(7)))

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Mathematica [C]  time = 0.0774692, size = 51, normalized size = 1.21 \[ -\frac{i F\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{1+\sqrt{7}}} x\right )|-\frac{4}{3}-\frac{\sqrt{7}}{3}\right )}{\sqrt{\sqrt{7}-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-I)*EllipticF[I*ArcSinh[Sqrt[3/(1 + Sqrt[7])]*x], -4/3 - Sqrt[7]/3])/Sqrt[-1 +
Sqrt[7]]

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Maple [B]  time = 0.099, size = 84, normalized size = 2. \[ 2\,{\frac{\sqrt{1- \left ( 1/2\,\sqrt{7}+1/2 \right ){x}^{2}}\sqrt{1- \left ( -1/2\,\sqrt{7}+1/2 \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{2+2\,\sqrt{7}},i/6\sqrt{42}-i/6\sqrt{6} \right ) }{\sqrt{2+2\,\sqrt{7}}\sqrt{-3\,{x}^{4}-2\,{x}^{2}+2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^4-2*x^2+2)^(1/2),x)

[Out]

2/(2+2*7^(1/2))^(1/2)*(1-(1/2*7^(1/2)+1/2)*x^2)^(1/2)*(1-(-1/2*7^(1/2)+1/2)*x^2)
^(1/2)/(-3*x^4-2*x^2+2)^(1/2)*EllipticF(1/2*x*(2+2*7^(1/2))^(1/2),1/6*I*42^(1/2)
-1/6*I*6^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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