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

Optimal. Leaf size=65 \[ \frac{\sqrt{x^2-1} \sqrt{3 x^2+2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{2}} x}{\sqrt{x^2-1}}\right )|\frac{2}{5}\right )}{\sqrt{5} \sqrt{3 x^4-x^2-2}} \]

[Out]

(Sqrt[-1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[ArcSin[(Sqrt[5/2]*x)/Sqrt[-1 + x^2]],
2/5])/(Sqrt[5]*Sqrt[-2 - x^2 + 3*x^4])

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Rubi [A]  time = 0.0247881, antiderivative size = 65, 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 \[ \frac{\sqrt{x^2-1} \sqrt{3 x^2+2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{2}} x}{\sqrt{x^2-1}}\right )|\frac{2}{5}\right )}{\sqrt{5} \sqrt{3 x^4-x^2-2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-2 - x^2 + 3*x^4],x]

[Out]

(Sqrt[-1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[ArcSin[(Sqrt[5/2]*x)/Sqrt[-1 + x^2]],
2/5])/(Sqrt[5]*Sqrt[-2 - x^2 + 3*x^4])

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Rubi in Sympy [A]  time = 3.12424, size = 68, normalized size = 1.05 \[ \frac{\sqrt{2} \sqrt{\frac{4 x^{2}}{5} - \frac{4}{5}} \sqrt{6 x^{2} + 4} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{\sqrt{\frac{4 x^{2}}{5} - \frac{4}{5}}} \right )}\middle | \frac{2}{5}\right )}{4 \sqrt{3 x^{4} - x^{2} - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*sqrt(4*x**2/5 - 4/5)*sqrt(6*x**2 + 4)*elliptic_f(asin(sqrt(2)*x/sqrt(4*x
**2/5 - 4/5)), 2/5)/(4*sqrt(3*x**4 - x**2 - 2))

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Mathematica [C]  time = 0.0427372, size = 60, normalized size = 0.92 \[ -\frac{i \sqrt{1-x^2} \sqrt{3 x^2+2} F\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{9 x^4-3 x^2-6}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*Sqrt[1 - x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], -2/3])/Sqr
t[-6 - 3*x^2 + 9*x^4]

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Maple [C]  time = 0.035, size = 53, normalized size = 0.8 \[{-{\frac{i}{6}}\sqrt{6}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{6},{\frac{i}{3}}\sqrt{6} \right ) \sqrt{6\,{x}^{2}+4}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{3\,{x}^{4}-{x}^{2}-2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^4-x^2-2)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(3*x^4 - x^2 - 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(3*x^4 - x^2 - 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(3*x^4 - x^2 - 2), x)