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

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

[Out]

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

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Rubi [A]  time = 0.0249164, 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 \[ \frac{\sqrt{x^2+1} \sqrt{2 x^2-3} F\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{2 x^2-3}}\right )|\frac{2}{5}\right )}{\sqrt{5} \sqrt{2 x^4-x^2-3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-3 - x^2 + 2*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 3.07947, size = 68, normalized size = 1.05 \[ \frac{\sqrt{3} \sqrt{\frac{4 x^{2}}{5} - \frac{6}{5}} \sqrt{6 x^{2} + 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{\sqrt{\frac{4 x^{2}}{5} - \frac{6}{5}}} \right )}\middle | \frac{2}{5}\right )}{6 \sqrt{2 x^{4} - x^{2} - 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0425312, size = 51, normalized size = 0.78 \[ \frac{\sqrt{3-2 x^2} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right )|-\frac{3}{2}\right )}{\sqrt{4 x^4-2 x^2-6}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 - 2*x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[2/3]*x], -3/2])/Sqrt[-6 - 2
*x^2 + 4*x^4]

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Maple [C]  time = 0.012, size = 45, normalized size = 0.7 \[{-{\frac{i}{3}}{\it EllipticF} \left ( ix,{\frac{i}{3}}\sqrt{6} \right ) \sqrt{{x}^{2}+1}\sqrt{-6\,{x}^{2}+9}{\frac{1}{\sqrt{2\,{x}^{4}-{x}^{2}-3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*x^4-x^2-3)^(1/2),x)

[Out]

-1/3*I*(x^2+1)^(1/2)*(-6*x^2+9)^(1/2)/(2*x^4-x^2-3)^(1/2)*EllipticF(I*x,1/3*I*6^
(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(2*x^4 - x^2 - 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(2*x^4 - x^2 - 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(2*x^4 - x^2 - 3), x)