3.348 \(\int \frac{1}{\left (2+x^2-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{x \left (5-x^2\right )}{18 \sqrt{-x^4+x^2+2}}+\frac{1}{6} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1}{18} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*(5 - x^2))/(18*Sqrt[2 + x^2 - x^4]) + EllipticE[ArcSin[x/Sqrt[2]], -2]/18 + E
llipticF[ArcSin[x/Sqrt[2]], -2]/6

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Rubi [A]  time = 0.137175, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{x \left (5-x^2\right )}{18 \sqrt{-x^4+x^2+2}}+\frac{1}{6} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1}{18} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + x^2 - x^4)^(-3/2),x]

[Out]

(x*(5 - x^2))/(18*Sqrt[2 + x^2 - x^4]) + EllipticE[ArcSin[x/Sqrt[2]], -2]/18 + E
llipticF[ArcSin[x/Sqrt[2]], -2]/6

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Rubi in Sympy [A]  time = 21.0142, size = 49, normalized size = 0.89 \[ \frac{x \left (- x^{2} + 5\right )}{18 \sqrt{- x^{4} + x^{2} + 2}} + \frac{E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{18} + \frac{F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(-x**2 + 5)/(18*sqrt(-x**4 + x**2 + 2)) + elliptic_e(asin(sqrt(2)*x/2), -2)/18
+ elliptic_f(asin(sqrt(2)*x/2), -2)/6

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Mathematica [C]  time = 0.10784, size = 79, normalized size = 1.44 \[ \frac{1}{18} \left (\frac{5 x}{\sqrt{-x^4+x^2+2}}-\frac{x^3}{\sqrt{-x^4+x^2+2}}-3 i \sqrt{2} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+i \sqrt{2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + x^2 - x^4)^(-3/2),x]

[Out]

((5*x)/Sqrt[2 + x^2 - x^4] - x^3/Sqrt[2 + x^2 - x^4] + I*Sqrt[2]*EllipticE[I*Arc
Sinh[x], -1/2] - (3*I)*Sqrt[2]*EllipticF[I*ArcSinh[x], -1/2])/18

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Maple [B]  time = 0.006, size = 133, normalized size = 2.4 \[ 2\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{5\,x}{36}}-1/36\,{x}^{3} \right ) }+{\frac{\sqrt{2}}{9}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{\sqrt{2}}{36}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^4+x^2+2)^(3/2),x)

[Out]

2*(5/36*x-1/36*x^3)/(-x^4+x^2+2)^(1/2)+1/9*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2
)/(-x^4+x^2+2)^(1/2)*EllipticF(1/2*2^(1/2)*x,I*2^(1/2))-1/36*2^(1/2)*(-2*x^2+4)^
(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*(EllipticF(1/2*2^(1/2)*x,I*2^(1/2))-Ellip
ticE(1/2*2^(1/2)*x,I*2^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-x^4 + x^2 + 2)^(-3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-x^4 + x^2 + 2)^(-3/2), x)