3.791 \(\int \left (1-x^2\right ) \sqrt{\frac{1}{2-x^2}} \, dx\)

Optimal. Leaf size=18 \[ \frac{x}{2 \sqrt{\frac{1}{2-x^2}}} \]

[Out]

x/(2*Sqrt[(2 - x^2)^(-1)])

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Rubi [A]  time = 0.085749, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{x}{2 \sqrt{\frac{1}{2-x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^2)*Sqrt[(2 - x^2)^(-1)],x]

[Out]

x/(2*Sqrt[(2 - x^2)^(-1)])

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Rubi in Sympy [A]  time = 4.51291, size = 17, normalized size = 0.94 \[ \frac{x \left (- x^{2} + 2\right ) \sqrt{\frac{1}{- x^{2} + 2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(-x**2 + 2)*sqrt(1/(-x**2 + 2))/2

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Mathematica [A]  time = 0.0207967, size = 18, normalized size = 1. \[ \frac{x}{2 \sqrt{\frac{1}{2-x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^2)*Sqrt[(2 - x^2)^(-1)],x]

[Out]

x/(2*Sqrt[(2 - x^2)^(-1)])

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Maple [A]  time = 0.008, size = 20, normalized size = 1.1 \[ -{\frac{x \left ({x}^{2}-2 \right ) }{2}\sqrt{- \left ({x}^{2}-2 \right ) ^{-1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)*(1/(-x^2+2))^(1/2),x)

[Out]

-1/2*(x^2-2)*x*(-1/(x^2-2))^(1/2)

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Maxima [A]  time = 0.776316, size = 16, normalized size = 0.89 \[ \frac{1}{2} \, \sqrt{-x^{2} + 2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-x^2 + 2)*x

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Fricas [A]  time = 0.27434, size = 19, normalized size = 1.06 \[ \frac{x}{2 \, \sqrt{-\frac{1}{x^{2} - 2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x/sqrt(-1/(x^2 - 2))

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Sympy [A]  time = 1.89166, size = 26, normalized size = 1.44 \[ - \frac{x^{3} \sqrt{\frac{1}{- x^{2} + 2}}}{2} + x \sqrt{\frac{1}{- x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**3*sqrt(1/(-x**2 + 2))/2 + x*sqrt(1/(-x**2 + 2))

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GIAC/XCAS [A]  time = 0.269351, size = 24, normalized size = 1.33 \[ -\frac{1}{2} \, \sqrt{-x^{2} + 2} x{\rm sign}\left (x^{2} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(-x^2 + 2)*x*sign(x^2 - 2)