3.264 \(\int \frac{x}{\sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=13 \[ -\sqrt{1-x^2} \]

[Out]

-Sqrt[1 - x^2]

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Rubi [A]  time = 0.0064787, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\sqrt{1-x^2} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2]

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Rubi in Sympy [A]  time = 0.884892, size = 8, normalized size = 0.62 \[ - \sqrt{- x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**2 + 1)

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Mathematica [A]  time = 0.00314063, size = 13, normalized size = 1. \[ -\sqrt{1-x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2]

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Maple [A]  time = 0., size = 17, normalized size = 1.3 \[{ \left ( -1+x \right ) \left ( 1+x \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.35058, size = 15, normalized size = 1.15 \[ -\sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1)

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Fricas [A]  time = 0.203735, size = 23, normalized size = 1.77 \[ \frac{x^{2}}{\sqrt{-x^{2} + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^2/(sqrt(-x^2 + 1) - 1)

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Sympy [A]  time = 0.274035, size = 8, normalized size = 0.62 \[ - \sqrt{- x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-x**2+1)**(1/2),x)

[Out]

-sqrt(-x**2 + 1)

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GIAC/XCAS [A]  time = 0.198465, size = 15, normalized size = 1.15 \[ -\sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1)