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

Optimal. Leaf size=14 \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]

[Out]

Sqrt[1 - x^2] + ArcSin[x]

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Rubi [A]  time = 0.0212491, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)/Sqrt[1 - x^2],x]

[Out]

Sqrt[1 - x^2] + ArcSin[x]

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Rubi in Sympy [A]  time = 3.7339, size = 10, normalized size = 0.71 \[ \sqrt{- x^{2} + 1} + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x**2 + 1) + asin(x)

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Mathematica [A]  time = 0.0118023, size = 14, normalized size = 1. \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 - x^2] + ArcSin[x]

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Maple [A]  time = 0.006, size = 13, normalized size = 0.9 \[ \arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)/(-x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.793986, size = 16, normalized size = 1.14 \[ \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x^2 + 1) + arcsin(x)

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Fricas [A]  time = 0.221777, size = 65, normalized size = 4.64 \[ -\frac{x^{2} + 2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right )}{\sqrt{-x^{2} + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.323521, size = 10, normalized size = 0.71 \[ \sqrt{- x^{2} + 1} + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x**2 + 1) + asin(x)

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GIAC/XCAS [A]  time = 0.212393, size = 16, normalized size = 1.14 \[ \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x^2 + 1) + arcsin(x)