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

Optimal. Leaf size=16 \[ \sin ^{-1}(x)-\sqrt{1-x^2} \]

[Out]

-Sqrt[1 - x^2] + ArcSin[x]

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Rubi [A]  time = 0.0266875, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \sin ^{-1}(x)-\sqrt{1-x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x^2]/(1 - x),x]

[Out]

-Sqrt[1 - x^2] + ArcSin[x]

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Rubi in Sympy [A]  time = 4.65262, size = 10, normalized size = 0.62 \[ - \sqrt{- x^{2} + 1} + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**2 + 1) + asin(x)

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Mathematica [B]  time = 0.046004, size = 54, normalized size = 3.38 \[ \sqrt{1-x^2} \left (\frac{2 \log \left (\sqrt{-x-1}+\sqrt{1-x}\right )}{\sqrt{-x-1} \sqrt{1-x}}-1\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 - x^2]*(-1 + (2*Log[Sqrt[-1 - x] + Sqrt[1 - x]])/(Sqrt[-1 - x]*Sqrt[1 - x
]))

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Maple [A]  time = 0.006, size = 20, normalized size = 1.3 \[ -\sqrt{- \left ( -1+x \right ) ^{2}-2\,x+2}+\arcsin \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)^(1/2)/(1-x),x)

[Out]

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

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Maxima [A]  time = 0.783426, size = 19, normalized size = 1.19 \[ -\sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1) + arcsin(x)

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Fricas [A]  time = 0.216297, size = 63, normalized size = 3.94 \[ \frac{x^{2} - 2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right )}{\sqrt{-x^{2} + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-x^2 + 1)/(x - 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 = 5.7653, size = 17, normalized size = 1.06 \[ - \begin{cases} \sqrt{- x^{2} + 1} - \operatorname{asin}{\left (x \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Piecewise((sqrt(-x**2 + 1) - asin(x), (x > -1) & (x < 1)))

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GIAC/XCAS [A]  time = 0.212457, size = 19, normalized size = 1.19 \[ -\sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1) + arcsin(x)