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

Optimal. Leaf size=25 \[ \frac{2 \sqrt{1-x^2}}{1-x}-\sin ^{-1}(x) \]

[Out]

(2*Sqrt[1 - x^2])/(1 - x) - ArcSin[x]

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Rubi [A]  time = 0.0284689, antiderivative size = 25, 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 \[ \frac{2 \sqrt{1-x^2}}{1-x}-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x^2]/(1 - x)^2,x]

[Out]

(2*Sqrt[1 - x^2])/(1 - x) - ArcSin[x]

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Rubi in Sympy [A]  time = 4.95426, size = 15, normalized size = 0.6 \[ - \operatorname{asin}{\left (x \right )} + \frac{2 \sqrt{- x^{2} + 1}}{- x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-asin(x) + 2*sqrt(-x**2 + 1)/(-x + 1)

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Mathematica [A]  time = 0.0835818, size = 50, normalized size = 2. \[ 2 \sqrt{1-x^2} \left (\frac{1}{1-x}+\frac{\sinh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )}{\sqrt{x-1} \sqrt{x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 40, normalized size = 1.6 \[{\frac{1}{ \left ( -1+x \right ) ^{2}} \left ( - \left ( -1+x \right ) ^{2}-2\,x+2 \right ) ^{{\frac{3}{2}}}}+\sqrt{- \left ( -1+x \right ) ^{2}-2\,x+2}-\arcsin \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)^(1/2)/(1-x)^2,x)

[Out]

1/(-1+x)^2*(-(-1+x)^2-2*x+2)^(3/2)+(-(-1+x)^2-2*x+2)^(1/2)-arcsin(x)

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Maxima [A]  time = 0.776576, size = 28, normalized size = 1.12 \[ -\frac{2 \, \sqrt{-x^{2} + 1}}{x - 1} - \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-x^2 + 1)/(x - 1) - arcsin(x)

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Fricas [A]  time = 0.215074, size = 66, normalized size = 2.64 \[ \frac{2 \,{\left ({\left (x + \sqrt{-x^{2} + 1} - 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + 2 \, x\right )}}{x + \sqrt{-x^{2} + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-(x - 1)*(x + 1))/(x - 1)**2, x)

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GIAC/XCAS [A]  time = 0.217012, size = 66, normalized size = 2.64 \[ 2 \,{\left (i - \arctan \left (i\right )\right )}{\rm sign}\left (\frac{1}{x - 1}\right ) - 2 \,{\left (\sqrt{-\frac{2}{x - 1} - 1} - \arctan \left (\sqrt{-\frac{2}{x - 1} - 1}\right )\right )}{\rm sign}\left (\frac{1}{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(i - arctan(i))*sign(1/(x - 1)) - 2*(sqrt(-2/(x - 1) - 1) - arctan(sqrt(-2/(x
- 1) - 1)))*sign(1/(x - 1))