3.814 \(\int \frac{\sqrt{1-x^2}}{(1-x)^3} \, dx\)

Optimal. Leaf size=22 \[ \frac{\left (1-x^2\right )^{3/2}}{3 (1-x)^3} \]

[Out]

(1 - x^2)^(3/2)/(3*(1 - x)^3)

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Rubi [A]  time = 0.0252435, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{\left (1-x^2\right )^{3/2}}{3 (1-x)^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x^2]/(1 - x)^3,x]

[Out]

(1 - x^2)^(3/2)/(3*(1 - x)^3)

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Rubi in Sympy [A]  time = 4.52736, size = 14, normalized size = 0.64 \[ \frac{\left (- x^{2} + 1\right )^{\frac{3}{2}}}{3 \left (- x + 1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+1)**(1/2)/(1-x)**3,x)

[Out]

(-x**2 + 1)**(3/2)/(3*(-x + 1)**3)

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Mathematica [A]  time = 0.0176503, size = 23, normalized size = 1.05 \[ \frac{(x+1) \sqrt{1-x^2}}{3 (x-1)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - x^2]/(1 - x)^3,x]

[Out]

((1 + x)*Sqrt[1 - x^2])/(3*(-1 + x)^2)

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Maple [A]  time = 0.003, size = 20, normalized size = 0.9 \[{\frac{1+x}{3\, \left ( -1+x \right ) ^{2}}\sqrt{-{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)^(1/2)/(1-x)^3,x)

[Out]

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

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Maxima [A]  time = 0.6973, size = 51, normalized size = 2.32 \[ \frac{2 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-x^2 + 1)/(x - 1)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.209818, size = 68, normalized size = 3.09 \[ \frac{2 \,{\left (x^{3} + 3 \, \sqrt{-x^{2} + 1} x - 3 \, x\right )}}{3 \,{\left (x^{3} -{\left (x^{2} - 3 \, x + 2\right )} \sqrt{-x^{2} + 1} - 3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-x^2 + 1)/(x - 1)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{- x^{2} + 1}}{x^{3} - 3 x^{2} + 3 x - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)**(1/2)/(1-x)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215037, size = 55, normalized size = 2.5 \[ \frac{2 \,{\left (\frac{3 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{3 \,{\left (\frac{\sqrt{-x^{2} + 1} - 1}{x} + 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-x^2 + 1)/(x - 1)^3,x, algorithm="giac")

[Out]

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