Optimal. Leaf size=22 \[ \frac{\left (1-x^2\right )^{3/2}}{3 (1-x)^3} \]
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Rubi [A] time = 0.0057352, 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, Rules used = {651} \[ \frac{\left (1-x^2\right )^{3/2}}{3 (1-x)^3} \]
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x^2}}{(1-x)^3} \, dx &=\frac{\left (1-x^2\right )^{3/2}}{3 (1-x)^3}\\ \end{align*}
Mathematica [A] time = 0.0167045, size = 23, normalized size = 1.05 \[ \frac{(x+1) \sqrt{1-x^2}}{3 (x-1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 20, normalized size = 0.9 \begin{align*}{\frac{1+x}{3\, \left ( -1+x \right ) ^{2}}\sqrt{-{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.994955, size = 51, normalized size = 2.32 \begin{align*} \frac{2 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11562, size = 84, normalized size = 3.82 \begin{align*} \frac{x^{2} + \sqrt{-x^{2} + 1}{\left (x + 1\right )} - 2 \, x + 1}{3 \,{\left (x^{2} - 2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\sqrt{1 - x^{2}}}{x^{3} - 3 x^{2} + 3 x - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2547, size = 55, normalized size = 2.5 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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