Optimal. Leaf size=22 \[ \frac {\left (1-x^2\right )^{3/2}}{3 (1-x)^3} \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\left (1-x^2\right )^{3/2}}{3 (1-x)^3} \end {gather*}
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*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 1.05 \begin {gather*} \frac {(x+1) \sqrt {1-x^2}}{3 (x-1)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 23, normalized size = 1.05 \begin {gather*} \frac {(x+1) \sqrt {1-x^2}}{3 (x-1)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 33, normalized size = 1.50 \begin {gather*} \frac {x^{2} + \sqrt {-x^{2} + 1} {\left (x + 1\right )} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 41, normalized size = 1.86 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 0.91 \begin {gather*} \frac {\left (x +1\right ) \sqrt {-x^{2}+1}}{3 \left (x -1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.32, size = 38, normalized size = 1.73 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 19, normalized size = 0.86 \begin {gather*} \frac {\sqrt {1-x^2}\,\left (x+1\right )}{3\,{\left (x-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {1 - x^{2}}}{x^{3} - 3 x^{2} + 3 x - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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