Optimal. Leaf size=25 \[ \frac {2 \sqrt {1-x^2}}{1-x}-\sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {663, 216} \begin {gather*} \frac {2 \sqrt {1-x^2}}{1-x}-\sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 663
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx &=\frac {2 \sqrt {1-x^2}}{1-x}-\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {2 \sqrt {1-x^2}}{1-x}-\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 50, normalized size = 2.00 \begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 39, normalized size = 1.56 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{x+1}\right )-\frac {2 \sqrt {1-x^2}}{x-1} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 41, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left ({\left (x - 1\right )} \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + x - \sqrt {-x^{2} + 1} - 1\right )}}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 1.60 \begin {gather*} -\arcsin \relax (x )+\frac {\left (-2 x -\left (x -1\right )^{2}+2\right )^{\frac {3}{2}}}{\left (x -1\right )^{2}}+\sqrt {-2 x -\left (x -1\right )^{2}+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 21, normalized size = 0.84 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1}}{x - 1} - \arcsin \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 21, normalized size = 0.84 \begin {gather*} -\mathrm {asin}\relax (x)-\frac {2\,\sqrt {1-x^2}}{x-1} \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 {- \left (x - 1\right ) \left (x + 1\right )}}{\left (x - 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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