Optimal. Leaf size=16 \[ \sin ^{-1}(x)-\sqrt{1-x^2} \]
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Rubi [A] time = 0.0064882, antiderivative size = 16, 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, Rules used = {665, 216} \[ \sin ^{-1}(x)-\sqrt{1-x^2} \]
Antiderivative was successfully verified.
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Rule 665
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x^2}}{1-x} \, dx &=-\sqrt{1-x^2}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\sqrt{1-x^2}+\sin ^{-1}(x)\\ \end{align*}
Mathematica [B] time = 0.0495471, size = 44, normalized size = 2.75 \[ \frac{x^2+2 \sqrt{1-x^2} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-1}{\sqrt{1-x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 20, normalized size = 1.3 \begin{align*} -\sqrt{- \left ( -1+x \right ) ^{2}+2-2\,x}+\arcsin \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70816, size = 19, normalized size = 1.19 \begin{align*} -\sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.21357, size = 70, normalized size = 4.38 \begin{align*} -\sqrt{-x^{2} + 1} - 2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.82402, size = 17, normalized size = 1.06 \begin{align*} - \begin{cases} \sqrt{1 - x^{2}} - \operatorname{asin}{\left (x \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21201, size = 19, normalized size = 1.19 \begin{align*} -\sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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