Optimal. Leaf size=16 \[ \sin ^{-1}(x)-\sqrt {1-x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 16, 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 = {665, 216} \[ \sin ^{-1}(x)-\sqrt {1-x^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 665
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*}
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Mathematica [B] time = 0.05, 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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fricas [B] time = 0.96, size = 30, normalized size = 1.88 \[ -\sqrt {-x^{2} + 1} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 14, normalized size = 0.88 \[ -\sqrt {-x^{2} + 1} + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 1.25 \[ \arcsin \relax (x )-\sqrt {-2 x -\left (x -1\right )^{2}+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.92, size = 14, normalized size = 0.88 \[ -\sqrt {-x^{2} + 1} + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 14, normalized size = 0.88 \[ \mathrm {asin}\relax (x)-\sqrt {1-x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.40, size = 17, normalized size = 1.06 \[ - \begin {cases} \sqrt {1 - x^{2}} - \operatorname {asin}{\relax (x )} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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