Integrand size = 19, antiderivative size = 16 \[ \int \frac {\sqrt {1-x^2}}{1-x} \, dx=-\sqrt {1-x^2}+\arcsin (x) \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {679, 222} \[ \int \frac {\sqrt {1-x^2}}{1-x} \, dx=\arcsin (x)-\sqrt {1-x^2} \]
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Rule 222
Rule 679
Rubi steps \begin{align*} \text {integral}& = -\sqrt {1-x^2}+\int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\sqrt {1-x^2}+\sin ^{-1}(x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(34\) vs. \(2(16)=32\).
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {1-x^2}}{1-x} \, dx=-\sqrt {1-x^2}-2 \arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right ) \]
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Time = 2.47 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19
method | result | size |
risch | \(\frac {x^{2}-1}{\sqrt {-x^{2}+1}}+\arcsin \left (x \right )\) | \(19\) |
default | \(-\sqrt {-\left (-1+x \right )^{2}+2-2 x}+\arcsin \left (x \right )\) | \(20\) |
trager | \(-\sqrt {-x^{2}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\) | \(39\) |
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Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (14) = 28\).
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {\sqrt {1-x^2}}{1-x} \, dx=-\sqrt {-x^{2} + 1} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
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Time = 1.17 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {1-x^2}}{1-x} \, dx=- \begin {cases} \sqrt {1 - x^{2}} - \operatorname {asin}{\left (x \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {1-x^2}}{1-x} \, dx=-\sqrt {-x^{2} + 1} + \arcsin \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {1-x^2}}{1-x} \, dx=-\sqrt {-x^{2} + 1} + \arcsin \left (x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {1-x^2}}{1-x} \, dx=\mathrm {asin}\left (x\right )-\sqrt {1-x^2} \]
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