Integrand size = 17, antiderivative size = 14 \[ \int \frac {1-x}{\sqrt {1-x^2}} \, dx=\sqrt {1-x^2}+\arcsin (x) \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {655, 222} \[ \int \frac {1-x}{\sqrt {1-x^2}} \, dx=\arcsin (x)+\sqrt {1-x^2} \]
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Rule 222
Rule 655
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. \(32\) vs. \(2(14)=28\).
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.29 \[ \int \frac {1-x}{\sqrt {1-x^2}} \, dx=\sqrt {1-x^2}-2 \arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right ) \]
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Time = 2.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(\arcsin \left (x \right )+\sqrt {-x^{2}+1}\) | \(13\) |
risch | \(-\frac {x^{2}-1}{\sqrt {-x^{2}+1}}+\arcsin \left (x \right )\) | \(20\) |
meijerg | \(\arcsin \left (x \right )+\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{2}+1}}{2 \sqrt {\pi }}\) | \(29\) |
trager | \(\sqrt {-x^{2}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\) | \(37\) |
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Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.00 \[ \int \frac {1-x}{\sqrt {1-x^2}} \, dx=\sqrt {-x^{2} + 1} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1-x}{\sqrt {1-x^2}} \, dx=\sqrt {1 - x^{2}} + \operatorname {asin}{\left (x \right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1-x}{\sqrt {1-x^2}} \, dx=\sqrt {-x^{2} + 1} + \arcsin \left (x\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1-x}{\sqrt {1-x^2}} \, dx=\sqrt {-x^{2} + 1} + \arcsin \left (x\right ) \]
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Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1-x}{\sqrt {1-x^2}} \, dx=\mathrm {asin}\left (x\right )+\sqrt {1-x^2} \]
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