Integrand size = 14, antiderivative size = 10 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=\arcsin \left (\frac {1+x}{\sqrt {2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {633, 222} \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=\arcsin \left (\frac {x+1}{\sqrt {2}}\right ) \]
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Rule 222
Rule 633
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{8}}} \, dx,x,-2-2 x\right )}{2 \sqrt {2}} \\ & = \arcsin \left (\frac {1+x}{\sqrt {2}}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(10)=20\).
Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.30 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=2 \arctan \left (\frac {x}{-1+\sqrt {1-2 x-x^2}}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
method | result | size |
default | \(\arcsin \left (\frac {\left (1+x \right ) \sqrt {2}}{2}\right )\) | \(10\) |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-2 x +1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )\) | \(39\) |
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Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (9) = 18\).
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 2.10 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=-2 \, \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 1} - 1}{x}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=\operatorname {asin}{\left (\frac {\sqrt {2} \left (x + 1\right )}{2} \right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=-\arcsin \left (-\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (9) = 18\).
Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.70 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=\frac {1}{2} \, \sqrt {-x^{2} - 2 \, x + 1} {\left (x + 1\right )} + \arcsin \left (\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=\mathrm {asin}\left (\frac {\sqrt {8}\,\left (2\,x+2\right )}{8}\right ) \]
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