Integrand size = 14, antiderivative size = 4 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=\arcsin (4+x) \]
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Time = 0.00 (sec) , antiderivative size = 4, 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} \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=\arcsin (x+4) \]
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Rule 222
Rule 633
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,-8-2 x\right )\right ) \\ & = \sin ^{-1}(4+x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(4)=8\).
Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 5.75 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=-2 \arctan \left (\frac {\sqrt {-15-8 x-x^2}}{5+x}\right ) \]
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Time = 1.26 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25
method | result | size |
default | \(\arcsin \left (x +4\right )\) | \(5\) |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-8 x -15}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )\) | \(39\) |
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (4) = 8\).
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 7.25 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=-\arctan \left (\frac {\sqrt {-x^{2} - 8 \, x - 15} {\left (x + 4\right )}}{x^{2} + 8 \, x + 15}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=\operatorname {asin}{\left (x + 4 \right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=-\arcsin \left (-x - 4\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (4) = 8\).
Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 6.00 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=\frac {1}{2} \, \sqrt {-x^{2} - 8 \, x - 15} {\left (x + 4\right )} + \frac {1}{2} \, \arcsin \left (x + 4\right ) \]
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Time = 21.15 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=\mathrm {asin}\left (x+4\right ) \]
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