Integrand size = 17, antiderivative size = 4 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=\arcsin (4+x) \]
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Time = 0.00 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {55, 633, 222} \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=\arcsin (x+4) \]
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Rule 55
Rule 222
Rule 633
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx \\ & = -\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. \(44\) vs. \(2(4)=8\).
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 11.00 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=\frac {2 \sqrt {3+x} \sqrt {5+x} \text {arctanh}\left (\frac {\sqrt {5+x}}{\sqrt {3+x}}\right )}{\sqrt {-((3+x) (5+x))}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs. \(2(4)=8\).
Time = 1.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 7.25
method | result | size |
default | \(\frac {\sqrt {\left (-3-x \right ) \left (5+x \right )}\, \arcsin \left (x +4\right )}{\sqrt {-3-x}\, \sqrt {5+x}}\) | \(29\) |
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (4) = 8\).
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 7.25 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=-\arctan \left (\frac {\sqrt {x + 5} {\left (x + 4\right )} \sqrt {-x - 3}}{x^{2} + 8 \, x + 15}\right ) \]
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Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 39, normalized size of antiderivative = 9.75 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=\begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 5}}{2} \right )} & \text {for}\: \left |{x + 5}\right | > 2 \\2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 5}}{2} \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=-\arcsin \left (-x - 4\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 13 vs. \(2 (4) = 8\).
Time = 0.43 (sec) , antiderivative size = 13, normalized size of antiderivative = 3.25 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 5}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 8.25 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=4\,\mathrm {atan}\left (\frac {-\sqrt {-x-3}+\sqrt {3}\,1{}\mathrm {i}}{\sqrt {x+5}-\sqrt {5}}\right ) \]
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