Integrand size = 19, antiderivative size = 13 \[ \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx=\frac {\arcsin (2 x)}{2 \sqrt {6}} \]
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Time = 0.00 (sec) , antiderivative size = 13, 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 = {41, 222} \[ \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx=\frac {\arcsin (2 x)}{2 \sqrt {6}} \]
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Rule 41
Rule 222
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {6-24 x^2}} \, dx \\ & = \frac {\sin ^{-1}(2 x)}{2 \sqrt {6}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(27\) vs. \(2(13)=26\).
Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.08 \[ \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx=-\frac {\arctan \left (\frac {\sqrt {1-4 x^2}}{1+2 x}\right )}{\sqrt {6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(36\) vs. \(2(9)=18\).
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.85
method | result | size |
default | \(\frac {\sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{12 \sqrt {2+4 x}\, \sqrt {3-6 x}}\) | \(37\) |
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Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (9) = 18\).
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.15 \[ \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx=-\frac {1}{12} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{12 \, x}\right ) \]
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Result contains complex when optimal does not.
Time = 1.37 (sec) , antiderivative size = 41, normalized size of antiderivative = 3.15 \[ \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx=\begin {cases} - \frac {\sqrt {6} i \operatorname {acosh}{\left (\sqrt {x + \frac {1}{2}} \right )}}{6} & \text {for}\: \left |{x + \frac {1}{2}}\right | > 1 \\\frac {\sqrt {6} \operatorname {asin}{\left (\sqrt {x + \frac {1}{2}} \right )}}{6} & \text {otherwise} \end {cases} \]
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none
Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx=\frac {1}{12} \, \sqrt {6} \arcsin \left (2 \, x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (9) = 18\).
Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx=\frac {1}{6} \, \sqrt {3} \sqrt {2} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {2 \, x + 1}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 3.08 \[ \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx=-\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {24}\,\left (\sqrt {3}-\sqrt {3-6\,x}\right )}{6\,\left (\sqrt {2}-\sqrt {4\,x+2}\right )}\right )}{3} \]
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