Integrand size = 19, antiderivative size = 20 \[ \int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx=-\frac {2 \text {arcsinh}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b} \]
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Time = 0.01 (sec) , antiderivative size = 20, 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 = {65, 221} \[ \int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx=-\frac {2 \text {arcsinh}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b} \]
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Rule 65
Rule 221
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {2+x^2}} \, dx,x,\sqrt {-b x}\right )}{b} \\ & = -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(20)=40\).
Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.60 \[ \int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx=-\frac {4 \sqrt {x} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2-b x}}\right )}{\sqrt {b} \sqrt {-b x}} \]
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Time = 0.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35
method | result | size |
meijerg | \(\frac {2 \sqrt {x}\, \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {b}\, \sqrt {-b x}}\) | \(27\) |
default | \(\frac {\sqrt {-b x \left (-b x +2\right )}\, \ln \left (\frac {b^{2} x -b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}-2 b x}\right )}{\sqrt {-b x}\, \sqrt {-b x +2}\, \sqrt {b^{2}}}\) | \(64\) |
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none
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx=-\frac {\log \left (-b x + \sqrt {-b x + 2} \sqrt {-b x} + 1\right )}{b} \]
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Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55 \[ \int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx=\begin {cases} - \frac {2 \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b} & \text {for}\: \left |{b x}\right | > 2 \\- \frac {2 i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx=\frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - 2 \, b x} b - 2 \, b\right )}{b} \]
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none
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx=\frac {2 \, \log \left (\sqrt {-b x + 2} - \sqrt {-b x}\right )}{b} \]
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Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {2-b\,x}\right )}{\sqrt {-b\,x}\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]
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