Integrand size = 21, antiderivative size = 22 \[ \int \frac {1}{\sqrt {-1-b x} \sqrt {2-b x}} \, dx=-\frac {2 \text {arcsinh}\left (\frac {\sqrt {-1-b x}}{\sqrt {3}}\right )}{b} \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {65, 221} \[ \int \frac {1}{\sqrt {-1-b x} \sqrt {2-b x}} \, dx=-\frac {2 \text {arcsinh}\left (\frac {\sqrt {-b x-1}}{\sqrt {3}}\right )}{b} \]
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Rule 65
Rule 221
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {3+x^2}} \, dx,x,\sqrt {-1-b x}\right )}{b} \\ & = -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-1-b x}}{\sqrt {3}}\right )}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(22)=44\).
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.73 \[ \int \frac {1}{\sqrt {-1-b x} \sqrt {2-b x}} \, dx=\frac {\log \left (\sqrt {-1-b x}-\sqrt {2-b x}\right )}{b}-\frac {\log \left (b \sqrt {-1-b x}+b \sqrt {2-b x}\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(19)=38\).
Time = 0.54 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.18
method | result | size |
default | \(\frac {\sqrt {\left (-b x -1\right ) \left (-b x +2\right )}\, \ln \left (\frac {-\frac {1}{2} b +b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}-b x -2}\right )}{\sqrt {-b x -1}\, \sqrt {-b x +2}\, \sqrt {b^{2}}}\) | \(70\) |
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none
Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt {-1-b x} \sqrt {2-b x}} \, dx=-\frac {\log \left (-2 \, b x + 2 \, \sqrt {-b x + 2} \sqrt {-b x - 1} + 1\right )}{b} \]
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\[ \int \frac {1}{\sqrt {-1-b x} \sqrt {2-b x}} \, dx=\int \frac {1}{\sqrt {- b x - 1} \sqrt {- b x + 2}}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\sqrt {-1-b x} \sqrt {2-b x}} \, dx=\frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - b x - 2} b - b\right )}{b} \]
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none
Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt {-1-b x} \sqrt {2-b x}} \, dx=\frac {2 \, \log \left (\sqrt {-b x + 2} - \sqrt {-b x - 1}\right )}{b} \]
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Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.09 \[ \int \frac {1}{\sqrt {-1-b x} \sqrt {2-b x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {-b\,x-1}-\mathrm {i}\right )}{\left (\sqrt {2}-\sqrt {2-b\,x}\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]
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