Integrand size = 20, antiderivative size = 26 \[ \int \frac {1}{\sqrt {-3-b x} \sqrt {2+b x}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-3-b x}}{\sqrt {2+b x}}\right )}{b} \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {65, 223, 209} \[ \int \frac {1}{\sqrt {-3-b x} \sqrt {2+b x}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-b x-3}}{\sqrt {b x+2}}\right )}{b} \]
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Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2}} \, dx,x,\sqrt {-3-b x}\right )}{b} \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {-3-b x}}{\sqrt {2+b x}}\right )}{b} \\ & = -\frac {2 \tan ^{-1}\left (\frac {\sqrt {-3-b x}}{\sqrt {2+b x}}\right )}{b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\sqrt {-3-b x} \sqrt {2+b x}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-3-b x} \sqrt {2+b x}}{3+b x}\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(22)=44\).
Time = 0.54 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54
method | result | size |
default | \(\frac {\sqrt {\left (-b x -3\right ) \left (b x +2\right )}\, \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {5}{2 b}\right )}{\sqrt {-b^{2} x^{2}-5 b x -6}}\right )}{\sqrt {-b x -3}\, \sqrt {b x +2}\, \sqrt {b^{2}}}\) | \(66\) |
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none
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\sqrt {-3-b x} \sqrt {2+b x}} \, dx=-\frac {\arctan \left (\frac {{\left (2 \, b x + 5\right )} \sqrt {b x + 2} \sqrt {-b x - 3}}{2 \, {\left (b^{2} x^{2} + 5 \, b x + 6\right )}}\right )}{b} \]
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\[ \int \frac {1}{\sqrt {-3-b x} \sqrt {2+b x}} \, dx=\int \frac {1}{\sqrt {- b x - 3} \sqrt {b x + 2}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {-3-b x} \sqrt {2+b x}} \, dx=-\frac {\arcsin \left (-\frac {2 \, b^{2} x + 5 \, b}{b}\right )}{b} \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {-3-b x} \sqrt {2+b x}} \, dx=\frac {2 i \, \log \left (\sqrt {b x + 3} - \sqrt {b x + 2}\right )}{b} \]
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Time = 0.39 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81 \[ \int \frac {1}{\sqrt {-3-b x} \sqrt {2+b x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {b\,\left (-\sqrt {-b\,x-3}+\sqrt {3}\,1{}\mathrm {i}\right )}{\left (\sqrt {2}-\sqrt {b\,x+2}\right )\,\sqrt {b^2}}\right )}{\sqrt {b^2}} \]
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