Integrand size = 20, antiderivative size = 16 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2+b x}} \, dx=-\frac {\arcsin \left (\frac {1}{3} (-1-2 b x)\right )}{b} \]
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Time = 0.01 (sec) , antiderivative size = 16, 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 = {55, 633, 222} \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2+b x}} \, dx=-\frac {\arcsin \left (\frac {1}{3} (-2 b x-1)\right )}{b} \]
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Rule 55
Rule 222
Rule 633
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {2-b x-b^2 x^2}} \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9 b^2}}} \, dx,x,-b-2 b^2 x\right )}{3 b^2} \\ & = -\frac {\sin ^{-1}\left (\frac {1}{3} (-1-2 b x)\right )}{b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2+b x}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {(1-b x) (2+b x)}}{-1+b x}\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(11)=22\).
Time = 0.54 (sec) , antiderivative size = 66, normalized size of antiderivative = 4.12
method | result | size |
default | \(\frac {\sqrt {\left (-b x +1\right ) \left (b x +2\right )}\, \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1}{2 b}\right )}{\sqrt {-b^{2} x^{2}-b x +2}}\right )}{\sqrt {-b x +1}\, \sqrt {b x +2}\, \sqrt {b^{2}}}\) | \(66\) |
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (11) = 22\).
Time = 0.22 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.69 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2+b x}} \, dx=-\frac {\arctan \left (\frac {{\left (2 \, b x + 1\right )} \sqrt {b x + 2} \sqrt {-b x + 1}}{2 \, {\left (b^{2} x^{2} + b x - 2\right )}}\right )}{b} \]
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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 = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2+b x}} \, dx=-\frac {\arcsin \left (-\frac {2 \, b^{2} x + b}{3 \, b}\right )}{b} \]
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none
Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2+b x}} \, dx=\frac {2 \, \arcsin \left (\frac {1}{3} \, \sqrt {3} \sqrt {b x + 2}\right )}{b} \]
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Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.50 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2+b x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {b\,x+2}\right )}{\left (\sqrt {1-b\,x}-1\right )\,\sqrt {b^2}}\right )}{\sqrt {b^2}} \]
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