Integrand size = 23, antiderivative size = 32 \[ \int \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \, dx=\frac {\sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 2717} \[ \int \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \, dx=\frac {\sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}} \]
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Rule 17
Rule 2717
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b \cos (c+d x)} \int \cos (c+d x) \, dx}{\sqrt {\cos (c+d x)}} \\ & = \frac {\sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \, dx=\frac {\sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
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Time = 3.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\sin \left (d x +c \right ) \sqrt {\cos \left (d x +c \right ) b}}{d \sqrt {\cos \left (d x +c \right )}}\) | \(29\) |
risch | \(-\frac {i \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {i \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}\) | \(85\) |
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \, dx=\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{d \sqrt {\cos \left (d x + c\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29) = 58\).
Time = 1.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \, dx=\begin {cases} x \sqrt {b \cos {\left (c \right )}} \sqrt {\cos {\left (c \right )}} & \text {for}\: d = 0 \\0 & \text {for}\: c = - d x + \frac {\pi }{2} \vee c = - d x + \frac {3 \pi }{2} \\\frac {\sqrt {b \cos {\left (c + d x \right )}} \sin {\left (c + d x \right )}}{d \sqrt {\cos {\left (c + d x \right )}}} & \text {otherwise} \end {cases} \]
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Time = 0.44 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.41 \[ \int \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \, dx=\frac {\sqrt {b} \sin \left (d x + c\right )}{d} \]
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Time = 0.54 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \, dx=\frac {2 \, \sqrt {b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + d} \]
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Time = 14.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \, dx=\frac {\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {b\,\cos \left (c+d\,x\right )}}{d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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