Integrand size = 23, antiderivative size = 32 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{d \sqrt {b \cos (c+d x)}} \]
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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} \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {b \cos (c+d x)}} \]
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Rule 17
Rule 2717
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\cos (c+d x)} \int \cos (c+d x) \, dx}{\sqrt {b \cos (c+d x)}} \\ & = \frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{d \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{d \sqrt {b \cos (c+d x)}} \]
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Time = 2.84 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{d \sqrt {\cos \left (d x +c \right ) b}}\) | \(29\) |
risch | \(\frac {\sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{d \sqrt {\cos \left (d x +c \right ) b}}\) | \(29\) |
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{b d \sqrt {\cos \left (d x + c\right )}} \]
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Time = 18.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\begin {cases} \frac {\sin {\left (c + d x \right )} \sqrt {\cos {\left (c + d x \right )}}}{d \sqrt {b \cos {\left (c + d x \right )}}} & \text {for}\: d \neq 0 \\\frac {x \cos ^{\frac {3}{2}}{\left (c \right )}}{\sqrt {b \cos {\left (c \right )}}} & \text {otherwise} \end {cases} \]
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Time = 0.38 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.41 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sin \left (d x + c\right )}{\sqrt {b} d} \]
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\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {3}{2}}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
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Time = 14.62 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {b\,\cos \left (c+d\,x\right )}}{b\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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