Integrand size = 23, antiderivative size = 35 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{b d \sqrt {b \cos (c+d x)}} \]
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Time = 0.01 (sec) , antiderivative size = 35, 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} \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {\sin (c+d x) \sqrt {\cos (c+d x)}}{b d \sqrt {b \cos (c+d x)}} \]
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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}{b \sqrt {b \cos (c+d x)}} \\ & = \frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{b d \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d (b \cos (c+d x))^{3/2}} \]
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Time = 2.87 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{b d \sqrt {\cos \left (d x +c \right ) b}}\) | \(32\) |
risch | \(\frac {\sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{b d \sqrt {\cos \left (d x +c \right ) b}}\) | \(32\) |
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{b^{2} d \sqrt {\cos \left (d x + c\right )}} \]
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Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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none
Time = 0.37 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.37 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {\sin \left (d x + c\right )}{b^{\frac {3}{2}} d} \]
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\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {5}{2}}}{\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Time = 0.35 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.34 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {b\,\cos \left (c+d\,x\right )}}{b^2\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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