Integrand size = 23, antiderivative size = 27 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\frac {x \sqrt {\cos (c+d x)}}{b^2 \sqrt {b \cos (c+d x)}} \]
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Time = 0.00 (sec) , antiderivative size = 27, 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, 8} \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\frac {x \sqrt {\cos (c+d x)}}{b^2 \sqrt {b \cos (c+d x)}} \]
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Rule 8
Rule 17
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\cos (c+d x)} \int 1 \, dx}{b^2 \sqrt {b \cos (c+d x)}} \\ & = \frac {x \sqrt {\cos (c+d x)}}{b^2 \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\frac {x \sqrt {b \cos (c+d x)}}{b^3 \sqrt {\cos (c+d x)}} \]
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Time = 2.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {x \left (\sqrt {\cos }\left (d x +c \right )\right )}{b^{2} \sqrt {\cos \left (d x +c \right ) b}}\) | \(24\) |
default | \(\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \left (d x +c \right )}{d \,b^{2} \sqrt {\cos \left (d x +c \right ) b}}\) | \(31\) |
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Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.59 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\left [-\frac {\sqrt {-b} \log \left (2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{2 \, b^{3} d}, \frac {\arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right )}{b^{\frac {5}{2}} d}\right ] \]
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Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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Time = 0.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{b^{\frac {5}{2}} d} \]
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\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {5}{2}}}{\left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\frac {2\,x\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {b\,\cos \left (c+d\,x\right )}}{b^3\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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