Integrand size = 23, antiderivative size = 32 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sqrt {b \cos (c+d x)} \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {17, 3852, 8} \[ \int \frac {\sqrt {b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sin (c+d x) \sqrt {b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)} \]
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Rule 8
Rule 17
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b \cos (c+d x)} \int \sec ^2(c+d x) \, dx}{\sqrt {\cos (c+d x)}} \\ & = -\frac {\sqrt {b \cos (c+d x)} \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d \sqrt {\cos (c+d x)}} \\ & = \frac {\sqrt {b \cos (c+d x)} \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sqrt {b \cos (c+d x)} \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)} \]
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Time = 2.92 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\sin \left (d x +c \right ) \sqrt {\cos \left (d x +c \right ) b}}{d \cos \left (d x +c \right )^{\frac {3}{2}}}\) | \(29\) |
risch | \(\frac {2 i \sqrt {\cos \left (d x +c \right ) b}}{\sqrt {\cos \left (d x +c \right )}\, d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(38\) |
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Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right )^{\frac {3}{2}}} \]
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\[ \int \frac {\sqrt {b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {\sqrt {b \cos {\left (c + d x \right )}}}{\cos ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]
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Time = 0.44 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {2 \, \sqrt {b} \sin \left (2 \, d x + 2 \, c\right )}{{\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} d} \]
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\[ \int \frac {\sqrt {b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {\sqrt {b \cos \left (d x + c\right )}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
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Time = 14.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.84 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (\cos \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+\sin \left (2\,c+2\,d\,x\right )+1{}\mathrm {i}\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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