Integrand size = 23, antiderivative size = 65 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \, dx=\frac {b x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {b \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{2 d} \]
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Time = 0.02 (sec) , antiderivative size = 65, 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, 2715, 8} \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \, dx=\frac {b x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {b \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{2 d} \]
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Rule 8
Rule 17
Rule 2715
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{\sqrt {\cos (c+d x)}} \\ & = \frac {b \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int 1 \, dx}{2 \sqrt {\cos (c+d x)}} \\ & = \frac {b x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {b \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{2 d} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \, dx=\frac {(b \cos (c+d x))^{3/2} (2 (c+d x)+\sin (2 (c+d x)))}{4 d \cos ^{\frac {3}{2}}(c+d x)} \]
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Time = 3.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.66
method | result | size |
default | \(\frac {b \sqrt {\cos \left (d x +c \right ) b}\, \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+d x +c \right )}{2 d \sqrt {\cos \left (d x +c \right )}}\) | \(43\) |
risch | \(\frac {b \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{i \left (d x +c \right )} x}{{\mathrm e}^{2 i \left (d x +c \right )}+1}-\frac {i b \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{3 i \left (d x +c \right )}}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {i b \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{-i \left (d x +c \right )}}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}\) | \(139\) |
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Time = 0.39 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.35 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \, dx=\left [\frac {2 \, \sqrt {b \cos \left (d x + c\right )} b \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \sqrt {-b} b \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{4 \, d}, \frac {\sqrt {b \cos \left (d x + c\right )} b \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right )}{2 \, d}\right ] \]
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\[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \, dx=\int \left (b \cos {\left (c + d x \right )}\right )^{\frac {3}{2}} \sqrt {\cos {\left (c + d x \right )}}\, dx \]
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Time = 0.39 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.43 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \, dx=\frac {{\left (2 \, {\left (d x + c\right )} b + b \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {b}}{4 \, d} \]
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Time = 1.20 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.63 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \, dx=\frac {{\left (\sqrt {b} d x \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, \sqrt {b} d x \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, \sqrt {b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \sqrt {b} d x + 2 \, \sqrt {b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} b}{2 \, {\left (d \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, d \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + d\right )}} \]
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Time = 14.50 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \, dx=\frac {b\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )+4\,d\,x\,\cos \left (c+d\,x\right )\right )}{4\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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