Integrand size = 23, antiderivative size = 107 \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {3 x \sqrt {\cos (c+d x)}}{8 b \sqrt {b \cos (c+d x)}}+\frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 b d \sqrt {b \cos (c+d x)}}+\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b d \sqrt {b \cos (c+d x)}} \]
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Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {17, 2715, 8} \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {3 x \sqrt {\cos (c+d x)}}{8 b \sqrt {b \cos (c+d x)}}+\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 b d \sqrt {b \cos (c+d x)}}+\frac {3 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 b d \sqrt {b \cos (c+d x)}} \]
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Rule 8
Rule 17
Rule 2715
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\cos (c+d x)} \int \cos ^4(c+d x) \, dx}{b \sqrt {b \cos (c+d x)}} \\ & = \frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b d \sqrt {b \cos (c+d x)}}+\frac {\left (3 \sqrt {\cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 b \sqrt {b \cos (c+d x)}} \\ & = \frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 b d \sqrt {b \cos (c+d x)}}+\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b d \sqrt {b \cos (c+d x)}}+\frac {\left (3 \sqrt {\cos (c+d x)}\right ) \int 1 \, dx}{8 b \sqrt {b \cos (c+d x)}} \\ & = \frac {3 x \sqrt {\cos (c+d x)}}{8 b \sqrt {b \cos (c+d x)}}+\frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 b d \sqrt {b \cos (c+d x)}}+\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b d \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.51 \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {\cos ^{\frac {3}{2}}(c+d x) (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))}{32 d (b \cos (c+d x))^{3/2}} \]
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Time = 2.85 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \left (2 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )+3 d x +3 c \right )}{8 d b \sqrt {\cos \left (d x +c \right ) b}}\) | \(65\) |
risch | \(\frac {3 x \left (\sqrt {\cos }\left (d x +c \right )\right )}{8 b \sqrt {\cos \left (d x +c \right ) b}}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \sin \left (4 d x +4 c \right )}{32 b \sqrt {\cos \left (d x +c \right ) b}\, d}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \sin \left (2 d x +2 c \right )}{4 b \sqrt {\cos \left (d x +c \right ) b}\, d}\) | \(96\) |
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Time = 0.33 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.70 \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\left [\frac {2 \, \sqrt {b \cos \left (d x + c\right )} {\left (2 \, \cos \left (d x + c\right )^{2} + 3\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 3 \, \sqrt {-b} \log \left (2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{16 \, b^{2} d}, \frac {\sqrt {b \cos \left (d x + c\right )} {\left (2 \, \cos \left (d x + c\right )^{2} + 3\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 3 \, \sqrt {b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right )}{8 \, b^{2} d}\right ] \]
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Timed out. \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Time = 0.41 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.46 \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )}{32 \, b^{\frac {3}{2}} d} \]
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\[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {11}{2}}}{\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Time = 15.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.73 \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (8\,\sin \left (c+d\,x\right )+9\,\sin \left (3\,c+3\,d\,x\right )+\sin \left (5\,c+5\,d\,x\right )+24\,d\,x\,\cos \left (c+d\,x\right )\right )}{32\,b^2\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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