Integrand size = 23, antiderivative size = 107 \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{d \sqrt {b \cos (c+d x)}}-\frac {2 \sqrt {\cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {b \cos (c+d x)}}+\frac {\sqrt {\cos (c+d x)} \sin ^5(c+d x)}{5 d \sqrt {b \cos (c+d x)}} \]
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Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 2713} \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sin ^5(c+d x) \sqrt {\cos (c+d x)}}{5 d \sqrt {b \cos (c+d x)}}-\frac {2 \sin ^3(c+d x) \sqrt {\cos (c+d x)}}{3 d \sqrt {b \cos (c+d x)}}+\frac {\sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {b \cos (c+d x)}} \]
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Rule 17
Rule 2713
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\cos (c+d x)} \int \cos ^5(c+d x) \, dx}{\sqrt {b \cos (c+d x)}} \\ & = -\frac {\sqrt {\cos (c+d x)} \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt {b \cos (c+d x)}} \\ & = \frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{d \sqrt {b \cos (c+d x)}}-\frac {2 \sqrt {\cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {b \cos (c+d x)}}+\frac {\sqrt {\cos (c+d x)} \sin ^5(c+d x)}{5 d \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.53 \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sqrt {\cos (c+d x)} \sin (c+d x) \left (15-10 \sin ^2(c+d x)+3 \sin ^4(c+d x)\right )}{15 d \sqrt {b \cos (c+d x)}} \]
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Time = 2.91 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.49
method | result | size |
default | \(\frac {\left (3 \left (\cos ^{4}\left (d x +c \right )\right )+4 \left (\cos ^{2}\left (d x +c \right )\right )+8\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{15 d \sqrt {\cos \left (d x +c \right ) b}}\) | \(52\) |
risch | \(\frac {5 \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{8 d \sqrt {\cos \left (d x +c \right ) b}}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \sin \left (5 d x +5 c \right )}{80 \sqrt {\cos \left (d x +c \right ) b}\, d}+\frac {5 \left (\sqrt {\cos }\left (d x +c \right )\right ) \sin \left (3 d x +3 c \right )}{48 \sqrt {\cos \left (d x +c \right ) b}\, d}\) | \(95\) |
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Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.50 \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {{\left (3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{15 \, b d \sqrt {\cos \left (d x + c\right )}} \]
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Timed out. \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\text {Timed out} \]
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Time = 0.41 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.64 \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {3 \, \sin \left (5 \, d x + 5 \, c\right ) + 25 \, \sin \left (\frac {3}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right ) + 150 \, \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right )}{240 \, \sqrt {b} d} \]
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\[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {11}{2}}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
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Time = 14.70 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (175\,\sin \left (2\,c+2\,d\,x\right )+28\,\sin \left (4\,c+4\,d\,x\right )+3\,\sin \left (6\,c+6\,d\,x\right )\right )}{240\,b\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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