Integrand size = 23, antiderivative size = 126 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=-\frac {\sqrt {2} \arcsin \left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right )}{d}+\frac {7 \arcsin \left (\frac {\sin (c+d x)}{\sqrt {1+\cos (c+d x)}}\right )}{4 d}-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{4 d \sqrt {1+\cos (c+d x)}}+\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt {1+\cos (c+d x)}} \]
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Time = 0.33 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2857, 3062, 3061, 2860, 222, 2853} \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=-\frac {\sqrt {2} \arcsin \left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right )}{d}+\frac {7 \arcsin \left (\frac {\sin (c+d x)}{\sqrt {\cos (c+d x)+1}}\right )}{4 d}+\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {\cos (c+d x)+1}}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)}}{4 d \sqrt {\cos (c+d x)+1}} \]
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Rule 222
Rule 2853
Rule 2857
Rule 2860
Rule 3061
Rule 3062
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt {1+\cos (c+d x)}}-\frac {1}{4} \int \frac {(-3+\cos (c+d x)) \sqrt {\cos (c+d x)}}{\sqrt {1+\cos (c+d x)}} \, dx \\ & = -\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{4 d \sqrt {1+\cos (c+d x)}}+\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt {1+\cos (c+d x)}}-\frac {1}{4} \int \frac {\frac {1}{2}-\frac {7}{2} \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx \\ & = -\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{4 d \sqrt {1+\cos (c+d x)}}+\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt {1+\cos (c+d x)}}+\frac {7}{8} \int \frac {\sqrt {1+\cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx-\int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx \\ & = -\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{4 d \sqrt {1+\cos (c+d x)}}+\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt {1+\cos (c+d x)}}-\frac {7 \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,-\frac {\sin (c+d x)}{\sqrt {1+\cos (c+d x)}}\right )}{4 d}+\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,-\frac {\sin (c+d x)}{1+\cos (c+d x)}\right )}{d} \\ & = -\frac {\sqrt {2} \arcsin \left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right )}{d}+\frac {7 \arcsin \left (\frac {\sin (c+d x)}{\sqrt {1+\cos (c+d x)}}\right )}{4 d}-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{4 d \sqrt {1+\cos (c+d x)}}+\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt {1+\cos (c+d x)}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=-\frac {\left (\arcsin \left (\sqrt {1-\cos (c+d x)}\right )+8 \arcsin \left (\sqrt {\cos (c+d x)}\right )-4 \sqrt {2} \arctan \left (\frac {\sqrt {\cos (c+d x)}}{\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right )-2 \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)+\sqrt {-((-1+\cos (c+d x)) \cos (c+d x))}\right ) \sin (c+d x)}{4 d \sqrt {\sin ^2(c+d x)}} \]
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Time = 12.56 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.34
method | result | size |
default | \(\frac {\left (2 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+4 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sqrt {2}-\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+7 \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {2+2 \cos \left (d x +c \right )}\, \sqrt {2}}{8 d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\) | \(169\) |
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Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\frac {{\left (2 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {\cos \left (d x + c\right ) + 1} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 4 \, {\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\cos \left (d x + c\right ) + 1} \sqrt {\cos \left (d x + c\right )}}{\sin \left (d x + c\right )}\right ) - 7 \, {\left (\cos \left (d x + c\right ) + 1\right )} \arctan \left (\frac {\sqrt {\cos \left (d x + c\right ) + 1} \sqrt {\cos \left (d x + c\right )}}{\sin \left (d x + c\right )}\right )}{4 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {5}{2}}}{\sqrt {\cos \left (d x + c\right ) + 1}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{5/2}}{\sqrt {\cos \left (c+d\,x\right )+1}} \,d x \]
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