Integrand size = 25, antiderivative size = 122 \[ \int \frac {1}{\sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sin (c+d x)}{\sqrt {2} \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}}\right )}{d}+\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \]
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Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2858, 3063, 12, 2861, 212} \[ \int \frac {1}{\sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sin (c+d x)}{\sqrt {2} \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}}\right )}{d}+\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \]
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Rule 12
Rule 212
Rule 2858
Rule 2861
Rule 3063
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{3} \int \frac {1+2 \cos (c+d x)}{\sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}}-\frac {2}{3} \int -\frac {3}{2 \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}}+\int \frac {1}{\sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}}-\frac {2 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {\sin (c+d x)}{\sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}}\right )}{d} \\ & = -\frac {\sqrt {2} \text {arctanh}\left (\frac {\sin (c+d x)}{\sqrt {2} \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}}\right )}{d}+\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {2 \left (-\frac {3 e^{-\frac {3}{2} i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^2 \text {arctanh}\left (\frac {1+e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )}{2 \sqrt {2}}+2 \sqrt {1+e^{2 i (c+d x)}} \cos \left (\frac {1}{2} (c+d x)\right ) (1+\cos (c+d x))\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \sqrt {1+e^{2 i (c+d x)}} \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \]
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Time = 4.64 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {\left (3 \cos \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\sqrt {2}}{2 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-2 \cos \left (d x +c \right )-2\right ) \sin \left (d x +c \right ) \sqrt {2}}{3 d \cos \left (d x +c \right )^{\frac {3}{2}} \sqrt {-2 \cos \left (d x +c \right )+2}}\) | \(100\) |
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none
Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {3 \, \sqrt {2} \cos \left (d x + c\right )^{2} \log \left (-\frac {2 \, {\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {-\cos \left (d x + c\right ) + 1} \sqrt {\cos \left (d x + c\right )} - {\left (3 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 4 \, {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {-\cos \left (d x + c\right ) + 1} \sqrt {\cos \left (d x + c\right )}}{6 \, d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )} \]
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\[ \int \frac {1}{\sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {1}{\sqrt {1 - \cos {\left (c + d x \right )}} \cos ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 563, normalized size of antiderivative = 4.61 \[ \int \frac {1}{\sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {3 \, {\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 )} \log \left (\frac {4 \, {\left ({\left | i \, e^{\left (i \, d x + i \, c\right )} - i \right |}^{2} + 2 \, \sqrt {\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} {\left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right )^{2}\right )} - 2 \, {\left (\sqrt {2} {\left | i \, e^{\left (i \, d x + i \, c\right )} - i \right |} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) - 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right )\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 )}^{\frac {1}{4}} + 4\right )}}{{\left | i \, e^{\left (i \, d x + i \, c\right )} - i \right |}^{2}}\right ) - 2 \, {\left (\sqrt {2} \sin \left (d x + c\right ) \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) + {\left (\sqrt {2} \cos \left (d x + c\right ) + 3 \, \sqrt {2}\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right )\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 )}^{\frac {3}{4}} - 4 \, {\left (\sqrt {2} \sin \left (d x + c\right ) \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) + {\left (\sqrt {2} \cos \left (d x + c\right ) - \sqrt {2}\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right )\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 )}^{\frac {1}{4}}}{3 \, {\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right )^{2} + \sqrt {2} \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) + \sqrt {2}\right )} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (103) = 206\).
Time = 0.54 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.90 \[ \int \frac {1}{\sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {\sqrt {2} {\left (\frac {8 \, {\left ({\left ({\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 3\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 3\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 1\right )}}{{\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1\right )}^{\frac {3}{2}}} - 3 \, \log \left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - \sqrt {\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1} + 1\right ) + 3 \, \log \left ({\left | -\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + \sqrt {\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1} + 3 \right |}\right ) + 3 \, \log \left ({\left | -\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + \sqrt {\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1} + 1 \right |}\right )\right )}}{6 \, d \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \]
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Timed out. \[ \int \frac {1}{\sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {1-\cos \left (c+d\,x\right )}} \,d x \]
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