Integrand size = 26, antiderivative size = 37 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 a \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \]
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Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {2850} \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 a \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \]
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Rule 2850
Rubi steps \begin{align*} \text {integral}& = \frac {2 a \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \sqrt {a-a \cos (c+d x)} \cot \left (\frac {1}{2} (c+d x)\right )}{d \sqrt {\cos (c+d x)}} \]
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Time = 5.46 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {2 \sqrt {-a \left (\cos \left (d x +c \right )-1\right )}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{d \sqrt {\cos \left (d x +c \right )}}\) | \(40\) |
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none
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \, \sqrt {-a \cos \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) + 1\right )}}{d \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )} \]
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\[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\sqrt {- a \left (\cos {\left (c + d x \right )} - 1\right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (33) = 66\).
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.22 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \, {\left (\sqrt {2} \sqrt {a} - \frac {\sqrt {2} \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \]
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none
Time = 0.40 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \, \sqrt {2} {\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 1\right )} \sqrt {a} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{\sqrt {\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1} d} \]
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Time = 14.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2\,\sin \left (c+d\,x\right )\,\sqrt {-a\,\left (\cos \left (c+d\,x\right )-1\right )}}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\left (\cos \left (c+d\,x\right )-1\right )} \]
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