Integrand size = 25, antiderivative size = 36 \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 a \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \]
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Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2850} \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 a \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}} \]
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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 = 39, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \sqrt {a (1+\cos (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{d \sqrt {\cos (c+d x)}} \]
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Time = 5.37 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17
method | result | size |
default | \(\frac {2 \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \sin \left (d x +c \right )}{d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}}\) | \(42\) |
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none
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )} \]
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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. 98 vs. \(2 (32) = 64\).
Time = 0.36 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.72 \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \, {\left (\frac {\sqrt {2} \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sqrt {2} \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \]
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Time = 0.35 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {4 \, \sqrt {2} \sqrt {a} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )}{\sqrt {\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1} d} \]
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Time = 14.47 (sec) , antiderivative size = 41, 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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