Integrand size = 25, antiderivative size = 36 \[ \int \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {2 a \sqrt {\sec (c+d x)} \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}} \]
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Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {4307, 2850} \[ \int \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {2 a \sin (c+d x) \sqrt {\sec (c+d x)}}{d \sqrt {a \cos (c+d x)+a}} \]
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Rule 2850
Rule 4307
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 a \sqrt {\sec (c+d x)} \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {2 \sqrt {a (1+\cos (c+d x))} \sqrt {\sec (c+d x)} \tan \left (\frac {1}{2} (c+d x)\right )}{d} \]
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Time = 5.78 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11
method | result | size |
default | \(-\frac {2 \cot \left (d x +c \right ) \left (\sec ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (\cos \left (d x +c \right )-1\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{d}\) | \(40\) |
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none
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {2 \, \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{{\left (d \cos \left (d x + c\right ) + d\right )} \sqrt {\cos \left (d x + c\right )}} \]
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Timed out. \[ \int \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (32) = 64\).
Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.72 \[ \int \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {2 \, {\left (\frac {\sqrt {2} \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sqrt {2} \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \]
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Time = 0.39 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {4 \, \sqrt {2} \sqrt {a} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )}{\sqrt {\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1} d} \]
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Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19 \[ \int \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {2\,\sin \left (c+d\,x\right )\,\sqrt {a\,\left (\cos \left (c+d\,x\right )+1\right )}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{d\,\left (\cos \left (c+d\,x\right )+1\right )} \]
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