Integrand size = 21, antiderivative size = 37 \[ \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d} \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2852, 212} \[ \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d} \]
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Rule 212
Rule 2852
Rubi steps \begin{align*} \text {integral}& = -\frac {(2 a) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d} \\ & = -\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(94\) vs. \(2(37)=74\).
Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.54 \[ \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\left (-\log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sqrt {a (1+\sin (c+d x))}}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(31)=62\).
Time = 0.51 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.84
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right )}{\cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(68\) |
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Time = 0.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 5.92 \[ \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\left [\frac {\sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right )}{2 \, d}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {a \sin \left (d x + c\right ) + a} \sqrt {-a} {\left (\sin \left (d x + c\right ) - 2\right )}}{2 \, a \cos \left (d x + c\right )}\right )}{d}\right ] \]
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\[ \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \csc {\left (c + d x \right )}\, dx \]
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\[ \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \csc \left (d x + c\right ) \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (31) = 62\).
Time = 0.46 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.81 \[ \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {a} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
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Timed out. \[ \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\sin \left (c+d\,x\right )} \,d x \]
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