Integrand size = 23, antiderivative size = 64 \[ \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a+a \sin (c+d x)}} \]
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Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2851, 2852, 212} \[ \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}} \]
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Rule 212
Rule 2851
Rule 2852
Rubi steps \begin{align*} \text {integral}& = -\frac {a \cot (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {1}{2} \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {a \cot (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\frac {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 {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a+a \sin (c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(178\) vs. \(2(64)=128\).
Time = 0.66 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.78 \[ \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )+\left (\log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin (c+d x)\right )}{d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc \left (\frac {1}{4} (c+d x)\right )-\sec \left (\frac {1}{4} (c+d x)\right )\right ) \left (\csc \left (\frac {1}{4} (c+d x)\right )+\sec \left (\frac {1}{4} (c+d x)\right )\right )} \]
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Time = 0.68 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.62
method | result | size |
default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (\sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {3}{2}}+\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right ) a^{2} \sin \left (d x +c \right )\right )}{\sin \left (d x +c \right ) a^{\frac {3}{2}} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(104\) |
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Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (56) = 112\).
Time = 0.30 (sec) , antiderivative size = 258, normalized size of antiderivative = 4.03 \[ \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {{\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{4 \, {\left (d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right ) - d\right )}} \]
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\[ \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \csc ^{2}{\left (c + d x \right )}\, dx \]
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\[ \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \csc \left (d x + c\right )^{2} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (56) = 112\).
Time = 0.61 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.91 \[ \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {2} {\left (\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}\right )} \sqrt {a}}{4 \, d} \]
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Timed out. \[ \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^2} \,d x \]
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