Integrand size = 10, antiderivative size = 26 \[ \int \sqrt {a+a \csc (x)} \, dx=-2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3859, 209} \[ \int \sqrt {a+a \csc (x)} \, dx=-2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right ) \]
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Rule 209
Rule 3859
Rubi steps \begin{align*} \text {integral}& = -\left ((2 a) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {a \cot (x)}{\sqrt {a+a \csc (x)}}\right )\right ) \\ & = -2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \sqrt {a+a \csc (x)} \, dx=-\frac {2 a \arctan \left (\sqrt {-1+\csc (x)}\right ) \cot (x)}{\sqrt {-1+\csc (x)} \sqrt {a (1+\csc (x))}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(165\) vs. \(2(20)=40\).
Time = 1.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 6.38
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {a \left (\csc \left (x \right )+1\right )}\, \sin \left (x \right ) \sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \left (\ln \left (\frac {\csc \left (x \right )-\cot \left (x \right )+\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1}{-\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+\csc \left (x \right )-\cot \left (x \right )+1}\right )+4 \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1\right )+4 \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}-1\right )+\ln \left (\frac {-\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+\csc \left (x \right )-\cot \left (x \right )+1}{\csc \left (x \right )-\cot \left (x \right )+\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1}\right )\right )}{2-2 \cos \left (x \right )+2 \sin \left (x \right )}\) | \(166\) |
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.62 \[ \int \sqrt {a+a \csc (x)} \, dx=\left [\sqrt {-a} \log \left (\frac {2 \, a \cos \left (x\right )^{2} - 2 \, {\left (\cos \left (x\right )^{2} + {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} + a \cos \left (x\right ) - {\left (2 \, a \cos \left (x\right ) + a\right )} \sin \left (x\right ) - a}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right ), 2 \, \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{a \cos \left (x\right ) + a \sin \left (x\right ) + a}\right )\right ] \]
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\[ \int \sqrt {a+a \csc (x)} \, dx=\int \sqrt {a \csc {\left (x \right )} + a}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (20) = 40\).
Time = 0.32 (sec) , antiderivative size = 148, normalized size of antiderivative = 5.69 \[ \int \sqrt {a+a \csc (x)} \, dx=-\frac {2}{3} \, \sqrt {2} \sqrt {a} \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac {3}{2}} + \sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right )\right )} \sqrt {a} - 2 \, \sqrt {2} \sqrt {a} \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}} + \frac {2 \, {\left (\frac {3 \, \sqrt {2} \sqrt {a} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {\sqrt {2} \sqrt {a} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}}{3 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (20) = 40\).
Time = 0.48 (sec) , antiderivative size = 353, normalized size of antiderivative = 13.58 \[ \int \sqrt {a+a \csc (x)} \, dx=\frac {1}{4} \, \sqrt {2} {\left (\frac {2 \, \sqrt {2} {\left (a \sqrt {{\left | a \right |}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right ) + {\left | a \right |}^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right )\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} + 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a} + \frac {2 \, \sqrt {2} {\left (a \sqrt {{\left | a \right |}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right ) + {\left | a \right |}^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right )\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} - 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a} + \frac {\sqrt {2} {\left (a \sqrt {{\left | a \right |}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right ) - {\left | a \right |}^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right )\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) + \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{a} - \frac {\sqrt {2} {\left (a \sqrt {{\left | a \right |}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right ) - {\left | a \right |}^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right )\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{a}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \]
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Timed out. \[ \int \sqrt {a+a \csc (x)} \, dx=\int \sqrt {a+\frac {a}{\sin \left (x\right )}} \,d x \]
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