Integrand size = 17, antiderivative size = 42 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cos (x)}{\sqrt {2} \sqrt {\sin (x)} \sqrt {a+a \sin (x)}}\right )}{\sqrt {a}} \]
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Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2861, 211} \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cos (x)}{\sqrt {2} \sqrt {\sin (x)} \sqrt {a \sin (x)+a}}\right )}{\sqrt {a}} \]
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Rule 211
Rule 2861
Rubi steps \begin{align*} \text {integral}& = -\left ((2 a) \text {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,\frac {a \cos (x)}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}}\right )\right ) \\ & = -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cos (x)}{\sqrt {2} \sqrt {\sin (x)} \sqrt {a+a \sin (x)}}\right )}{\sqrt {a}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\frac {2 \arctan \left (\sqrt {\tan \left (\frac {x}{2}\right )}\right ) \sqrt {\sin (x)} \left (1+\tan \left (\frac {x}{2}\right )\right )}{\sqrt {a (1+\sin (x))} \sqrt {\tan \left (\frac {x}{2}\right )}} \]
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Time = 2.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {2 \left (\cos \left (x \right )+1+\sin \left (x \right )\right ) \sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\right )}{\sqrt {\sin \left (x \right )}\, \sqrt {a \left (1+\sin \left (x \right )\right )}}\) | \(40\) |
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none
Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.88 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\left [\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {1}{a}} \log \left (\frac {17 \, \cos \left (x\right )^{3} - 4 \, \sqrt {2} {\left (3 \, \cos \left (x\right )^{2} + {\left (3 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - \cos \left (x\right ) - 4\right )} \sqrt {a \sin \left (x\right ) + a} \sqrt {-\frac {1}{a}} \sqrt {\sin \left (x\right )} + 3 \, \cos \left (x\right )^{2} + {\left (17 \, \cos \left (x\right )^{2} + 14 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 18 \, \cos \left (x\right ) - 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ), \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a \sin \left (x\right ) + a} {\left (3 \, \sin \left (x\right ) - 1\right )}}{4 \, \sqrt {a} \cos \left (x\right ) \sqrt {\sin \left (x\right )}}\right )}{2 \, \sqrt {a}}\right ] \]
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\[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\int \frac {1}{\sqrt {a \left (\sin {\left (x \right )} + 1\right )} \sqrt {\sin {\left (x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (x\right ) + a} \sqrt {\sin \left (x\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (x\right ) + a} \sqrt {\sin \left (x\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\int \frac {1}{\sqrt {\sin \left (x\right )}\,\sqrt {a+a\,\sin \left (x\right )}} \,d x \]
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