Integrand size = 8, antiderivative size = 12 \[ \int \sqrt {1+\sin (x)} \, dx=-\frac {2 \cos (x)}{\sqrt {1+\sin (x)}} \]
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Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2725} \[ \int \sqrt {1+\sin (x)} \, dx=-\frac {2 \cos (x)}{\sqrt {\sin (x)+1}} \]
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Rule 2725
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (x)}{\sqrt {1+\sin (x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(40\) vs. \(2(12)=24\).
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 3.33 \[ \int \sqrt {1+\sin (x)} \, dx=\frac {2 \left (-\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \sqrt {1+\sin (x)}}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )} \]
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Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.42
method | result | size |
default | \(\frac {2 \left (-1+\sin \left (x \right )\right ) \sqrt {1+\sin \left (x \right )}}{\cos \left (x \right )}\) | \(17\) |
risch | \(-\frac {i \sqrt {2}\, \sqrt {2+2 \sin \left (x \right )}\, \left ({\mathrm e}^{i x}-i\right ) \left (i+{\mathrm e}^{i x}\right )}{{\mathrm e}^{2 i x}-1+2 i {\mathrm e}^{i x}}\) | \(48\) |
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Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (10) = 20\).
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 2.00 \[ \int \sqrt {1+\sin (x)} \, dx=-\frac {2 \, {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )} \sqrt {\sin \left (x\right ) + 1}}{\cos \left (x\right ) + \sin \left (x\right ) + 1} \]
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\[ \int \sqrt {1+\sin (x)} \, dx=\int \sqrt {\sin {\left (x \right )} + 1}\, dx \]
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\[ \int \sqrt {1+\sin (x)} \, dx=\int { \sqrt {\sin \left (x\right ) + 1} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (10) = 20\).
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \sqrt {1+\sin (x)} \, dx=2 \, \sqrt {2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right ) \]
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Time = 13.45 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33 \[ \int \sqrt {1+\sin (x)} \, dx=\frac {2\,\left (\sin \left (x\right )-1\right )\,\sqrt {\sin \left (x\right )+1}}{\cos \left (x\right )} \]
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