Integrand size = 10, antiderivative size = 21 \[ \int \sqrt {\sin (a+b x)} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b} \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2719} \[ \int \sqrt {\sin (a+b x)} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b} \]
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Rule 2719
Rubi steps \begin{align*} \text {integral}& = \frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \sqrt {\sin (a+b x)} \, dx=-\frac {2 E\left (\left .\frac {1}{2} \left (-a+\frac {\pi }{2}-b x\right )\right |2\right )}{b} \]
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Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 4.33
method | result | size |
default | \(-\frac {\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \left (2 E\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-F\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\) | \(91\) |
risch | \(-\frac {i \sqrt {2}\, \sqrt {-i \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{-i \left (b x +a \right )}}}{b}+\frac {i \left (\frac {2 i \left (i-i {\mathrm e}^{2 i \left (b x +a \right )}\right )}{\sqrt {{\mathrm e}^{i \left (b x +a \right )} \left (i-i {\mathrm e}^{2 i \left (b x +a \right )}\right )}}-\frac {\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (b x +a \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (b x +a \right )}}\, \left (-2 E\left (\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}, \frac {\sqrt {2}}{2}\right )+F\left (\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-i {\mathrm e}^{3 i \left (b x +a \right )}+i {\mathrm e}^{i \left (b x +a \right )}}}\right ) \sqrt {2}\, \sqrt {-i \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{-i \left (b x +a \right )}}\, \sqrt {-i \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{i \left (b x +a \right )}}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}\) | \(283\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.00 \[ \int \sqrt {\sin (a+b x)} \, dx=\frac {i \, \sqrt {2} \sqrt {-i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - i \, \sqrt {2} \sqrt {i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b} \]
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\[ \int \sqrt {\sin (a+b x)} \, dx=\int \sqrt {\sin {\left (a + b x \right )}}\, dx \]
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\[ \int \sqrt {\sin (a+b x)} \, dx=\int { \sqrt {\sin \left (b x + a\right )} \,d x } \]
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\[ \int \sqrt {\sin (a+b x)} \, dx=\int { \sqrt {\sin \left (b x + a\right )} \,d x } \]
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Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \sqrt {\sin (a+b x)} \, dx=\frac {2\,\mathrm {E}\left (\frac {a}{2}-\frac {\pi }{4}+\frac {b\,x}{2}\middle |2\right )}{b} \]
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