Integrand size = 21, antiderivative size = 56 \[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d} \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2830, 2725} \[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}} \]
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Rule 2725
Rule 2830
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+\frac {1}{3} \int \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {2 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.45 \[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {3}{2} (c+d x)\right )-4 \sin ^3\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {a (1+\sin (c+d x))}}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Time = 0.53 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) a \left (\sin \left (d x +c \right )-1\right ) \left (\sin \left (d x +c \right )+2\right )}{3 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(51\) |
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Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.20 \[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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\[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \sin {\left (c + d x \right )}\, dx \]
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\[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \sin \left (d x + c\right ) \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.16 \[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} {\left (3 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right )\right )} \sqrt {a}}{3 \, d} \]
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Timed out. \[ \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]
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