Integrand size = 21, antiderivative size = 86 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {8 a^2 \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d} \]
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Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2830, 2726, 2725} \[ \int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {8 a^2 \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d} \]
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Rule 2725
Rule 2726
Rule 2830
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {3}{5} \int (a+a \sin (c+d x))^{3/2} \, dx \\ & = -\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {1}{5} (4 a) \int \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {8 a^2 \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.83 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {\left (-4 \cos \left (\frac {1}{2} (c+d x)\right ) \left (-8+5 \sqrt {2} \sqrt {1+\cos (c+d x)}\right )+\sqrt {2} \sqrt {1+\cos (c+d x)} \left (-5 \cos \left (\frac {3}{2} (c+d x)\right )+\cos \left (\frac {5}{2} (c+d x)\right )+8 (4+\cos (c+d x)) \sin ^3\left (\frac {1}{2} (c+d x)\right )\right )\right ) (a (1+\sin (c+d x)))^{3/2}}{20 d \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )} \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
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Time = 0.54 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{2} \left (\sin \left (d x +c \right )-1\right ) \left (\sin ^{2}\left (d x +c \right )+3 \sin \left (d x +c \right )+6\right )}{5 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(63\) |
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Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.15 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {2 \, {\left (a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} - 7 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) - 4 \, a\right )} \sin \left (d x + c\right ) - 4 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{5 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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\[ \int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \sin {\left (c + d x \right )}\, dx \]
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\[ \int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sin \left (d x + c\right ) \,d x } \]
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Time = 0.41 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.10 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {\sqrt {2} {\left (20 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )} \sqrt {a}}{10 \, d} \]
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Timed out. \[ \int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \sin \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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