Integrand size = 23, antiderivative size = 122 \[ \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {4 a \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d} \]
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Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2849, 2838, 2830, 2725} \[ \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac {8 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{35 d}-\frac {4 a \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}} \]
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Rule 2725
Rule 2830
Rule 2838
Rule 2849
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {6}{7} \int \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}+\frac {12 \int \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{35 a} \\ & = -\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}+\frac {2}{5} \int \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {4 a \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.16 \[ \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {a (1+\sin (c+d x))} \left (-105 \cos \left (\frac {1}{2} (c+d x)\right )-35 \cos \left (\frac {3}{2} (c+d x)\right )+7 \cos \left (\frac {5}{2} (c+d x)\right )+5 \cos \left (\frac {7}{2} (c+d x)\right )+105 \sin \left (\frac {1}{2} (c+d x)\right )-35 \sin \left (\frac {3}{2} (c+d x)\right )-7 \sin \left (\frac {5}{2} (c+d x)\right )+5 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{140 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Time = 0.50 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) a \left (\sin \left (d x +c \right )-1\right ) \left (5 \left (\sin ^{3}\left (d x +c \right )\right )+6 \left (\sin ^{2}\left (d x +c \right )\right )+8 \sin \left (d x +c \right )+16\right )}{35 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(73\) |
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Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.91 \[ \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} - 12 \, \cos \left (d x + c\right )^{2} + {\left (5 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - 13 \, \cos \left (d x + c\right ) + 9\right )} \sin \left (d x + c\right ) - 22 \, \cos \left (d x + c\right ) - 9\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{35 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \sin \left (d x + c\right )^{3} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98 \[ \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} {\left (105 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 35 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 7 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )} \sqrt {a}}{140 \, d} \]
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Timed out. \[ \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int {\sin \left (c+d\,x\right )}^3\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]
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