Integrand size = 23, antiderivative size = 203 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {284 a^3 \cos (c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}-\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}+\frac {568 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{693 d}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}-\frac {284 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{231 d} \]
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Time = 0.26 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2842, 3060, 2849, 2838, 2830, 2725} \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {46 a^3 \sin ^4(c+d x) \cos (c+d x)}{99 d \sqrt {a \sin (c+d x)+a}}-\frac {710 a^3 \sin ^3(c+d x) \cos (c+d x)}{693 d \sqrt {a \sin (c+d x)+a}}-\frac {284 a^3 \cos (c+d x)}{99 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}+\frac {568 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{693 d}-\frac {284 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{231 d} \]
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Rule 2725
Rule 2830
Rule 2838
Rule 2842
Rule 2849
Rule 3060
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {2}{11} \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {19 a^2}{2}+\frac {23}{2} a^2 \sin (c+d x)\right ) \, dx \\ & = -\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {1}{99} \left (355 a^2\right ) \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}-\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {1}{231} \left (710 a^2\right ) \int \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}-\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}-\frac {284 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{231 d}+\frac {1}{231} (284 a) \int \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}-\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}+\frac {568 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{693 d}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}-\frac {284 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{231 d}+\frac {1}{99} \left (142 a^2\right ) \int \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {284 a^3 \cos (c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}-\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}+\frac {568 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{693 d}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}-\frac {284 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{231 d} \\ \end{align*}
Time = 1.36 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.92 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-31878-40656 \cos (c+d x)+36352 \sqrt {2} \sqrt {1+\cos (c+d x)}-5313 \cos (2 (c+d x))+4752 \cos (3 (c+d x))+902 \cos (4 (c+d x))-448 \cos (5 (c+d x))-63 \cos (6 (c+d x))+23100 \sin (c+d x)-12243 \sin (2 (c+d x))-2178 \sin (3 (c+d x))+1672 \sin (4 (c+d x))+322 \sin (5 (c+d x))-63 \sin (6 (c+d x))\right )}{22176 d \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Time = 3.57 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.47
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{3} \left (\sin \left (d x +c \right )-1\right ) \left (63 \left (\sin ^{5}\left (d x +c \right )\right )+224 \left (\sin ^{4}\left (d x +c \right )\right )+355 \left (\sin ^{3}\left (d x +c \right )\right )+426 \left (\sin ^{2}\left (d x +c \right )\right )+568 \sin \left (d x +c \right )+1136\right )}{693 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(95\) |
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Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.95 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {2 \, {\left (63 \, a^{2} \cos \left (d x + c\right )^{6} + 224 \, a^{2} \cos \left (d x + c\right )^{5} - 320 \, a^{2} \cos \left (d x + c\right )^{4} - 874 \, a^{2} \cos \left (d x + c\right )^{3} + 593 \, a^{2} \cos \left (d x + c\right )^{2} + 1786 \, a^{2} \cos \left (d x + c\right ) + 800 \, a^{2} + {\left (63 \, a^{2} \cos \left (d x + c\right )^{5} - 161 \, a^{2} \cos \left (d x + c\right )^{4} - 481 \, a^{2} \cos \left (d x + c\right )^{3} + 393 \, a^{2} \cos \left (d x + c\right )^{2} + 986 \, a^{2} \cos \left (d x + c\right ) - 800 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{693 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sin \left (d x + c\right )^{3} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.95 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {\sqrt {2} {\left (31878 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8778 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3465 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1287 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 385 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 63 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )} \sqrt {a}}{11088 \, d} \]
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Timed out. \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int {\sin \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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