Integrand size = 29, antiderivative size = 92 \[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {8 a^2 \cos ^3(c+d x)}{105 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^3(c+d x)}{35 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d} \]
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Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2935, 2753, 2752} \[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {8 a^2 \cos ^3(c+d x)}{105 d (a \sin (c+d x)+a)^{3/2}}-\frac {2 \cos ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {2 a \cos ^3(c+d x)}{35 d \sqrt {a \sin (c+d x)+a}} \]
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Rule 2752
Rule 2753
Rule 2935
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}+\frac {1}{7} \int \cos ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {2 a \cos ^3(c+d x)}{35 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}+\frac {1}{35} (4 a) \int \frac {\cos ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {8 a^2 \cos ^3(c+d x)}{105 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^3(c+d x)}{35 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.97 \[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \sqrt {a (1+\sin (c+d x))} (59-15 \cos (2 (c+d x))+66 \sin (c+d x))}{105 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.71
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a \left (\sin \left (d x +c \right )-1\right )^{2} \left (15 \left (\sin ^{2}\left (d x +c \right )\right )+33 \sin \left (d x +c \right )+22\right )}{105 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(65\) |
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Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.21 \[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} + 18 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + {\left (15 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} - 4 \, \cos \left (d x + c\right ) - 8\right )} \sin \left (d x + c\right ) + 4 \, \cos \left (d x + c\right ) + 8\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{105 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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\[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}\, dx \]
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\[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.08 \[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {8 \, \sqrt {2} {\left (30 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 63 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 35 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )} \sqrt {a}}{105 \, d} \]
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Timed out. \[ \int \cos ^2(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]
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