Integrand size = 31, antiderivative size = 124 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {8 a^2 \cos ^3(c+d x)}{63 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^3(c+d x)}{21 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{21 d}-\frac {2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 a d} \]
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Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2957, 2935, 2753, 2752} \[ \int \cos ^2(c+d x) \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {8 a^2 \cos ^3(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}-\frac {2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 a d}+\frac {4 \cos ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{21 d}-\frac {2 a \cos ^3(c+d x)}{21 d \sqrt {a \sin (c+d x)+a}} \]
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Rule 2752
Rule 2753
Rule 2935
Rule 2957
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 a d}+\frac {2 \int \cos ^2(c+d x) \left (\frac {3 a}{2}-3 a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{9 a} \\ & = \frac {4 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{21 d}-\frac {2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 a d}+\frac {5}{21} \int \cos ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {2 a \cos ^3(c+d x)}{21 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{21 d}-\frac {2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 a d}+\frac {1}{21} (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)}{63 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^3(c+d x)}{21 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{21 d}-\frac {2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 a d} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.80 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \sqrt {a (1+\sin (c+d x))} (-62+30 \cos (2 (c+d x))-69 \sin (c+d x)+7 \sin (3 (c+d x)))}{126 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Time = 1.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.60
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a \left (\sin \left (d x +c \right )-1\right )^{2} \left (7 \left (\sin ^{3}\left (d x +c \right )\right )+15 \left (\sin ^{2}\left (d x +c \right )\right )+12 \sin \left (d x +c \right )+8\right )}{63 d \cos \left (d x +c \right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}\) | \(75\) |
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Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.05 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {2 \, {\left (7 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 11 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - {\left (7 \, \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} - 4 \, \cos \left (d x + c\right ) - 8\right )} \sin \left (d x + c\right ) - 4 \, \cos \left (d x + c\right ) - 8\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{63 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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\[ \int \cos ^2(c+d x) \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}\, dx \]
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\[ \int \cos ^2(c+d x) \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )^{2} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {8 \, \sqrt {2} {\left (28 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 72 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )} \sqrt {a}}{63 \, d} \]
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Timed out. \[ \int \cos ^2(c+d x) \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^2\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]
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