Integrand size = 29, antiderivative size = 109 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {4 (a+a \sin (c+d x))^5}{5 a^3 d}-\frac {2 (a+a \sin (c+d x))^6}{a^4 d}+\frac {13 (a+a \sin (c+d x))^7}{7 a^5 d}-\frac {3 (a+a \sin (c+d x))^8}{4 a^6 d}+\frac {(a+a \sin (c+d x))^9}{9 a^7 d} \]
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Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 90} \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {(a \sin (c+d x)+a)^9}{9 a^7 d}-\frac {3 (a \sin (c+d x)+a)^8}{4 a^6 d}+\frac {13 (a \sin (c+d x)+a)^7}{7 a^5 d}-\frac {2 (a \sin (c+d x)+a)^6}{a^4 d}+\frac {4 (a \sin (c+d x)+a)^5}{5 a^3 d} \]
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Rule 12
Rule 90
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a-x)^2 x^2 (a+x)^4}{a^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int (a-x)^2 x^2 (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \left (4 a^4 (a+x)^4-12 a^3 (a+x)^5+13 a^2 (a+x)^6-6 a (a+x)^7+(a+x)^8\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {4 (a+a \sin (c+d x))^5}{5 a^3 d}-\frac {2 (a+a \sin (c+d x))^6}{a^4 d}+\frac {13 (a+a \sin (c+d x))^7}{7 a^5 d}-\frac {3 (a+a \sin (c+d x))^8}{4 a^6 d}+\frac {(a+a \sin (c+d x))^9}{9 a^7 d} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 (7560 \cos (2 (c+d x))+1260 \cos (4 (c+d x))-840 \cos (6 (c+d x))-315 \cos (8 (c+d x))-16380 \sin (c+d x)+1680 \sin (3 (c+d x))+2016 \sin (5 (c+d x))+270 \sin (7 (c+d x))-70 \sin (9 (c+d x)))}{161280 d} \]
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Time = 0.43 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {2 \left (\sin ^{6}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(79\) |
default | \(\frac {a^{2} \left (\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {2 \left (\sin ^{6}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(79\) |
parallelrisch | \(\frac {a^{2} \left (-\sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (2406 \cos \left (2 d x +2 c \right )+315 \sin \left (5 d x +5 c \right )-70 \cos \left (6 d x +6 c \right )+3150 \sin \left (d x +c \right )+1785 \sin \left (3 d x +3 c \right )+60 \cos \left (4 d x +4 c \right )+4324\right ) \left (\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40320 d}\) | \(118\) |
risch | \(\frac {13 a^{2} \sin \left (d x +c \right )}{128 d}+\frac {a^{2} \sin \left (9 d x +9 c \right )}{2304 d}+\frac {a^{2} \cos \left (8 d x +8 c \right )}{512 d}-\frac {3 a^{2} \sin \left (7 d x +7 c \right )}{1792 d}+\frac {a^{2} \cos \left (6 d x +6 c \right )}{192 d}-\frac {a^{2} \sin \left (5 d x +5 c \right )}{80 d}-\frac {a^{2} \cos \left (4 d x +4 c \right )}{128 d}-\frac {a^{2} \sin \left (3 d x +3 c \right )}{96 d}-\frac {3 a^{2} \cos \left (2 d x +2 c \right )}{64 d}\) | \(152\) |
norman | \(\frac {\frac {8 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {48 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {136 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {11104 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 d}-\frac {136 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {48 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 a^{2} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {8 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {8 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\) | \(265\) |
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Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.90 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {315 \, a^{2} \cos \left (d x + c\right )^{8} - 420 \, a^{2} \cos \left (d x + c\right )^{6} + 4 \, {\left (35 \, a^{2} \cos \left (d x + c\right )^{8} - 95 \, a^{2} \cos \left (d x + c\right )^{6} + 12 \, a^{2} \cos \left (d x + c\right )^{4} + 16 \, a^{2} \cos \left (d x + c\right )^{2} + 32 \, a^{2}\right )} \sin \left (d x + c\right )}{1260 \, d} \]
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Time = 0.91 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.74 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\begin {cases} \frac {8 a^{2} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {8 a^{2} \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {4 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \cos ^{8}{\left (c + d x \right )}}{12 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \sin ^{2}{\left (c \right )} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.89 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {140 \, a^{2} \sin \left (d x + c\right )^{9} + 315 \, a^{2} \sin \left (d x + c\right )^{8} - 180 \, a^{2} \sin \left (d x + c\right )^{7} - 840 \, a^{2} \sin \left (d x + c\right )^{6} - 252 \, a^{2} \sin \left (d x + c\right )^{5} + 630 \, a^{2} \sin \left (d x + c\right )^{4} + 420 \, a^{2} \sin \left (d x + c\right )^{3}}{1260 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.39 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} + \frac {a^{2} \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {a^{2} \cos \left (4 \, d x + 4 \, c\right )}{128 \, d} - \frac {3 \, a^{2} \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {a^{2} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {3 \, a^{2} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {a^{2} \sin \left (3 \, d x + 3 \, c\right )}{96 \, d} + \frac {13 \, a^{2} \sin \left (d x + c\right )}{128 \, d} \]
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Time = 9.47 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {\frac {a^2\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {a^2\,{\sin \left (c+d\,x\right )}^8}{4}-\frac {a^2\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {2\,a^2\,{\sin \left (c+d\,x\right )}^6}{3}-\frac {a^2\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a^2\,{\sin \left (c+d\,x\right )}^4}{2}+\frac {a^2\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \]
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