Integrand size = 29, antiderivative size = 111 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {2 (a+a \sin (c+d x))^6}{3 a^3 d}-\frac {12 (a+a \sin (c+d x))^7}{7 a^4 d}+\frac {13 (a+a \sin (c+d x))^8}{8 a^5 d}-\frac {2 (a+a \sin (c+d x))^9}{3 a^6 d}+\frac {(a+a \sin (c+d x))^{10}}{10 a^7 d} \]
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Time = 0.09 (sec) , antiderivative size = 111, 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} \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {(a \sin (c+d x)+a)^{10}}{10 a^7 d}-\frac {2 (a \sin (c+d x)+a)^9}{3 a^6 d}+\frac {13 (a \sin (c+d x)+a)^8}{8 a^5 d}-\frac {12 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac {2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]
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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)^5}{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)^5 \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \left (4 a^4 (a+x)^5-12 a^3 (a+x)^6+13 a^2 (a+x)^7-6 a (a+x)^8+(a+x)^9\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {2 (a+a \sin (c+d x))^6}{3 a^3 d}-\frac {12 (a+a \sin (c+d x))^7}{7 a^4 d}+\frac {13 (a+a \sin (c+d x))^8}{8 a^5 d}-\frac {2 (a+a \sin (c+d x))^9}{3 a^6 d}+\frac {(a+a \sin (c+d x))^{10}}{10 a^7 d} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.99 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 (-2835+34440 \cos (2 (c+d x))+5040 \cos (4 (c+d x))-4060 \cos (6 (c+d x))-1260 \cos (8 (c+d x))+84 \cos (10 (c+d x))-63840 \sin (c+d x)+8960 \sin (3 (c+d x))+8064 \sin (5 (c+d x))+240 \sin (7 (c+d x))-560 \sin (9 (c+d x)))}{430080 d} \]
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Time = 0.63 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {\left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{3}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {5 \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(89\) |
default | \(\frac {a^{3} \left (\frac {\left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{3}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {5 \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(89\) |
parallelrisch | \(-\frac {a^{3} \left (\sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (1356 \cos \left (2 d x +2 c \right )-21 \sin \left (7 d x +7 c \right )+252 \sin \left (5 d x +5 c \right )-140 \cos \left (6 d x +6 c \right )+3465 \sin \left (d x +c \right )+1834 \sin \left (3 d x +3 c \right )-360 \cos \left (4 d x +4 c \right )+3624\right ) \left (\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{26880 d}\) | \(127\) |
risch | \(\frac {19 a^{3} \sin \left (d x +c \right )}{128 d}-\frac {a^{3} \cos \left (10 d x +10 c \right )}{5120 d}+\frac {a^{3} \sin \left (9 d x +9 c \right )}{768 d}+\frac {3 a^{3} \cos \left (8 d x +8 c \right )}{1024 d}-\frac {a^{3} \sin \left (7 d x +7 c \right )}{1792 d}+\frac {29 a^{3} \cos \left (6 d x +6 c \right )}{3072 d}-\frac {3 a^{3} \sin \left (5 d x +5 c \right )}{160 d}-\frac {3 a^{3} \cos \left (4 d x +4 c \right )}{256 d}-\frac {a^{3} \sin \left (3 d x +3 c \right )}{48 d}-\frac {41 a^{3} \cos \left (2 d x +2 c \right )}{512 d}\) | \(169\) |
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Time = 0.31 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {84 \, a^{3} \cos \left (d x + c\right )^{10} - 525 \, a^{3} \cos \left (d x + c\right )^{8} + 560 \, a^{3} \cos \left (d x + c\right )^{6} - 8 \, {\left (35 \, a^{3} \cos \left (d x + c\right )^{8} - 65 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} + 8 \, a^{3} \cos \left (d x + c\right )^{2} + 16 \, a^{3}\right )} \sin \left (d x + c\right )}{840 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (99) = 198\).
Time = 1.26 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.30 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {8 a^{3} \sin ^{9}{\left (c + d x \right )}}{105 d} + \frac {12 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {8 a^{3} \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {4 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} - \frac {a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{12 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{2 d} - \frac {a^{3} \cos ^{10}{\left (c + d x \right )}}{60 d} - \frac {a^{3} \cos ^{8}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{2}{\left (c \right )} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.99 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {84 \, a^{3} \sin \left (d x + c\right )^{10} + 280 \, a^{3} \sin \left (d x + c\right )^{9} + 105 \, a^{3} \sin \left (d x + c\right )^{8} - 600 \, a^{3} \sin \left (d x + c\right )^{7} - 700 \, a^{3} \sin \left (d x + c\right )^{6} + 168 \, a^{3} \sin \left (d x + c\right )^{5} + 630 \, a^{3} \sin \left (d x + c\right )^{4} + 280 \, a^{3} \sin \left (d x + c\right )^{3}}{840 \, d} \]
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Time = 0.63 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.51 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^{3} \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {3 \, a^{3} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {29 \, a^{3} \cos \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {3 \, a^{3} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {41 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )}{512 \, d} + \frac {a^{3} \sin \left (9 \, d x + 9 \, c\right )}{768 \, d} - \frac {a^{3} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {3 \, a^{3} \sin \left (5 \, d x + 5 \, c\right )}{160 \, d} - \frac {a^{3} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {19 \, a^{3} \sin \left (d x + c\right )}{128 \, d} \]
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Time = 9.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.98 \[ \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^{10}}{10}+\frac {a^3\,{\sin \left (c+d\,x\right )}^9}{3}+\frac {a^3\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {5\,a^3\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {5\,a^3\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a^3\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^4}{4}+\frac {a^3\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \]
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