Integrand size = 21, antiderivative size = 45 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {(a+a \sin (c+d x))^{10}}{5 a^2 d}-\frac {(a+a \sin (c+d x))^{11}}{11 a^3 d} \]
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Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 45} \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {(a \sin (c+d x)+a)^{10}}{5 a^2 d}-\frac {(a \sin (c+d x)+a)^{11}}{11 a^3 d} \]
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Rule 45
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a-x) (a+x)^9 \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \left (2 a (a+x)^9-(a+x)^{10}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {(a+a \sin (c+d x))^{10}}{5 a^2 d}-\frac {(a+a \sin (c+d x))^{11}}{11 a^3 d} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=-\frac {a^8 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{20} (-6+5 \sin (c+d x))}{55 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(107\) vs. \(2(41)=82\).
Time = 1.42 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.40
method | result | size |
derivativedivides | \(-\frac {a^{8} \left (\frac {\left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {4 \left (\sin ^{10}\left (d x +c \right )\right )}{5}+3 \left (\sin ^{9}\left (d x +c \right )\right )+6 \left (\sin ^{8}\left (d x +c \right )\right )+6 \left (\sin ^{7}\left (d x +c \right )\right )-\frac {42 \left (\sin ^{5}\left (d x +c \right )\right )}{5}-12 \left (\sin ^{4}\left (d x +c \right )\right )-9 \left (\sin ^{3}\left (d x +c \right )\right )-4 \left (\sin ^{2}\left (d x +c \right )\right )-\sin \left (d x +c \right )\right )}{d}\) | \(108\) |
default | \(-\frac {a^{8} \left (\frac {\left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {4 \left (\sin ^{10}\left (d x +c \right )\right )}{5}+3 \left (\sin ^{9}\left (d x +c \right )\right )+6 \left (\sin ^{8}\left (d x +c \right )\right )+6 \left (\sin ^{7}\left (d x +c \right )\right )-\frac {42 \left (\sin ^{5}\left (d x +c \right )\right )}{5}-12 \left (\sin ^{4}\left (d x +c \right )\right )-9 \left (\sin ^{3}\left (d x +c \right )\right )-4 \left (\sin ^{2}\left (d x +c \right )\right )-\sin \left (d x +c \right )\right )}{d}\) | \(108\) |
parallelrisch | \(\frac {a^{8} \left (5 \sin \left (11 d x +11 c \right )-715 \sin \left (9 d x +9 c \right )-3520 \cos \left (8 d x +8 c \right )+88 \cos \left (10 d x +10 c \right )+11495 \sin \left (7 d x +7 c \right )+25080 \cos \left (6 d x +6 c \right )-31977 \sin \left (5 d x +5 c \right )-284240 \cos \left (2 d x +2 c \right )-106590 \sin \left (3 d x +3 c \right )+461890 \sin \left (d x +c \right )+262592\right )}{56320 d}\) | \(118\) |
risch | \(\frac {4199 a^{8} \sin \left (d x +c \right )}{512 d}+\frac {a^{8} \sin \left (11 d x +11 c \right )}{11264 d}+\frac {a^{8} \cos \left (10 d x +10 c \right )}{640 d}-\frac {13 a^{8} \sin \left (9 d x +9 c \right )}{1024 d}-\frac {a^{8} \cos \left (8 d x +8 c \right )}{16 d}+\frac {209 a^{8} \sin \left (7 d x +7 c \right )}{1024 d}+\frac {57 a^{8} \cos \left (6 d x +6 c \right )}{128 d}-\frac {2907 a^{8} \sin \left (5 d x +5 c \right )}{5120 d}-\frac {969 a^{8} \sin \left (3 d x +3 c \right )}{512 d}-\frac {323 a^{8} \cos \left (2 d x +2 c \right )}{64 d}\) | \(169\) |
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Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (41) = 82\).
Time = 0.31 (sec) , antiderivative size = 136, normalized size of antiderivative = 3.02 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {44 \, a^{8} \cos \left (d x + c\right )^{10} - 550 \, a^{8} \cos \left (d x + c\right )^{8} + 1760 \, a^{8} \cos \left (d x + c\right )^{6} - 1760 \, a^{8} \cos \left (d x + c\right )^{4} + {\left (5 \, a^{8} \cos \left (d x + c\right )^{10} - 190 \, a^{8} \cos \left (d x + c\right )^{8} + 1040 \, a^{8} \cos \left (d x + c\right )^{6} - 1568 \, a^{8} \cos \left (d x + c\right )^{4} + 256 \, a^{8} \cos \left (d x + c\right )^{2} + 512 \, a^{8}\right )} \sin \left (d x + c\right )}{55 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (36) = 72\).
Time = 2.05 (sec) , antiderivative size = 422, normalized size of antiderivative = 9.38 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\begin {cases} \frac {2 a^{8} \sin ^{11}{\left (c + d x \right )}}{99 d} + \frac {a^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{9 d} + \frac {8 a^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac {4 a^{8} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {4 a^{8} \sin ^{7}{\left (c + d x \right )}}{d} - \frac {2 a^{8} \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {14 a^{8} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {56 a^{8} \sin ^{5}{\left (c + d x \right )}}{15 d} - \frac {2 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {14 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {28 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {2 a^{8} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {28 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {14 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a^{8} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {a^{8} \cos ^{10}{\left (c + d x \right )}}{5 d} - \frac {7 a^{8} \cos ^{8}{\left (c + d x \right )}}{3 d} - \frac {14 a^{8} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {2 a^{8} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{8} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (41) = 82\).
Time = 0.19 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.98 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=-\frac {5 \, a^{8} \sin \left (d x + c\right )^{11} + 44 \, a^{8} \sin \left (d x + c\right )^{10} + 165 \, a^{8} \sin \left (d x + c\right )^{9} + 330 \, a^{8} \sin \left (d x + c\right )^{8} + 330 \, a^{8} \sin \left (d x + c\right )^{7} - 462 \, a^{8} \sin \left (d x + c\right )^{5} - 660 \, a^{8} \sin \left (d x + c\right )^{4} - 495 \, a^{8} \sin \left (d x + c\right )^{3} - 220 \, a^{8} \sin \left (d x + c\right )^{2} - 55 \, a^{8} \sin \left (d x + c\right )}{55 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (41) = 82\).
Time = 0.38 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.98 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=-\frac {5 \, a^{8} \sin \left (d x + c\right )^{11} + 44 \, a^{8} \sin \left (d x + c\right )^{10} + 165 \, a^{8} \sin \left (d x + c\right )^{9} + 330 \, a^{8} \sin \left (d x + c\right )^{8} + 330 \, a^{8} \sin \left (d x + c\right )^{7} - 462 \, a^{8} \sin \left (d x + c\right )^{5} - 660 \, a^{8} \sin \left (d x + c\right )^{4} - 495 \, a^{8} \sin \left (d x + c\right )^{3} - 220 \, a^{8} \sin \left (d x + c\right )^{2} - 55 \, a^{8} \sin \left (d x + c\right )}{55 \, d} \]
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Time = 0.13 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.93 \[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {-\frac {a^8\,{\sin \left (c+d\,x\right )}^{11}}{11}-\frac {4\,a^8\,{\sin \left (c+d\,x\right )}^{10}}{5}-3\,a^8\,{\sin \left (c+d\,x\right )}^9-6\,a^8\,{\sin \left (c+d\,x\right )}^8-6\,a^8\,{\sin \left (c+d\,x\right )}^7+\frac {42\,a^8\,{\sin \left (c+d\,x\right )}^5}{5}+12\,a^8\,{\sin \left (c+d\,x\right )}^4+9\,a^8\,{\sin \left (c+d\,x\right )}^3+4\,a^8\,{\sin \left (c+d\,x\right )}^2+a^8\,\sin \left (c+d\,x\right )}{d} \]
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