Integrand size = 27, antiderivative size = 125 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a x}{128}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^9(c+d x)}{9 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d} \]
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Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2917, 2648, 2715, 8, 2645, 14} \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5 a x}{128} \]
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Rule 8
Rule 14
Rule 2645
Rule 2648
Rule 2715
Rule 2917
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+a \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx \\ & = -\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{8} a \int \cos ^6(c+d x) \, dx-\frac {a \text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{48} (5 a) \int \cos ^4(c+d x) \, dx-\frac {a \text {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^9(c+d x)}{9 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{64} (5 a) \int \cos ^2(c+d x) \, dx \\ & = -\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^9(c+d x)}{9 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{128} (5 a) \int 1 \, dx \\ & = \frac {5 a x}{128}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^9(c+d x)}{9 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (2520 d x-1512 \cos (c+d x)-672 \cos (3 (c+d x))+108 \cos (7 (c+d x))+28 \cos (9 (c+d x))+1008 \sin (2 (c+d x))-504 \sin (4 (c+d x))-336 \sin (6 (c+d x))-63 \sin (8 (c+d x)))}{64512 d} \]
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Time = 0.46 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.77
method | result | size |
parallelrisch | \(-\frac {\left (-5 d x +\sin \left (4 d x +4 c \right )+\frac {2 \sin \left (6 d x +6 c \right )}{3}+\frac {\sin \left (8 d x +8 c \right )}{8}+3 \cos \left (d x +c \right )+\frac {4 \cos \left (3 d x +3 c \right )}{3}-\frac {3 \cos \left (7 d x +7 c \right )}{14}-\frac {\cos \left (9 d x +9 c \right )}{18}-2 \sin \left (2 d x +2 c \right )+\frac {256}{63}\right ) a}{128 d}\) | \(96\) |
derivativedivides | \(\frac {a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\) | \(98\) |
default | \(\frac {a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\) | \(98\) |
risch | \(\frac {5 a x}{128}-\frac {3 a \cos \left (d x +c \right )}{128 d}+\frac {a \cos \left (9 d x +9 c \right )}{2304 d}-\frac {a \sin \left (8 d x +8 c \right )}{1024 d}+\frac {3 a \cos \left (7 d x +7 c \right )}{1792 d}-\frac {a \sin \left (6 d x +6 c \right )}{192 d}-\frac {a \sin \left (4 d x +4 c \right )}{128 d}-\frac {a \cos \left (3 d x +3 c \right )}{96 d}+\frac {a \sin \left (2 d x +2 c \right )}{64 d}\) | \(123\) |
norman | \(\frac {-\frac {145 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {45 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {45 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {5 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {83 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {191 a \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {20 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 a x}{128}-\frac {4 a}{63 d}-\frac {4 a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {12 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d}-\frac {83 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {191 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {12 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d}+\frac {20 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {105 a x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {105 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {315 a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {315 a x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {45 a x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {45 a x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {145 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {12 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 a \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {5 a x \left (\tan ^{18}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\) | \(416\) |
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Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.67 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {896 \, a \cos \left (d x + c\right )^{9} - 1152 \, a \cos \left (d x + c\right )^{7} + 315 \, a d x - 21 \, {\left (48 \, a \cos \left (d x + c\right )^{7} - 8 \, a \cos \left (d x + c\right )^{5} - 10 \, a \cos \left (d x + c\right )^{3} - 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (116) = 232\).
Time = 0.93 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.98 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {5 a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {5 a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {5 a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 a \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {73 a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {5 a \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {2 a \cos ^{9}{\left (c + d x \right )}}{63 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{2}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1024 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a + 21 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{64512 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5}{128} \, a x + \frac {a \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {3 \, a \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{96 \, d} - \frac {3 \, a \cos \left (d x + c\right )}{128 \, d} - \frac {a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
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Time = 14.52 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.09 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5\,a\,x}{128}+\frac {\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}-\frac {191\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{96}+\left (\frac {a\,\left (11340\,c+11340\,d\,x-32256\right )}{8064}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {83\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\left (\frac {a\,\left (26460\,c+26460\,d\,x+53760\right )}{8064}-\frac {105\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {145\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}+\left (\frac {a\,\left (39690\,c+39690\,d\,x-161280\right )}{8064}-\frac {315\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\left (\frac {a\,\left (39690\,c+39690\,d\,x+96768\right )}{8064}-\frac {315\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {145\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\left (\frac {a\,\left (26460\,c+26460\,d\,x-96768\right )}{8064}-\frac {105\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {83\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}+\left (\frac {a\,\left (11340\,c+11340\,d\,x+13824\right )}{8064}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {191\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\left (\frac {a\,\left (2835\,c+2835\,d\,x-4608\right )}{8064}-\frac {45\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {a\,\left (315\,c+315\,d\,x-512\right )}{8064}-\frac {5\,a\,\left (c+d\,x\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]
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