Integrand size = 27, antiderivative size = 143 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a x}{128}-\frac {a \cos ^5(c+d x)}{5 d}+\frac {2 a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^9(c+d x)}{9 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d} \]
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Time = 0.14 (sec) , antiderivative size = 143, 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, 276} \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^9(c+d x)}{9 d}+\frac {2 a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^5(c+d x)}{5 d}-\frac {a \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {a \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 a x}{128} \]
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Rule 8
Rule 276
Rule 2645
Rule 2648
Rule 2715
Rule 2917
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+a \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx \\ & = -\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{8} (3 a) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {a \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{16} a \int \cos ^4(c+d x) \, dx-\frac {a \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a \cos ^5(c+d x)}{5 d}+\frac {2 a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^9(c+d x)}{9 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{64} (3 a) \int \cos ^2(c+d x) \, dx \\ & = -\frac {a \cos ^5(c+d x)}{5 d}+\frac {2 a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^9(c+d x)}{9 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{128} (3 a) \int 1 \, dx \\ & = \frac {3 a x}{128}-\frac {a \cos ^5(c+d x)}{5 d}+\frac {2 a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^9(c+d x)}{9 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.59 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (7560 c+7560 d x-7560 \cos (c+d x)-1680 \cos (3 (c+d x))+1008 \cos (5 (c+d x))+180 \cos (7 (c+d x))-140 \cos (9 (c+d x))-2520 \sin (4 (c+d x))+315 \sin (8 (c+d x)))}{322560 d} \]
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Time = 0.50 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.59
method | result | size |
parallelrisch | \(-\frac {\left (-3 d x +\sin \left (4 d x +4 c \right )-\frac {\sin \left (8 d x +8 c \right )}{8}+3 \cos \left (d x +c \right )+\frac {2 \cos \left (3 d x +3 c \right )}{3}-\frac {2 \cos \left (5 d x +5 c \right )}{5}-\frac {\cos \left (7 d x +7 c \right )}{14}+\frac {\cos \left (9 d x +9 c \right )}{18}+\frac {1024}{315}\right ) a}{128 d}\) | \(85\) |
risch | \(\frac {3 a x}{128}-\frac {3 a \cos \left (d x +c \right )}{128 d}-\frac {a \cos \left (9 d x +9 c \right )}{2304 d}+\frac {a \sin \left (8 d x +8 c \right )}{1024 d}+\frac {a \cos \left (7 d x +7 c \right )}{1792 d}+\frac {a \cos \left (5 d x +5 c \right )}{320 d}-\frac {a \sin \left (4 d x +4 c \right )}{128 d}-\frac {a \cos \left (3 d x +3 c \right )}{192 d}\) | \(108\) |
derivativedivides | \(\frac {a \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\) | \(124\) |
default | \(\frac {a \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\) | \(124\) |
norman | \(\frac {\frac {169 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {27 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {27 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {3 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {155 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {13 a \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {16 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 a x}{128}-\frac {16 a}{315 d}-\frac {64 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {155 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {13 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {32 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {16 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {3 a \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {32 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {63 a x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {63 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {189 a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {189 a x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {27 a x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {27 a x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {3 a x \left (\tan ^{18}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {169 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {112 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\) | \(399\) |
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Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {4480 \, a \cos \left (d x + c\right )^{9} - 11520 \, a \cos \left (d x + c\right )^{7} + 8064 \, a \cos \left (d x + c\right )^{5} - 945 \, a d x - 315 \, {\left (16 \, a \cos \left (d x + c\right )^{7} - 24 \, a \cos \left (d x + c\right )^{5} + 2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (131) = 262\).
Time = 0.98 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.90 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {3 a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {9 a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {3 a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {11 a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {4 a \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {3 a \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {8 a \cos ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{4}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.50 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {1024 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a - 315 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{322560 \, d} \]
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Time = 0.52 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.75 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3}{128} \, a x - \frac {a \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {a \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {a \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {3 \, a \cos \left (d x + c\right )}{128 \, d} + \frac {a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} \]
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Time = 13.58 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.47 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3\,a\,x}{128}+\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{32}-\frac {155\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\left (\frac {a\,\left (79380\,c+79380\,d\,x-430080\right )}{40320}-\frac {63\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\frac {169\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}+\left (\frac {a\,\left (119070\,c+119070\,d\,x+645120\right )}{40320}-\frac {189\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\left (\frac {a\,\left (119070\,c+119070\,d\,x-903168\right )}{40320}-\frac {189\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {169\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\left (\frac {a\,\left (79380\,c+79380\,d\,x+258048\right )}{40320}-\frac {63\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {155\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}+\left (\frac {a\,\left (34020\,c+34020\,d\,x-73728\right )}{40320}-\frac {27\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}+\left (\frac {a\,\left (8505\,c+8505\,d\,x-18432\right )}{40320}-\frac {27\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {a\,\left (945\,c+945\,d\,x-2048\right )}{40320}-\frac {3\,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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