Integrand size = 27, antiderivative size = 157 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27 a^3 x}{128}-\frac {9 a^3 \cos ^5(c+d x)}{80 d}+\frac {27 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {9 a^3 \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {3 a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {9 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d} \]
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Time = 0.15 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2939, 2757, 2748, 2715, 8} \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {9 a^3 \cos ^5(c+d x)}{80 d}-\frac {9 \cos ^5(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{112 d}+\frac {9 a^3 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {27 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {27 a^3 x}{128}-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}-\frac {3 a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{56 d} \]
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Rule 8
Rule 2715
Rule 2748
Rule 2757
Rule 2939
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}+\frac {3}{8} \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx \\ & = -\frac {3 a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}+\frac {1}{56} (27 a) \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {3 a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {9 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac {1}{16} \left (9 a^2\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {9 a^3 \cos ^5(c+d x)}{80 d}-\frac {3 a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {9 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac {1}{16} \left (9 a^3\right ) \int \cos ^4(c+d x) \, dx \\ & = -\frac {9 a^3 \cos ^5(c+d x)}{80 d}+\frac {9 a^3 \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {3 a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {9 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac {1}{64} \left (27 a^3\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {9 a^3 \cos ^5(c+d x)}{80 d}+\frac {27 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {9 a^3 \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {3 a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {9 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac {1}{128} \left (27 a^3\right ) \int 1 \, dx \\ & = \frac {27 a^3 x}{128}-\frac {9 a^3 \cos ^5(c+d x)}{80 d}+\frac {27 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {9 a^3 \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {3 a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {9 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.61 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (8400 c+7560 d x-9520 \cos (c+d x)-3920 \cos (3 (c+d x))-112 \cos (5 (c+d x))+240 \cos (7 (c+d x))+1680 \sin (2 (c+d x))-1960 \sin (4 (c+d x))-560 \sin (6 (c+d x))+35 \sin (8 (c+d x)))}{35840 d} \]
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Time = 0.48 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(-\frac {a^{3} \left (-7560 d x +9520 \cos \left (d x +c \right )-35 \sin \left (8 d x +8 c \right )+1960 \sin \left (4 d x +4 c \right )+560 \sin \left (6 d x +6 c \right )-1680 \sin \left (2 d x +2 c \right )+112 \cos \left (5 d x +5 c \right )-240 \cos \left (7 d x +7 c \right )+3920 \cos \left (3 d x +3 c \right )+13312\right )}{35840 d}\) | \(100\) |
risch | \(\frac {27 a^{3} x}{128}-\frac {17 a^{3} \cos \left (d x +c \right )}{64 d}+\frac {a^{3} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {3 a^{3} \cos \left (7 d x +7 c \right )}{448 d}-\frac {a^{3} \sin \left (6 d x +6 c \right )}{64 d}-\frac {a^{3} \cos \left (5 d x +5 c \right )}{320 d}-\frac {7 a^{3} \sin \left (4 d x +4 c \right )}{128 d}-\frac {7 a^{3} \cos \left (3 d x +3 c \right )}{64 d}+\frac {3 a^{3} \sin \left (2 d x +2 c \right )}{64 d}\) | \(141\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+3 a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}}{d}\) | \(178\) |
default | \(\frac {a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+3 a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}}{d}\) | \(178\) |
norman | \(\frac {-\frac {14 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {27 a^{3} x}{128}-\frac {26 a^{3}}{35 d}+\frac {305 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {437 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {27 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {189 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {945 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {919 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {18 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {26 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {437 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {919 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {27 a^{3} x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {2 a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {138 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}-\frac {10 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {158 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {189 a^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {189 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {305 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {27 a^{3} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {27 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {189 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {27 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(451\) |
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Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.62 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {1920 \, a^{3} \cos \left (d x + c\right )^{7} - 3584 \, a^{3} \cos \left (d x + c\right )^{5} + 945 \, a^{3} d x + 35 \, {\left (16 \, a^{3} \cos \left (d x + c\right )^{7} - 88 \, a^{3} \cos \left (d x + c\right )^{5} + 18 \, a^{3} \cos \left (d x + c\right )^{3} + 27 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4480 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (148) = 296\).
Time = 0.74 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.80 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {3 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {3 a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {9 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {9 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {11 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {3 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {6 a^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin {\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.73 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {7168 \, a^{3} \cos \left (d x + c\right )^{5} - 3072 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 560 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 35 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{35840 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.89 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27}{128} \, a^{3} x + \frac {3 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {a^{3} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {7 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {17 \, a^{3} \cos \left (d x + c\right )}{64 \, d} + \frac {a^{3} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} - \frac {7 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
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Time = 11.97 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.94 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27\,a^3\,x}{128}-\frac {\frac {919\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{64}-\frac {437\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{64}-\frac {305\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}-\frac {919\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {437\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {305\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}-\frac {27\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+\frac {a^3\,\left (945\,c+945\,d\,x\right )}{4480}-\frac {a^3\,\left (945\,c+945\,d\,x-3328\right )}{4480}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {a^3\,\left (945\,c+945\,d\,x\right )}{560}-\frac {a^3\,\left (7560\,c+7560\,d\,x-8960\right )}{4480}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (945\,c+945\,d\,x\right )}{560}-\frac {a^3\,\left (7560\,c+7560\,d\,x-17664\right )}{4480}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3\,\left (945\,c+945\,d\,x\right )}{160}-\frac {a^3\,\left (26460\,c+26460\,d\,x-12544\right )}{4480}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {a^3\,\left (945\,c+945\,d\,x\right )}{160}-\frac {a^3\,\left (26460\,c+26460\,d\,x-80640\right )}{4480}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {a^3\,\left (945\,c+945\,d\,x\right )}{80}-\frac {a^3\,\left (52920\,c+52920\,d\,x-44800\right )}{4480}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^3\,\left (945\,c+945\,d\,x\right )}{80}-\frac {a^3\,\left (52920\,c+52920\,d\,x-141568\right )}{4480}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^3\,\left (945\,c+945\,d\,x\right )}{64}-\frac {a^3\,\left (66150\,c+66150\,d\,x-116480\right )}{4480}\right )+\frac {27\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \]
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