Integrand size = 27, antiderivative size = 127 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a x}{128}-\frac {a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^7(c+d x)}{7 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d} \]
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Time = 0.13 (sec) , antiderivative size = 127, 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, 2645, 14, 2648, 2715, 8} \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^5(c+d x)}{5 d}-\frac {a \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {a \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 a x}{128} \]
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Rule 8
Rule 14
Rule 2645
Rule 2648
Rule 2715
Rule 2917
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^4(c+d x) \sin ^3(c+d x) \, dx+a \int \cos ^4(c+d x) \sin ^4(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 ) \, 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-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^7(c+d x)}{7 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 {a \cos ^7(c+d x)}{7 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 {a \cos ^7(c+d x)}{7 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 = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.56 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (840 d x-1680 \cos (c+d x)-560 \cos (3 (c+d x))+112 \cos (5 (c+d x))+80 \cos (7 (c+d x))-280 \sin (4 (c+d x))+35 \sin (8 (c+d x)))}{35840 d} \]
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Time = 0.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.60
method | result | size |
parallelrisch | \(-\frac {a \left (-840 d x +1680 \cos \left (d x +c \right )-112 \cos \left (5 d x +5 c \right )-80 \cos \left (7 d x +7 c \right )+560 \cos \left (3 d x +3 c \right )-35 \sin \left (8 d x +8 c \right )+280 \sin \left (4 d x +4 c \right )+2048\right )}{35840 d}\) | \(76\) |
risch | \(\frac {3 a x}{128}-\frac {3 a \cos \left (d x +c \right )}{64 d}+\frac {a \sin \left (8 d x +8 c \right )}{1024 d}+\frac {a \cos \left (7 d x +7 c \right )}{448 d}+\frac {a \cos \left (5 d x +5 c \right )}{320 d}-\frac {a \sin \left (4 d x +4 c \right )}{128 d}-\frac {a \cos \left (3 d x +3 c \right )}{64 d}\) | \(93\) |
derivativedivides | \(\frac {a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+a \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(106\) |
default | \(\frac {a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+a \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(106\) |
norman | \(\frac {-\frac {3 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {333 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {23 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {3 a x}{128}-\frac {4 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {32 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {4 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {32 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {4 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a}{35 d}+\frac {671 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {671 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {3 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {21 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {105 a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {21 a x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {3 a x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 a x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {333 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {23 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {3 a \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(367\) |
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Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.66 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {640 \, a \cos \left (d x + c\right )^{7} - 896 \, a \cos \left (d x + c\right )^{5} + 105 \, a d x + 35 \, {\left (16 \, a \cos \left (d x + c\right )^{7} - 24 \, a \cos \left (d x + c\right )^{5} + 2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4480 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (116) = 232\).
Time = 0.67 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.95 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {3 a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {9 a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {3 a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {11 a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {2 a \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{3}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1024 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a + 35 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{35840 \, d} \]
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Time = 0.76 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.72 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3}{128} \, a x + \frac {a \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {a \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {3 \, a \cos \left (d x + c\right )}{64 \, d} + \frac {a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} \]
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Time = 13.54 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.52 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3\,a\,x}{128}+\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+\frac {23\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\left (\frac {a\,\left (2940\,c+2940\,d\,x-17920\right )}{4480}-\frac {21\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {333\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {671\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\left (\frac {a\,\left (7350\,c+7350\,d\,x-17920\right )}{4480}-\frac {105\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {671\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{64}+\left (\frac {a\,\left (5880\,c+5880\,d\,x-28672\right )}{4480}-\frac {21\,a\,\left (c+d\,x\right )}{16}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {333\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{64}+\left (\frac {a\,\left (2940\,c+2940\,d\,x+3584\right )}{4480}-\frac {21\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {23\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}+\left (\frac {a\,\left (840\,c+840\,d\,x-4096\right )}{4480}-\frac {3\,a\,\left (c+d\,x\right )}{16}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {a\,\left (105\,c+105\,d\,x-512\right )}{4480}-\frac {3\,a\,\left (c+d\,x\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \]
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