Integrand size = 27, antiderivative size = 97 \[ \int \cos ^5(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a \sin ^5(c+d x)}{5 d}+\frac {b \sin ^6(c+d x)}{6 d}-\frac {2 a \sin ^7(c+d x)}{7 d}-\frac {b \sin ^8(c+d x)}{4 d}+\frac {a \sin ^9(c+d x)}{9 d}+\frac {b \sin ^{10}(c+d x)}{10 d} \]
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Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2916, 12, 780} \[ \int \cos ^5(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a \sin ^9(c+d x)}{9 d}-\frac {2 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^5(c+d x)}{5 d}+\frac {b \sin ^{10}(c+d x)}{10 d}-\frac {b \sin ^8(c+d x)}{4 d}+\frac {b \sin ^6(c+d x)}{6 d} \]
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Rule 12
Rule 780
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4 (a+x) \left (b^2-x^2\right )^2}{b^4} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int x^4 (a+x) \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^9 d} \\ & = \frac {\text {Subst}\left (\int \left (a b^4 x^4+b^4 x^5-2 a b^2 x^6-2 b^2 x^7+a x^8+x^9\right ) \, dx,x,b \sin (c+d x)\right )}{b^9 d} \\ & = \frac {a \sin ^5(c+d x)}{5 d}+\frac {b \sin ^6(c+d x)}{6 d}-\frac {2 a \sin ^7(c+d x)}{7 d}-\frac {b \sin ^8(c+d x)}{4 d}+\frac {a \sin ^9(c+d x)}{9 d}+\frac {b \sin ^{10}(c+d x)}{10 d} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \cos ^5(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {-3150 b \cos (2 (c+d x))+525 b \cos (6 (c+d x))-63 b \cos (10 (c+d x))+7560 a \sin (c+d x)-1680 a \sin (3 (c+d x))-1008 a \sin (5 (c+d x))+180 a \sin (7 (c+d x))+140 a \sin (9 (c+d x))}{322560 d} \]
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Time = 0.78 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {\frac {b \left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {a \left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {b \left (\sin ^{8}\left (d x +c \right )\right )}{4}-\frac {2 a \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\sin ^{6}\left (d x +c \right )\right ) b}{6}+\frac {\left (\sin ^{5}\left (d x +c \right )\right ) a}{5}}{d}\) | \(72\) |
default | \(\frac {\frac {b \left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {a \left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {b \left (\sin ^{8}\left (d x +c \right )\right )}{4}-\frac {2 a \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\sin ^{6}\left (d x +c \right )\right ) b}{6}+\frac {\left (\sin ^{5}\left (d x +c \right )\right ) a}{5}}{d}\) | \(72\) |
risch | \(\frac {3 a \sin \left (d x +c \right )}{128 d}-\frac {b \cos \left (10 d x +10 c \right )}{5120 d}+\frac {a \sin \left (9 d x +9 c \right )}{2304 d}+\frac {a \sin \left (7 d x +7 c \right )}{1792 d}+\frac {5 b \cos \left (6 d x +6 c \right )}{3072 d}-\frac {a \sin \left (5 d x +5 c \right )}{320 d}-\frac {a \sin \left (3 d x +3 c \right )}{192 d}-\frac {5 b \cos \left (2 d x +2 c \right )}{512 d}\) | \(119\) |
parallelrisch | \(\frac {\left (\sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-5 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+10 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (63 b \sin \left (5 d x +5 c \right )+880 a \cos \left (2 d x +2 c \right )+140 \cos \left (4 d x +4 c \right ) a +315 b \sin \left (3 d x +3 c \right )+420 b \sin \left (d x +c \right )+996 a \right ) \left (\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+5 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80640 d}\) | \(131\) |
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Time = 0.39 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98 \[ \int \cos ^5(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {126 \, b \cos \left (d x + c\right )^{10} - 315 \, b \cos \left (d x + c\right )^{8} + 210 \, b \cos \left (d x + c\right )^{6} - 4 \, {\left (35 \, a \cos \left (d x + c\right )^{8} - 50 \, a \cos \left (d x + c\right )^{6} + 3 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right )}{1260 \, d} \]
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Time = 1.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.40 \[ \int \cos ^5(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx=\begin {cases} \frac {8 a \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 a \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} - \frac {b \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac {b \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{12 d} - \frac {b \cos ^{10}{\left (c + d x \right )}}{60 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right ) \sin ^{4}{\left (c \right )} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74 \[ \int \cos ^5(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {126 \, b \sin \left (d x + c\right )^{10} + 140 \, a \sin \left (d x + c\right )^{9} - 315 \, b \sin \left (d x + c\right )^{8} - 360 \, a \sin \left (d x + c\right )^{7} + 210 \, b \sin \left (d x + c\right )^{6} + 252 \, a \sin \left (d x + c\right )^{5}}{1260 \, d} \]
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Time = 0.46 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.22 \[ \int \cos ^5(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {5 \, b \cos \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {5 \, b \cos \left (2 \, d x + 2 \, c\right )}{512 \, d} + \frac {a \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {a \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {a \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {3 \, a \sin \left (d x + c\right )}{128 \, d} \]
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Time = 11.47 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73 \[ \int \cos ^5(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {\frac {b\,{\sin \left (c+d\,x\right )}^{10}}{10}+\frac {a\,{\sin \left (c+d\,x\right )}^9}{9}-\frac {b\,{\sin \left (c+d\,x\right )}^8}{4}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {b\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}}{d} \]
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