Integrand size = 27, antiderivative size = 113 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {3 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^9(c+d x)}{3 d}-\frac {a \sin ^{11}(c+d x)}{11 d} \]
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Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2913, 2644, 276, 2645, 272, 45} \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin ^{11}(c+d x)}{11 d}+\frac {a \sin ^9(c+d x)}{3 d}-\frac {3 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^8(c+d x)}{8 d} \]
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Rule 45
Rule 272
Rule 276
Rule 2644
Rule 2645
Rule 2913
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^7(c+d x) \sin ^4(c+d x) \, dx+a \int \cos ^7(c+d x) \sin ^5(c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int x^7 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}+\frac {a \text {Subst}\left (\int x^4 \left (1-x^2\right )^3 \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {a \text {Subst}\left (\int (1-x)^2 x^3 \, dx,x,\cos ^2(c+d x)\right )}{2 d}+\frac {a \text {Subst}\left (\int \left (x^4-3 x^6+3 x^8-x^{10}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {a \sin ^5(c+d x)}{5 d}-\frac {3 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^9(c+d x)}{3 d}-\frac {a \sin ^{11}(c+d x)}{11 d}-\frac {a \text {Subst}\left (\int \left (x^3-2 x^4+x^5\right ) \, dx,x,\cos ^2(c+d x)\right )}{2 d} \\ & = -\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {3 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^9(c+d x)}{3 d}-\frac {a \sin ^{11}(c+d x)}{11 d} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.12 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a (46200 \cos (2 (c+d x))+5775 \cos (4 (c+d x))-7700 \cos (6 (c+d x))-2310 \cos (8 (c+d x))+924 \cos (10 (c+d x))+385 \cos (12 (c+d x))-129360 \sin (c+d x)+18480 \sin (3 (c+d x))+20328 \sin (5 (c+d x))+1320 \sin (7 (c+d x))-3080 \sin (9 (c+d x))-840 \sin (11 (c+d x)))}{9461760 d} \]
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Time = 0.72 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\sin ^{12}\left (d x +c \right )\right )}{12}+\frac {\left (\sin ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\sin ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\) | \(88\) |
default | \(-\frac {a \left (\frac {\left (\sin ^{12}\left (d x +c \right )\right )}{12}+\frac {\left (\sin ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\sin ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\) | \(88\) |
parallelrisch | \(\frac {a \left (\sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-5 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+10 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (26680 \cos \left (2 d x +2 c \right )+385 \sin \left (7 d x +7 c \right )+2849 \sin \left (5 d x +5 c \right )+840 \cos \left (6 d x +6 c \right )+8085 \sin \left (d x +c \right )+8085 \sin \left (3 d x +3 c \right )+7280 \cos \left (4 d x +4 c \right )+24336\right ) \left (\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+5 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2365440 d}\) | \(147\) |
risch | \(\frac {a \sin \left (11 d x +11 c \right )}{11264 d}-\frac {a \cos \left (12 d x +12 c \right )}{24576 d}+\frac {7 a \sin \left (d x +c \right )}{512 d}-\frac {a \cos \left (10 d x +10 c \right )}{10240 d}+\frac {a \sin \left (9 d x +9 c \right )}{3072 d}+\frac {a \cos \left (8 d x +8 c \right )}{4096 d}-\frac {a \sin \left (7 d x +7 c \right )}{7168 d}+\frac {5 a \cos \left (6 d x +6 c \right )}{6144 d}-\frac {11 a \sin \left (5 d x +5 c \right )}{5120 d}-\frac {5 a \cos \left (4 d x +4 c \right )}{8192 d}-\frac {a \sin \left (3 d x +3 c \right )}{512 d}-\frac {5 a \cos \left (2 d x +2 c \right )}{1024 d}\) | \(179\) |
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Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.94 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {770 \, a \cos \left (d x + c\right )^{12} - 1848 \, a \cos \left (d x + c\right )^{10} + 1155 \, a \cos \left (d x + c\right )^{8} - 8 \, {\left (105 \, a \cos \left (d x + c\right )^{10} - 140 \, a \cos \left (d x + c\right )^{8} + 5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{9240 \, d} \]
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Time = 2.63 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.42 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {16 a \sin ^{11}{\left (c + d x \right )}}{1155 d} + \frac {8 a \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{105 d} + \frac {6 a \sin ^{7}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{35 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{5 d} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{20 d} - \frac {a \cos ^{12}{\left (c + d x \right )}}{120 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{4}{\left (c \right )} \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {770 \, a \sin \left (d x + c\right )^{12} + 840 \, a \sin \left (d x + c\right )^{11} - 2772 \, a \sin \left (d x + c\right )^{10} - 3080 \, a \sin \left (d x + c\right )^{9} + 3465 \, a \sin \left (d x + c\right )^{8} + 3960 \, a \sin \left (d x + c\right )^{7} - 1540 \, a \sin \left (d x + c\right )^{6} - 1848 \, a \sin \left (d x + c\right )^{5}}{9240 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.58 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos \left (12 \, d x + 12 \, c\right )}{24576 \, d} - \frac {a \cos \left (10 \, d x + 10 \, c\right )}{10240 \, d} + \frac {a \cos \left (8 \, d x + 8 \, c\right )}{4096 \, d} + \frac {5 \, a \cos \left (6 \, d x + 6 \, c\right )}{6144 \, d} - \frac {5 \, a \cos \left (4 \, d x + 4 \, c\right )}{8192 \, d} - \frac {5 \, a \cos \left (2 \, d x + 2 \, c\right )}{1024 \, d} + \frac {a \sin \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {a \sin \left (9 \, d x + 9 \, c\right )}{3072 \, d} - \frac {a \sin \left (7 \, d x + 7 \, c\right )}{7168 \, d} - \frac {11 \, a \sin \left (5 \, d x + 5 \, c\right )}{5120 \, d} - \frac {a \sin \left (3 \, d x + 3 \, c\right )}{512 \, d} + \frac {7 \, a \sin \left (d x + c\right )}{512 \, d} \]
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Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^{12}}{12}-\frac {a\,{\sin \left (c+d\,x\right )}^{11}}{11}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^{10}}{10}+\frac {a\,{\sin \left (c+d\,x\right )}^9}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}}{d} \]
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