Integrand size = 27, antiderivative size = 113 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^9(c+d x)}{3 d}+\frac {3 a \sin ^{11}(c+d x)}{11 d}-\frac {a \sin ^{13}(c+d x)}{13 d} \]
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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, 2645, 272, 45, 2644, 276} \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin ^{13}(c+d x)}{13 d}+\frac {3 a \sin ^{11}(c+d x)}{11 d}-\frac {a \sin ^9(c+d x)}{3 d}+\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^8(c+d x)}{8 d} \]
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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 ^5(c+d x) \, dx+a \int \cos ^7(c+d x) \sin ^6(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^6 \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^6-3 x^8+3 x^{10}-x^{12}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^9(c+d x)}{3 d}+\frac {3 a \sin ^{11}(c+d x)}{11 d}-\frac {a \sin ^{13}(c+d x)}{13 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 ^7(c+d x)}{7 d}-\frac {a \sin ^9(c+d x)}{3 d}+\frac {3 a \sin ^{11}(c+d x)}{11 d}-\frac {a \sin ^{13}(c+d x)}{13 d} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.21 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a (600600 \cos (2 (c+d x))+75075 \cos (4 (c+d x))-100100 \cos (6 (c+d x))-30030 \cos (8 (c+d x))+12012 \cos (10 (c+d x))+5005 \cos (12 (c+d x))-600600 \sin (c+d x)+150150 \sin (3 (c+d x))+90090 \sin (5 (c+d x))-25740 \sin (7 (c+d x))-20020 \sin (9 (c+d x))+2730 \sin (11 (c+d x))+2310 \sin (13 (c+d x)))}{123002880 d} \]
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Time = 0.86 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\sin ^{13}\left (d x +c \right )\right )}{13}+\frac {\left (\sin ^{12}\left (d x +c \right )\right )}{12}-\frac {3 \left (\sin ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}\right )}{d}\) | \(88\) |
default | \(-\frac {a \left (\frac {\left (\sin ^{13}\left (d x +c \right )\right )}{13}+\frac {\left (\sin ^{12}\left (d x +c \right )\right )}{12}-\frac {3 \left (\sin ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}\right )}{d}\) | \(88\) |
parallelrisch | \(-\frac {a \left (-10+\cos \left (3 d x +3 c \right )-6 \cos \left (2 d x +2 c \right )+15 \cos \left (d x +c \right )\right ) \left (126126+147147 \cos \left (2 d x +2 c \right )+2310 \sin \left (7 d x +7 c \right )+16590 \sin \left (5 d x +5 c \right )+5005 \cos \left (6 d x +6 c \right )+40830 \sin \left (d x +c \right )+44870 \sin \left (3 d x +3 c \right )+42042 \cos \left (4 d x +4 c \right )\right ) \left (\cos \left (3 d x +3 c \right )+6 \cos \left (2 d x +2 c \right )+15 \cos \left (d x +c \right )+10\right )}{30750720 d}\) | \(143\) |
risch | \(-\frac {a \sin \left (11 d x +11 c \right )}{45056 d}-\frac {a \cos \left (10 d x +10 c \right )}{10240 d}-\frac {a \cos \left (12 d x +12 c \right )}{24576 d}+\frac {5 a \sin \left (d x +c \right )}{1024 d}-\frac {a \sin \left (13 d x +13 c \right )}{53248 d}+\frac {a \sin \left (9 d x +9 c \right )}{6144 d}+\frac {a \cos \left (8 d x +8 c \right )}{4096 d}+\frac {3 a \sin \left (7 d x +7 c \right )}{14336 d}+\frac {5 a \cos \left (6 d x +6 c \right )}{6144 d}-\frac {3 a \sin \left (5 d x +5 c \right )}{4096 d}-\frac {5 a \cos \left (4 d x +4 c \right )}{8192 d}-\frac {5 a \sin \left (3 d x +3 c \right )}{4096 d}-\frac {5 a \cos \left (2 d x +2 c \right )}{1024 d}\) | \(194\) |
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Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {10010 \, a \cos \left (d x + c\right )^{12} - 24024 \, a \cos \left (d x + c\right )^{10} + 15015 \, a \cos \left (d x + c\right )^{8} + 40 \, {\left (231 \, a \cos \left (d x + c\right )^{12} - 567 \, a \cos \left (d x + c\right )^{10} + 371 \, a \cos \left (d x + c\right )^{8} - 5 \, a \cos \left (d x + c\right )^{6} - 6 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right )}{120120 \, d} \]
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Time = 3.57 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.42 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {16 a \sin ^{13}{\left (c + d x \right )}}{3003 d} + \frac {8 a \sin ^{11}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{231 d} + \frac {2 a \sin ^{9}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{21 d} + \frac {a \sin ^{7}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{7 d} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{20 d} - \frac {a \cos ^{12}{\left (c + d x \right )}}{120 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{5}{\left (c \right )} \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {9240 \, a \sin \left (d x + c\right )^{13} + 10010 \, a \sin \left (d x + c\right )^{12} - 32760 \, a \sin \left (d x + c\right )^{11} - 36036 \, a \sin \left (d x + c\right )^{10} + 40040 \, a \sin \left (d x + c\right )^{9} + 45045 \, a \sin \left (d x + c\right )^{8} - 17160 \, a \sin \left (d x + c\right )^{7} - 20020 \, a \sin \left (d x + c\right )^{6}}{120120 \, d} \]
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Time = 0.44 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.71 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos \left (12 \, d x + 12 \, c\right )}{24576 \, d} - \frac {a \cos \left (10 \, d x + 10 \, c\right )}{10240 \, d} + \frac {a \cos \left (8 \, d x + 8 \, c\right )}{4096 \, d} + \frac {5 \, a \cos \left (6 \, d x + 6 \, c\right )}{6144 \, d} - \frac {5 \, a \cos \left (4 \, d x + 4 \, c\right )}{8192 \, d} - \frac {5 \, a \cos \left (2 \, d x + 2 \, c\right )}{1024 \, d} - \frac {a \sin \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac {a \sin \left (11 \, d x + 11 \, c\right )}{45056 \, d} + \frac {a \sin \left (9 \, d x + 9 \, c\right )}{6144 \, d} + \frac {3 \, a \sin \left (7 \, d x + 7 \, c\right )}{14336 \, d} - \frac {3 \, a \sin \left (5 \, d x + 5 \, c\right )}{4096 \, d} - \frac {5 \, a \sin \left (3 \, d x + 3 \, c\right )}{4096 \, d} + \frac {5 \, a \sin \left (d x + c\right )}{1024 \, d} \]
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Time = 10.01 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^{13}}{13}-\frac {a\,{\sin \left (c+d\,x\right )}^{12}}{12}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^{11}}{11}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^{10}}{10}-\frac {a\,{\sin \left (c+d\,x\right )}^9}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}}{d} \]
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