Integrand size = 29, antiderivative size = 209 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {41 a^3 x}{1024}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {7 a^3 \cos ^9(c+d x)}{9 d}-\frac {3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {41 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {41 a^3 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {41 a^3 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d} \]
41/1024*a^3*x-4/7*a^3*cos(d*x+c)^7/d+7/9*a^3*cos(d*x+c)^9/d-3/11*a^3*cos(d *x+c)^11/d+41/1024*a^3*cos(d*x+c)*sin(d*x+c)/d+41/1536*a^3*cos(d*x+c)^3*si n(d*x+c)/d+41/1920*a^3*cos(d*x+c)^5*sin(d*x+c)/d-41/320*a^3*cos(d*x+c)^7*s in(d*x+c)/d-41/120*a^3*cos(d*x+c)^7*sin(d*x+c)^3/d-1/12*a^3*cos(d*x+c)^7*s in(d*x+c)^5/d
Time = 0.85 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.65 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1247400 c+1136520 d x-1496880 \cos (c+d x)-572880 \cos (3 (c+d x))+83160 \cos (5 (c+d x))+106920 \cos (7 (c+d x))+3080 \cos (9 (c+d x))-7560 \cos (11 (c+d x))+166320 \sin (2 (c+d x))-384615 \sin (4 (c+d x))-83160 \sin (6 (c+d x))+51975 \sin (8 (c+d x))+16632 \sin (10 (c+d x))-1155 \sin (12 (c+d x)))}{28385280 d} \]
(a^3*(1247400*c + 1136520*d*x - 1496880*Cos[c + d*x] - 572880*Cos[3*(c + d *x)] + 83160*Cos[5*(c + d*x)] + 106920*Cos[7*(c + d*x)] + 3080*Cos[9*(c + d*x)] - 7560*Cos[11*(c + d*x)] + 166320*Sin[2*(c + d*x)] - 384615*Sin[4*(c + d*x)] - 83160*Sin[6*(c + d*x)] + 51975*Sin[8*(c + d*x)] + 16632*Sin[10* (c + d*x)] - 1155*Sin[12*(c + d*x)]))/(28385280*d)
Time = 0.61 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3352, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sin ^3(c+d x) \cos ^6(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^6 (a \sin (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \sin ^6(c+d x) \cos ^6(c+d x)+3 a^3 \sin ^5(c+d x) \cos ^6(c+d x)+3 a^3 \sin ^4(c+d x) \cos ^6(c+d x)+a^3 \sin ^3(c+d x) \cos ^6(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {7 a^3 \cos ^9(c+d x)}{9 d}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac {41 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac {41 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac {41 a^3 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac {41 a^3 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac {41 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {41 a^3 x}{1024}\) |
(41*a^3*x)/1024 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (7*a^3*Cos[c + d*x]^9)/(9 *d) - (3*a^3*Cos[c + d*x]^11)/(11*d) + (41*a^3*Cos[c + d*x]*Sin[c + d*x])/ (1024*d) + (41*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + (41*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(1920*d) - (41*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(320 *d) - (41*a^3*Cos[c + d*x]^7*Sin[c + d*x]^3)/(120*d) - (a^3*Cos[c + d*x]^7 *Sin[c + d*x]^5)/(12*d)
3.7.6.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Time = 1.09 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.68
| method | result | size |
| parallelrisch | \(\frac {3 \left (\frac {41 d x}{6}+\sin \left (2 d x +2 c \right )-\frac {37 \sin \left (4 d x +4 c \right )}{16}-\frac {\sin \left (6 d x +6 c \right )}{2}+\frac {5 \sin \left (8 d x +8 c \right )}{16}-\frac {\cos \left (11 d x +11 c \right )}{22}+\frac {\sin \left (10 d x +10 c \right )}{10}-\frac {\sin \left (12 d x +12 c \right )}{144}-9 \cos \left (d x +c \right )-\frac {31 \cos \left (3 d x +3 c \right )}{9}+\frac {\cos \left (5 d x +5 c \right )}{2}+\frac {9 \cos \left (7 d x +7 c \right )}{14}+\frac {\cos \left (9 d x +9 c \right )}{54}-\frac {23552}{2079}\right ) a^{3}}{512 d}\) | \(142\) |
| risch | \(\frac {3 a^{3} \sin \left (10 d x +10 c \right )}{5120 d}-\frac {3 a^{3} \cos \left (11 d x +11 c \right )}{11264 d}+\frac {41 a^{3} x}{1024}-\frac {27 a^{3} \cos \left (d x +c \right )}{512 d}-\frac {a^{3} \sin \left (12 d x +12 c \right )}{24576 d}+\frac {a^{3} \cos \left (9 d x +9 c \right )}{9216 d}+\frac {15 a^{3} \sin \left (8 d x +8 c \right )}{8192 d}+\frac {27 a^{3} \cos \left (7 d x +7 c \right )}{7168 d}-\frac {3 a^{3} \sin \left (6 d x +6 c \right )}{1024 d}+\frac {3 a^{3} \cos \left (5 d x +5 c \right )}{1024 d}-\frac {111 a^{3} \sin \left (4 d x +4 c \right )}{8192 d}-\frac {31 a^{3} \cos \left (3 d x +3 c \right )}{1536 d}+\frac {3 a^{3} \sin \left (2 d x +2 c \right )}{512 d}\) | \(209\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )}{d}\) | \(272\) |
| default | \(\frac {a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )}{d}\) | \(272\) |
3/512*(41/6*d*x+sin(2*d*x+2*c)-37/16*sin(4*d*x+4*c)-1/2*sin(6*d*x+6*c)+5/1 6*sin(8*d*x+8*c)-1/22*cos(11*d*x+11*c)+1/10*sin(10*d*x+10*c)-1/144*sin(12* d*x+12*c)-9*cos(d*x+c)-31/9*cos(3*d*x+3*c)+1/2*cos(5*d*x+5*c)+9/14*cos(7*d *x+7*c)+1/54*cos(9*d*x+9*c)-23552/2079)*a^3/d
Time = 0.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.66 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {967680 \, a^{3} \cos \left (d x + c\right )^{11} - 2759680 \, a^{3} \cos \left (d x + c\right )^{9} + 2027520 \, a^{3} \cos \left (d x + c\right )^{7} - 142065 \, a^{3} d x + 231 \, {\left (1280 \, a^{3} \cos \left (d x + c\right )^{11} - 7808 \, a^{3} \cos \left (d x + c\right )^{9} + 8496 \, a^{3} \cos \left (d x + c\right )^{7} - 328 \, a^{3} \cos \left (d x + c\right )^{5} - 410 \, a^{3} \cos \left (d x + c\right )^{3} - 615 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \]
-1/3548160*(967680*a^3*cos(d*x + c)^11 - 2759680*a^3*cos(d*x + c)^9 + 2027 520*a^3*cos(d*x + c)^7 - 142065*a^3*d*x + 231*(1280*a^3*cos(d*x + c)^11 - 7808*a^3*cos(d*x + c)^9 + 8496*a^3*cos(d*x + c)^7 - 328*a^3*cos(d*x + c)^5 - 410*a^3*cos(d*x + c)^3 - 615*a^3*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 699 vs. \(2 (201) = 402\).
Time = 2.65 (sec) , antiderivative size = 699, normalized size of antiderivative = 3.34 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {5 a^{3} x \sin ^{12}{\left (c + d x \right )}}{1024} + \frac {15 a^{3} x \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{512} + \frac {9 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {75 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{1024} + \frac {45 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {25 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{256} + \frac {45 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {75 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{1024} + \frac {45 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{512} + \frac {45 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {5 a^{3} x \cos ^{12}{\left (c + d x \right )}}{1024} + \frac {9 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {5 a^{3} \sin ^{11}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{1024 d} + \frac {85 a^{3} \sin ^{9}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3072 d} + \frac {9 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {33 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{512 d} + \frac {21 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {33 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{512 d} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {85 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{3072 d} - \frac {21 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {4 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{21 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {5 a^{3} \sin {\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{1024 d} - \frac {9 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {8 a^{3} \cos ^{11}{\left (c + d x \right )}}{231 d} - \frac {2 a^{3} \cos ^{9}{\left (c + d x \right )}}{63 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{3}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((5*a**3*x*sin(c + d*x)**12/1024 + 15*a**3*x*sin(c + d*x)**10*cos (c + d*x)**2/512 + 9*a**3*x*sin(c + d*x)**10/256 + 75*a**3*x*sin(c + d*x)* *8*cos(c + d*x)**4/1024 + 45*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 25*a**3*x*sin(c + d*x)**6*cos(c + d*x)**6/256 + 45*a**3*x*sin(c + d*x)**6* cos(c + d*x)**4/128 + 75*a**3*x*sin(c + d*x)**4*cos(c + d*x)**8/1024 + 45* a**3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 15*a**3*x*sin(c + d*x)**2*cos (c + d*x)**10/512 + 45*a**3*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 5*a**3 *x*cos(c + d*x)**12/1024 + 9*a**3*x*cos(c + d*x)**10/256 + 5*a**3*sin(c + d*x)**11*cos(c + d*x)/(1024*d) + 85*a**3*sin(c + d*x)**9*cos(c + d*x)**3/( 3072*d) + 9*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 33*a**3*sin(c + d* x)**7*cos(c + d*x)**5/(512*d) + 21*a**3*sin(c + d*x)**7*cos(c + d*x)**3/(1 28*d) - 33*a**3*sin(c + d*x)**5*cos(c + d*x)**7/(512*d) + 3*a**3*sin(c + d *x)**5*cos(c + d*x)**5/(10*d) - 3*a**3*sin(c + d*x)**4*cos(c + d*x)**7/(7* d) - 85*a**3*sin(c + d*x)**3*cos(c + d*x)**9/(3072*d) - 21*a**3*sin(c + d* x)**3*cos(c + d*x)**7/(128*d) - 4*a**3*sin(c + d*x)**2*cos(c + d*x)**9/(21 *d) - a**3*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 5*a**3*sin(c + d*x)*cos (c + d*x)**11/(1024*d) - 9*a**3*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 8*a **3*cos(c + d*x)**11/(231*d) - 2*a**3*cos(c + d*x)**9/(63*d), Ne(d, 0)), ( x*(a*sin(c) + a)**3*sin(c)**3*cos(c)**6, True))
Time = 0.22 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.78 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {122880 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 450560 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 8316 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 1155 \, {\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{28385280 \, d} \]
-1/28385280*(122880*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^3 - 450560*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^3 - 8316*(32* sin(2*d*x + 2*c)^5 + 120*d*x + 120*c + 5*sin(8*d*x + 8*c) - 40*sin(4*d*x + 4*c))*a^3 - 1155*(4*sin(4*d*x + 4*c)^3 + 120*d*x + 120*c + 9*sin(8*d*x + 8*c) - 48*sin(4*d*x + 4*c))*a^3)/d
Time = 0.35 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {41}{1024} \, a^{3} x - \frac {3 \, a^{3} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {a^{3} \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {27 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {3 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac {31 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac {27 \, a^{3} \cos \left (d x + c\right )}{512 \, d} - \frac {a^{3} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} + \frac {3 \, a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {15 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {3 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {111 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac {3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
41/1024*a^3*x - 3/11264*a^3*cos(11*d*x + 11*c)/d + 1/9216*a^3*cos(9*d*x + 9*c)/d + 27/7168*a^3*cos(7*d*x + 7*c)/d + 3/1024*a^3*cos(5*d*x + 5*c)/d - 31/1536*a^3*cos(3*d*x + 3*c)/d - 27/512*a^3*cos(d*x + c)/d - 1/24576*a^3*s in(12*d*x + 12*c)/d + 3/5120*a^3*sin(10*d*x + 10*c)/d + 15/8192*a^3*sin(8* d*x + 8*c)/d - 3/1024*a^3*sin(6*d*x + 6*c)/d - 111/8192*a^3*sin(4*d*x + 4* c)/d + 3/512*a^3*sin(2*d*x + 2*c)/d
Time = 13.30 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.27 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Too large to display} \]
(41*a^3*x)/1024 - ((1435*a^3*tan(c/2 + (d*x)/2)^3)/1536 - (36401*a^3*tan(c /2 + (d*x)/2)^5)/2560 + (1263*a^3*tan(c/2 + (d*x)/2)^7)/2560 + (184331*a^3 *tan(c/2 + (d*x)/2)^9)/3840 - (35387*a^3*tan(c/2 + (d*x)/2)^11)/256 + (353 87*a^3*tan(c/2 + (d*x)/2)^13)/256 - (184331*a^3*tan(c/2 + (d*x)/2)^15)/384 0 - (1263*a^3*tan(c/2 + (d*x)/2)^17)/2560 + (36401*a^3*tan(c/2 + (d*x)/2)^ 19)/2560 - (1435*a^3*tan(c/2 + (d*x)/2)^21)/1536 - (41*a^3*tan(c/2 + (d*x) /2)^23)/512 + a^3*((41*c)/1024 + (41*d*x)/1024) - a^3*((41*c)/1024 + (41*d *x)/1024 - 92/693) - tan(c/2 + (d*x)/2)^22*(a^3*((123*c)/256 + (123*d*x)/2 56) - 12*a^3*((41*c)/1024 + (41*d*x)/1024)) - tan(c/2 + (d*x)/2)^2*(a^3*(( 123*c)/256 + (123*d*x)/256 - 368/231) - 12*a^3*((41*c)/1024 + (41*d*x)/102 4)) + tan(c/2 + (d*x)/2)^20*(66*a^3*((41*c)/1024 + (41*d*x)/1024) - a^3*(( 1353*c)/512 + (1353*d*x)/512 - 4)) + tan(c/2 + (d*x)/2)^4*(66*a^3*((41*c)/ 1024 + (41*d*x)/1024) - a^3*((1353*c)/512 + (1353*d*x)/512 - 100/21)) + ta n(c/2 + (d*x)/2)^18*(220*a^3*((41*c)/1024 + (41*d*x)/1024) - a^3*((2255*c) /256 + (2255*d*x)/256 - 112/3)) + tan(c/2 + (d*x)/2)^6*(220*a^3*((41*c)/10 24 + (41*d*x)/1024) - a^3*((2255*c)/256 + (2255*d*x)/256 + 512/63)) + tan( c/2 + (d*x)/2)^14*(792*a^3*((41*c)/1024 + (41*d*x)/1024) - a^3*((4059*c)/1 28 + (4059*d*x)/128 - 128)) + tan(c/2 + (d*x)/2)^10*(792*a^3*((41*c)/1024 + (41*d*x)/1024) - a^3*((4059*c)/128 + (4059*d*x)/128 + 160/7)) + tan(c/2 + (d*x)/2)^12*(924*a^3*((41*c)/1024 + (41*d*x)/1024) - a^3*((9471*c)/25...