Integrand size = 29, antiderivative size = 203 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {15 a^3 x}{256}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {5 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d} \]
15/256*a^3*x-4/5*a^3*cos(d*x+c)^5/d+9/7*a^3*cos(d*x+c)^7/d-2/3*a^3*cos(d*x +c)^9/d+1/11*a^3*cos(d*x+c)^11/d+15/256*a^3*cos(d*x+c)*sin(d*x+c)/d+5/128* a^3*cos(d*x+c)^3*sin(d*x+c)/d-5/32*a^3*cos(d*x+c)^5*sin(d*x+c)/d-5/16*a^3* cos(d*x+c)^5*sin(d*x+c)^3/d-3/10*a^3*cos(d*x+c)^5*sin(d*x+c)^5/d
Time = 12.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.62 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (138600 c+138600 d x-198660 \cos (c+d x)-41580 \cos (3 (c+d x))+27258 \cos (5 (c+d x))+3630 \cos (7 (c+d x))-3850 \cos (9 (c+d x))+210 \cos (11 (c+d x))-13860 \sin (2 (c+d x))-46200 \sin (4 (c+d x))+6930 \sin (6 (c+d x))+5775 \sin (8 (c+d x))-1386 \sin (10 (c+d x)))}{2365440 d} \]
(a^3*(138600*c + 138600*d*x - 198660*Cos[c + d*x] - 41580*Cos[3*(c + d*x)] + 27258*Cos[5*(c + d*x)] + 3630*Cos[7*(c + d*x)] - 3850*Cos[9*(c + d*x)] + 210*Cos[11*(c + d*x)] - 13860*Sin[2*(c + d*x)] - 46200*Sin[4*(c + d*x)] + 6930*Sin[6*(c + d*x)] + 5775*Sin[8*(c + d*x)] - 1386*Sin[10*(c + d*x)])) /(2365440*d)
Time = 0.60 (sec) , antiderivative size = 203, 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.
| \(\displaystyle \int \sin ^4(c+d x) \cos ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^4 \cos (c+d x)^4 (a \sin (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \sin ^7(c+d x) \cos ^4(c+d x)+3 a^3 \sin ^6(c+d x) \cos ^4(c+d x)+3 a^3 \sin ^5(c+d x) \cos ^4(c+d x)+a^3 \sin ^4(c+d x) \cos ^4(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \cos ^{11}(c+d x)}{11 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac {5 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac {5 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {5 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {15 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {15 a^3 x}{256}\) |
(15*a^3*x)/256 - (4*a^3*Cos[c + d*x]^5)/(5*d) + (9*a^3*Cos[c + d*x]^7)/(7* d) - (2*a^3*Cos[c + d*x]^9)/(3*d) + (a^3*Cos[c + d*x]^11)/(11*d) + (15*a^3 *Cos[c + d*x]*Sin[c + d*x])/(256*d) + (5*a^3*Cos[c + d*x]^3*Sin[c + d*x])/ (128*d) - (5*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(32*d) - (5*a^3*Cos[c + d*x] ^5*Sin[c + d*x]^3)/(16*d) - (3*a^3*Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*d)
3.4.92.3.1 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 = 0.78 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.65
| method | result | size |
| parallelrisch | \(-\frac {3 \left (-10 d x +\sin \left (2 d x +2 c \right )+\frac {10 \sin \left (4 d x +4 c \right )}{3}-\frac {\sin \left (6 d x +6 c \right )}{2}-\frac {5 \sin \left (8 d x +8 c \right )}{12}-\frac {\cos \left (11 d x +11 c \right )}{66}+\frac {\sin \left (10 d x +10 c \right )}{10}+\frac {43 \cos \left (d x +c \right )}{3}+3 \cos \left (3 d x +3 c \right )-\frac {59 \cos \left (5 d x +5 c \right )}{30}-\frac {11 \cos \left (7 d x +7 c \right )}{42}+\frac {5 \cos \left (9 d x +9 c \right )}{18}+\frac {53248}{3465}\right ) a^{3}}{512 d}\) | \(131\) |
| risch | \(\frac {a^{3} \cos \left (11 d x +11 c \right )}{11264 d}+\frac {15 a^{3} x}{256}-\frac {43 a^{3} \cos \left (d x +c \right )}{512 d}-\frac {3 a^{3} \sin \left (10 d x +10 c \right )}{5120 d}-\frac {5 a^{3} \cos \left (9 d x +9 c \right )}{3072 d}+\frac {5 a^{3} \sin \left (8 d x +8 c \right )}{2048 d}+\frac {11 a^{3} \cos \left (7 d x +7 c \right )}{7168 d}+\frac {3 a^{3} \sin \left (6 d x +6 c \right )}{1024 d}+\frac {59 a^{3} \cos \left (5 d x +5 c \right )}{5120 d}-\frac {5 a^{3} \sin \left (4 d x +4 c \right )}{256 d}-\frac {9 a^{3} \cos \left (3 d x +3 c \right )}{512 d}-\frac {3 a^{3} \sin \left (2 d x +2 c \right )}{512 d}\) | \(192\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{11}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{33}-\frac {8 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{231}-\frac {16 \left (\cos ^{5}\left (d x +c \right )\right )}{1155}\right )+3 a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\) | \(288\) |
| default | \(\frac {a^{3} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{11}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{33}-\frac {8 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{231}-\frac {16 \left (\cos ^{5}\left (d x +c \right )\right )}{1155}\right )+3 a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\) | \(288\) |
-3/512*(-10*d*x+sin(2*d*x+2*c)+10/3*sin(4*d*x+4*c)-1/2*sin(6*d*x+6*c)-5/12 *sin(8*d*x+8*c)-1/66*cos(11*d*x+11*c)+1/10*sin(10*d*x+10*c)+43/3*cos(d*x+c )+3*cos(3*d*x+3*c)-59/30*cos(5*d*x+5*c)-11/42*cos(7*d*x+7*c)+5/18*cos(9*d* x+9*c)+53248/3465)*a^3/d
Time = 0.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.67 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {26880 \, a^{3} \cos \left (d x + c\right )^{11} - 197120 \, a^{3} \cos \left (d x + c\right )^{9} + 380160 \, a^{3} \cos \left (d x + c\right )^{7} - 236544 \, a^{3} \cos \left (d x + c\right )^{5} + 17325 \, a^{3} d x - 231 \, {\left (384 \, a^{3} \cos \left (d x + c\right )^{9} - 1168 \, a^{3} \cos \left (d x + c\right )^{7} + 984 \, a^{3} \cos \left (d x + c\right )^{5} - 50 \, a^{3} \cos \left (d x + c\right )^{3} - 75 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{295680 \, d} \]
1/295680*(26880*a^3*cos(d*x + c)^11 - 197120*a^3*cos(d*x + c)^9 + 380160*a ^3*cos(d*x + c)^7 - 236544*a^3*cos(d*x + c)^5 + 17325*a^3*d*x - 231*(384*a ^3*cos(d*x + c)^9 - 1168*a^3*cos(d*x + c)^7 + 984*a^3*cos(d*x + c)^5 - 50* a^3*cos(d*x + c)^3 - 75*a^3*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 648 vs. \(2 (194) = 388\).
Time = 1.97 (sec) , antiderivative size = 648, normalized size of antiderivative = 3.19 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {9 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {45 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {3 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {45 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {3 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {45 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {9 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {45 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {9 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {9 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {21 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {3 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} - \frac {a^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {11 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {6 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {21 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {11 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {8 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{105 d} - \frac {12 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {9 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {16 a^{3} \cos ^{11}{\left (c + d x \right )}}{1155 d} - \frac {8 a^{3} \cos ^{9}{\left (c + d x \right )}}{105 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{4}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((9*a**3*x*sin(c + d*x)**10/256 + 45*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 3*a**3*x*sin(c + d*x)**8/128 + 45*a**3*x*sin(c + d*x)**6* cos(c + d*x)**4/128 + 3*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 45*a** 3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 9*a**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 45*a**3*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 3*a**3*x*sin (c + d*x)**2*cos(c + d*x)**6/32 + 9*a**3*x*cos(c + d*x)**10/256 + 3*a**3*x *cos(c + d*x)**8/128 + 9*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 21*a* *3*sin(c + d*x)**7*cos(c + d*x)**3/(128*d) + 3*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) - a**3*sin(c + d*x)**6*cos(c + d*x)**5/(5*d) - 3*a**3*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 11*a**3*sin(c + d*x)**5*cos(c + d*x)** 3/(128*d) - 6*a**3*sin(c + d*x)**4*cos(c + d*x)**7/(35*d) - 3*a**3*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 21*a**3*sin(c + d*x)**3*cos(c + d*x)**7/( 128*d) - 11*a**3*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 8*a**3*sin(c + d*x)**2*cos(c + d*x)**9/(105*d) - 12*a**3*sin(c + d*x)**2*cos(c + d*x)**7/ (35*d) - 9*a**3*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 3*a**3*sin(c + d*x) *cos(c + d*x)**7/(128*d) - 16*a**3*cos(c + d*x)**11/(1155*d) - 8*a**3*cos( c + d*x)**9/(105*d), Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)**4*cos(c)**4, True))
Time = 0.24 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.83 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {2048 \, {\left (105 \, \cos \left (d x + c\right )^{11} - 385 \, \cos \left (d x + c\right )^{9} + 495 \, \cos \left (d x + c\right )^{7} - 231 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 22528 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 693 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 2310 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{2365440 \, d} \]
1/2365440*(2048*(105*cos(d*x + c)^11 - 385*cos(d*x + c)^9 + 495*cos(d*x + c)^7 - 231*cos(d*x + c)^5)*a^3 - 22528*(35*cos(d*x + c)^9 - 90*cos(d*x + c )^7 + 63*cos(d*x + c)^5)*a^3 - 693*(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 + 2310*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*a^3)/d
Time = 0.70 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.94 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {15}{256} \, a^{3} x + \frac {a^{3} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} - \frac {5 \, a^{3} \cos \left (9 \, d x + 9 \, c\right )}{3072 \, d} + \frac {11 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {59 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{5120 \, d} - \frac {9 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{512 \, d} - \frac {43 \, a^{3} \cos \left (d x + c\right )}{512 \, d} - \frac {3 \, a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {5 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {3 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {5 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
15/256*a^3*x + 1/11264*a^3*cos(11*d*x + 11*c)/d - 5/3072*a^3*cos(9*d*x + 9 *c)/d + 11/7168*a^3*cos(7*d*x + 7*c)/d + 59/5120*a^3*cos(5*d*x + 5*c)/d - 9/512*a^3*cos(3*d*x + 3*c)/d - 43/512*a^3*cos(d*x + c)/d - 3/5120*a^3*sin( 10*d*x + 10*c)/d + 5/2048*a^3*sin(8*d*x + 8*c)/d + 3/1024*a^3*sin(6*d*x + 6*c)/d - 5/256*a^3*sin(4*d*x + 4*c)/d - 3/512*a^3*sin(2*d*x + 2*c)/d
Time = 13.18 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.49 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {15\,a^3\,x}{256}-\frac {\frac {15\,a^3\,\left (c+d\,x\right )}{256}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-\frac {231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640}-\frac {242\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}+\frac {3987\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {3987\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {242\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\frac {231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{640}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{4}-\frac {15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{128}-\frac {a^3\,\left (17325\,c+17325\,d\,x-53248\right )}{295680}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {165\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (190575\,c+190575\,d\,x-585728\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {825\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (952875\,c+952875\,d\,x-2928640\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (2858625\,c+2858625\,d\,x+675840\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (5717250\,c+5717250\,d\,x-3379200\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (2858625\,c+2858625\,d\,x-9461760\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (5717250\,c+5717250\,d\,x-14192640\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {3465\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (8004150\,c+8004150\,d\,x+16084992\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {3465\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (8004150\,c+8004150\,d\,x-40685568\right )}{295680}\right )+\frac {15\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \]
(15*a^3*x)/256 - ((15*a^3*(c + d*x))/256 + (5*a^3*tan(c/2 + (d*x)/2)^3)/4 - (231*a^3*tan(c/2 + (d*x)/2)^5)/640 - (242*a^3*tan(c/2 + (d*x)/2)^7)/5 + (3987*a^3*tan(c/2 + (d*x)/2)^9)/64 - (3987*a^3*tan(c/2 + (d*x)/2)^13)/64 + (242*a^3*tan(c/2 + (d*x)/2)^15)/5 + (231*a^3*tan(c/2 + (d*x)/2)^17)/640 - (5*a^3*tan(c/2 + (d*x)/2)^19)/4 - (15*a^3*tan(c/2 + (d*x)/2)^21)/128 - (a ^3*(17325*c + 17325*d*x - 53248))/295680 + tan(c/2 + (d*x)/2)^2*((165*a^3* (c + d*x))/256 - (a^3*(190575*c + 190575*d*x - 585728))/295680) + tan(c/2 + (d*x)/2)^4*((825*a^3*(c + d*x))/256 - (a^3*(952875*c + 952875*d*x - 2928 640))/295680) + tan(c/2 + (d*x)/2)^6*((2475*a^3*(c + d*x))/256 - (a^3*(285 8625*c + 2858625*d*x + 675840))/295680) + tan(c/2 + (d*x)/2)^8*((2475*a^3* (c + d*x))/128 - (a^3*(5717250*c + 5717250*d*x - 3379200))/295680) + tan(c /2 + (d*x)/2)^16*((2475*a^3*(c + d*x))/256 - (a^3*(2858625*c + 2858625*d*x - 9461760))/295680) + tan(c/2 + (d*x)/2)^14*((2475*a^3*(c + d*x))/128 - ( a^3*(5717250*c + 5717250*d*x - 14192640))/295680) + tan(c/2 + (d*x)/2)^12* ((3465*a^3*(c + d*x))/128 - (a^3*(8004150*c + 8004150*d*x + 16084992))/295 680) + tan(c/2 + (d*x)/2)^10*((3465*a^3*(c + d*x))/128 - (a^3*(8004150*c + 8004150*d*x - 40685568))/295680) + (15*a^3*tan(c/2 + (d*x)/2))/128)/(d*(t an(c/2 + (d*x)/2)^2 + 1)^11)