Integrand size = 27, antiderivative size = 165 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a x}{256}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {2 a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^{11}(c+d x)}{11 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{256 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d} \]
3/256*a*x-1/7*a*cos(d*x+c)^7/d+2/9*a*cos(d*x+c)^9/d-1/11*a*cos(d*x+c)^11/d +3/256*a*cos(d*x+c)*sin(d*x+c)/d+1/128*a*cos(d*x+c)^3*sin(d*x+c)/d+1/160*a *cos(d*x+c)^5*sin(d*x+c)/d-3/80*a*cos(d*x+c)^7*sin(d*x+c)/d-1/10*a*cos(d*x +c)^7*sin(d*x+c)^3/d
Time = 0.37 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.73 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (83160 d x-69300 \cos (c+d x)-23100 \cos (3 (c+d x))+6930 \cos (5 (c+d x))+4950 \cos (7 (c+d x))-770 \cos (9 (c+d x))-630 \cos (11 (c+d x))+13860 \sin (2 (c+d x))-27720 \sin (4 (c+d x))-6930 \sin (6 (c+d x))+3465 \sin (8 (c+d x))+1386 \sin (10 (c+d x)))}{7096320 d} \]
(a*(83160*d*x - 69300*Cos[c + d*x] - 23100*Cos[3*(c + d*x)] + 6930*Cos[5*( c + d*x)] + 4950*Cos[7*(c + d*x)] - 770*Cos[9*(c + d*x)] - 630*Cos[11*(c + d*x)] + 13860*Sin[2*(c + d*x)] - 27720*Sin[4*(c + d*x)] - 6930*Sin[6*(c + d*x)] + 3465*Sin[8*(c + d*x)] + 1386*Sin[10*(c + d*x)]))/(7096320*d)
Time = 0.79 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {3042, 3317, 3042, 3045, 244, 2009, 3048, 3042, 3048, 3042, 3115, 3042, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \sin ^4(c+d x) \cos ^6(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^4 \cos (c+d x)^6 (a \sin (c+d x)+a)dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \cos ^6(c+d x) \sin ^5(c+d x)dx+a \int \cos ^6(c+d x) \sin ^4(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^4dx+a \int \cos (c+d x)^6 \sin (c+d x)^5dx\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^4dx-\frac {a \int \cos ^6(c+d x) \left (1-\cos ^2(c+d x)\right )^2d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^4dx-\frac {a \int \left (\cos ^{10}(c+d x)-2 \cos ^8(c+d x)+\cos ^6(c+d x)\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^4dx-\frac {a \left (\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle a \left (\frac {3}{10} \int \cos ^6(c+d x) \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3}{10} \int \cos (c+d x)^6 \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \int \cos ^6(c+d x)dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)\right )}{d}\) |
-((a*(Cos[c + d*x]^7/7 - (2*Cos[c + d*x]^9)/9 + Cos[c + d*x]^11/11))/d) + a*(-1/10*(Cos[c + d*x]^7*Sin[c + d*x]^3)/d + (3*(-1/8*(Cos[c + d*x]^7*Sin[ c + d*x])/d + ((Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + (5*((Cos[c + d*x]^3*S in[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/6)/ 8))/10)
3.6.72.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Time = 0.73 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.78
| method | result | size |
| parallelrisch | \(\frac {\left (6 d x +\sin \left (2 d x +2 c \right )-2 \sin \left (4 d x +4 c \right )-\frac {\sin \left (6 d x +6 c \right )}{2}+\frac {\sin \left (8 d x +8 c \right )}{4}-\frac {\cos \left (11 d x +11 c \right )}{22}+\frac {\sin \left (10 d x +10 c \right )}{10}-5 \cos \left (d x +c \right )-\frac {5 \cos \left (3 d x +3 c \right )}{3}+\frac {\cos \left (5 d x +5 c \right )}{2}+\frac {5 \cos \left (7 d x +7 c \right )}{14}-\frac {\cos \left (9 d x +9 c \right )}{18}-\frac {4096}{693}\right ) a}{512 d}\) | \(129\) |
| derivativedivides | \(\frac {a \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d}\) | \(134\) |
| default | \(\frac {a \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d}\) | \(134\) |
| risch | \(\frac {3 a x}{256}-\frac {5 a \cos \left (d x +c \right )}{512 d}-\frac {a \cos \left (11 d x +11 c \right )}{11264 d}+\frac {a \sin \left (10 d x +10 c \right )}{5120 d}-\frac {a \cos \left (9 d x +9 c \right )}{9216 d}+\frac {a \sin \left (8 d x +8 c \right )}{2048 d}+\frac {5 a \cos \left (7 d x +7 c \right )}{7168 d}-\frac {a \sin \left (6 d x +6 c \right )}{1024 d}+\frac {a \cos \left (5 d x +5 c \right )}{1024 d}-\frac {a \sin \left (4 d x +4 c \right )}{256 d}-\frac {5 a \cos \left (3 d x +3 c \right )}{1536 d}+\frac {a \sin \left (2 d x +2 c \right )}{512 d}\) | \(168\) |
1/512*(6*d*x+sin(2*d*x+2*c)-2*sin(4*d*x+4*c)-1/2*sin(6*d*x+6*c)+1/4*sin(8* d*x+8*c)-1/22*cos(11*d*x+11*c)+1/10*sin(10*d*x+10*c)-5*cos(d*x+c)-5/3*cos( 3*d*x+3*c)+1/2*cos(5*d*x+5*c)+5/14*cos(7*d*x+7*c)-1/18*cos(9*d*x+9*c)-4096 /693)*a/d
Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.64 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {80640 \, a \cos \left (d x + c\right )^{11} - 197120 \, a \cos \left (d x + c\right )^{9} + 126720 \, a \cos \left (d x + c\right )^{7} - 10395 \, a d x - 693 \, {\left (128 \, a \cos \left (d x + c\right )^{9} - 176 \, a \cos \left (d x + c\right )^{7} + 8 \, a \cos \left (d x + c\right )^{5} + 10 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, d} \]
-1/887040*(80640*a*cos(d*x + c)^11 - 197120*a*cos(d*x + c)^9 + 126720*a*co s(d*x + c)^7 - 10395*a*d*x - 693*(128*a*cos(d*x + c)^9 - 176*a*cos(d*x + c )^7 + 8*a*cos(d*x + c)^5 + 10*a*cos(d*x + c)^3 + 15*a*cos(d*x + c))*sin(d* x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (153) = 306\).
Time = 1.79 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.93 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {3 a x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {7 a \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {4 a \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {3 a \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {8 a \cos ^{11}{\left (c + d x \right )}}{693 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{4}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*a*x*sin(c + d*x)**10/256 + 15*a*x*sin(c + d*x)**8*cos(c + d*x )**2/256 + 15*a*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 15*a*x*sin(c + d*x )**4*cos(c + d*x)**6/128 + 15*a*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 3* a*x*cos(c + d*x)**10/256 + 3*a*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a* sin(c + d*x)**7*cos(c + d*x)**3/(128*d) + a*sin(c + d*x)**5*cos(c + d*x)** 5/(10*d) - a*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 7*a*sin(c + d*x)**3*c os(c + d*x)**7/(128*d) - 4*a*sin(c + d*x)**2*cos(c + d*x)**9/(63*d) - 3*a* sin(c + d*x)*cos(c + d*x)**9/(256*d) - 8*a*cos(c + d*x)**11/(693*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**4*cos(c)**6, True))
Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.52 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {10240 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a - 693 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{7096320 \, d} \]
-1/7096320*(10240*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a - 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)/d
Time = 0.39 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.01 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3}{256} \, a x - \frac {a \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} - \frac {a \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {5 \, a \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {a \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac {5 \, a \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac {5 \, a \cos \left (d x + c\right )}{512 \, d} + \frac {a \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {a \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
3/256*a*x - 1/11264*a*cos(11*d*x + 11*c)/d - 1/9216*a*cos(9*d*x + 9*c)/d + 5/7168*a*cos(7*d*x + 7*c)/d + 1/1024*a*cos(5*d*x + 5*c)/d - 5/1536*a*cos( 3*d*x + 3*c)/d - 5/512*a*cos(d*x + c)/d + 1/5120*a*sin(10*d*x + 10*c)/d + 1/2048*a*sin(8*d*x + 8*c)/d - 1/1024*a*sin(6*d*x + 6*c)/d - 1/256*a*sin(4* d*x + 4*c)/d + 1/512*a*sin(2*d*x + 2*c)/d
Time = 14.02 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.71 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3\,a\,x}{256}+\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{128}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{4}-\frac {3323\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{640}+\left (\frac {a\,\left (1715175\,c+1715175\,d\,x-9461760\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+\frac {54\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\left (\frac {a\,\left (3430350\,c+3430350\,d\,x+23654400\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-\frac {841\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\left (\frac {a\,\left (4802490\,c+4802490\,d\,x-52039680\right )}{887040}-\frac {693\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\left (\frac {a\,\left (4802490\,c+4802490\,d\,x+42577920\right )}{887040}-\frac {693\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {841\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\left (\frac {a\,\left (3430350\,c+3430350\,d\,x-30412800\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {54\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}+\left (\frac {a\,\left (1715175\,c+1715175\,d\,x+6082560\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {3323\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640}+\left (\frac {a\,\left (571725\,c+571725\,d\,x-1126400\right )}{887040}-\frac {165\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\left (\frac {a\,\left (114345\,c+114345\,d\,x-225280\right )}{887040}-\frac {33\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {a\,\left (10395\,c+10395\,d\,x-20480\right )}{887040}-\frac {3\,a\,\left (c+d\,x\right )}{256}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \]
(3*a*x)/256 + ((a*(10395*c + 10395*d*x - 20480))/887040 - (3*a*tan(c/2 + ( d*x)/2))/128 - (3*a*(c + d*x))/256 + tan(c/2 + (d*x)/2)^2*((a*(114345*c + 114345*d*x - 225280))/887040 - (33*a*(c + d*x))/256) + tan(c/2 + (d*x)/2)^ 4*((a*(571725*c + 571725*d*x - 1126400))/887040 - (165*a*(c + d*x))/256) + tan(c/2 + (d*x)/2)^6*((a*(1715175*c + 1715175*d*x + 6082560))/887040 - (4 95*a*(c + d*x))/256) + tan(c/2 + (d*x)/2)^16*((a*(1715175*c + 1715175*d*x - 9461760))/887040 - (495*a*(c + d*x))/256) + tan(c/2 + (d*x)/2)^14*((a*(3 430350*c + 3430350*d*x + 23654400))/887040 - (495*a*(c + d*x))/128) + tan( c/2 + (d*x)/2)^8*((a*(3430350*c + 3430350*d*x - 30412800))/887040 - (495*a *(c + d*x))/128) + tan(c/2 + (d*x)/2)^10*((a*(4802490*c + 4802490*d*x + 42 577920))/887040 - (693*a*(c + d*x))/128) + tan(c/2 + (d*x)/2)^12*((a*(4802 490*c + 4802490*d*x - 52039680))/887040 - (693*a*(c + d*x))/128) - (a*tan( c/2 + (d*x)/2)^3)/4 + (3323*a*tan(c/2 + (d*x)/2)^5)/640 - (54*a*tan(c/2 + (d*x)/2)^7)/5 + (841*a*tan(c/2 + (d*x)/2)^9)/64 - (841*a*tan(c/2 + (d*x)/2 )^13)/64 + (54*a*tan(c/2 + (d*x)/2)^15)/5 - (3323*a*tan(c/2 + (d*x)/2)^17) /640 + (a*tan(c/2 + (d*x)/2)^19)/4 + (3*a*tan(c/2 + (d*x)/2)^21)/128)/(d*( tan(c/2 + (d*x)/2)^2 + 1)^11)