Integrand size = 27, antiderivative size = 181 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {33 a^3 x}{256}-\frac {33 a^3 \cos ^7(c+d x)}{560 d}+\frac {33 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {11 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {11 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d} \] Output:
33/256*a^3*x-33/560*a^3*cos(d*x+c)^7/d+33/256*a^3*cos(d*x+c)*sin(d*x+c)/d+ 11/128*a^3*cos(d*x+c)^3*sin(d*x+c)/d+11/160*a^3*cos(d*x+c)^5*sin(d*x+c)/d- 1/30*a*cos(d*x+c)^7*(a+a*sin(d*x+c))^2/d-1/10*cos(d*x+c)^7*(a+a*sin(d*x+c) )^3/d-11/240*cos(d*x+c)^7*(a^3+a^3*sin(d*x+c))/d
Time = 0.75 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.64 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (31500 c+27720 d x-31920 \cos (c+d x)-16800 \cos (3 (c+d x))-3360 \cos (5 (c+d x))+600 \cos (7 (c+d x))+280 \cos (9 (c+d x))+10500 \sin (2 (c+d x))-5880 \sin (4 (c+d x))-3570 \sin (6 (c+d x))-525 \sin (8 (c+d x))+42 \sin (10 (c+d x)))}{215040 d} \] Input:
Integrate[Cos[c + d*x]^6*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(31500*c + 27720*d*x - 31920*Cos[c + d*x] - 16800*Cos[3*(c + d*x)] - 3360*Cos[5*(c + d*x)] + 600*Cos[7*(c + d*x)] + 280*Cos[9*(c + d*x)] + 1050 0*Sin[2*(c + d*x)] - 5880*Sin[4*(c + d*x)] - 3570*Sin[6*(c + d*x)] - 525*S in[8*(c + d*x)] + 42*Sin[10*(c + d*x)]))/(215040*d)
Time = 0.91 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3339, 3042, 3157, 3042, 3157, 3042, 3148, 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.
| \(\displaystyle \int \sin (c+d x) \cos ^6(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x) \cos (c+d x)^6 (a \sin (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3339 |
| \(\displaystyle \frac {3}{10} \int \cos ^6(c+d x) (\sin (c+d x) a+a)^3dx-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \int \cos (c+d x)^6 (\sin (c+d x) a+a)^3dx-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \int \cos ^6(c+d x) (\sin (c+d x) a+a)^2dx-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \int \cos (c+d x)^6 (\sin (c+d x) a+a)^2dx-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \left (\frac {9}{8} a \int \cos ^6(c+d x) (\sin (c+d x) a+a)dx-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \left (\frac {9}{8} a \int \cos (c+d x)^6 (\sin (c+d x) a+a)dx-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \int \cos ^6(c+d x)dx-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {3}{10} \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}\) |
Input:
Int[Cos[c + d*x]^6*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]
Output:
-1/10*(Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3)/d + (3*(-1/9*(a*Cos[c + d*x] ^7*(a + a*Sin[c + d*x])^2)/d + (11*a*(-1/8*(Cos[c + d*x]^7*(a^2 + a^2*Sin[ c + d*x]))/d + (9*a*(-1/7*(a*Cos[c + d*x]^7)/d + a*((Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + (5*((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/6)))/8))/9))/10
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_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[ a^2 - b^2, 0] && NeQ[m + p + 1, 0]
Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.09
\[\frac {a^{3} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{7}}{10}-\frac {3 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{80}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a^{3} \cos \left (d x +c \right )^{7}}{7}}{d}\]
Input:
int(cos(d*x+c)^6*sin(d*x+c)*(a+a*sin(d*x+c))^3,x)
Output:
1/d*(a^3*(-1/10*sin(d*x+c)^3*cos(d*x+c)^7-3/80*cos(d*x+c)^7*sin(d*x+c)+1/1 60*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+3/256*d*x+3/ 256*c)+3*a^3*(-1/9*cos(d*x+c)^7*sin(d*x+c)^2-2/63*cos(d*x+c)^7)+3*a^3*(-1/ 8*cos(d*x+c)^7*sin(d*x+c)+1/48*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x +c))*sin(d*x+c)+5/128*d*x+5/128*c)-1/7*a^3*cos(d*x+c)^7)
Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {8960 \, a^{3} \cos \left (d x + c\right )^{9} - 15360 \, a^{3} \cos \left (d x + c\right )^{7} + 3465 \, a^{3} d x + 21 \, {\left (128 \, a^{3} \cos \left (d x + c\right )^{9} - 656 \, a^{3} \cos \left (d x + c\right )^{7} + 88 \, a^{3} \cos \left (d x + c\right )^{5} + 110 \, a^{3} \cos \left (d x + c\right )^{3} + 165 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{26880 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
1/26880*(8960*a^3*cos(d*x + c)^9 - 15360*a^3*cos(d*x + c)^7 + 3465*a^3*d*x + 21*(128*a^3*cos(d*x + c)^9 - 656*a^3*cos(d*x + c)^7 + 88*a^3*cos(d*x + c)^5 + 110*a^3*cos(d*x + c)^3 + 165*a^3*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (170) = 340\).
Time = 1.40 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.99 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {3 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {45 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {15 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {55 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {7 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {73 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {3 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {15 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {2 a^{3} \cos ^{9}{\left (c + d x \right )}}{21 d} - \frac {a^{3} \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin {\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)*(a+a*sin(d*x+c))**3,x)
Output:
Piecewise((3*a**3*x*sin(c + d*x)**10/256 + 15*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 15*a**3*x*sin(c + d*x)**8/128 + 15*a**3*x*sin(c + d*x)**6 *cos(c + d*x)**4/128 + 15*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15*a **3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 45*a**3*x*sin(c + d*x)**4*cos( c + d*x)**4/64 + 15*a**3*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 15*a**3*x *sin(c + d*x)**2*cos(c + d*x)**6/32 + 3*a**3*x*cos(c + d*x)**10/256 + 15*a **3*x*cos(c + d*x)**8/128 + 3*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a**3*sin(c + d*x)**7*cos(c + d*x)**3/(128*d) + 15*a**3*sin(c + d*x)**7*c os(c + d*x)/(128*d) + a**3*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 55*a** 3*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) - 7*a**3*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) + 73*a**3*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 3*a** 3*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 3*a**3*sin(c + d*x)*cos(c + d*x) **9/(256*d) - 15*a**3*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 2*a**3*cos(c + d*x)**9/(21*d) - a**3*cos(c + d*x)**7/(7*d), Ne(d, 0)), (x*(a*sin(c) + a )**3*sin(c)*cos(c)**6, True))
Time = 0.04 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {30720 \, a^{3} \cos \left (d x + c\right )^{7} - 10240 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 21 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 210 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{215040 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
-1/215040*(30720*a^3*cos(d*x + c)^7 - 10240*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^3 - 21*(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 - 210*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 1 20*c - 3*sin(8*d*x + 8*c) - 24*sin(4*d*x + 4*c))*a^3)/d
Time = 0.21 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.96 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {33}{256} \, a^{3} x + \frac {a^{3} \cos \left (9 \, d x + 9 \, c\right )}{768 \, d} + \frac {5 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a^{3} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {5 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {19 \, a^{3} \cos \left (d x + c\right )}{128 \, d} + \frac {a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac {5 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {17 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {7 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {25 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
33/256*a^3*x + 1/768*a^3*cos(9*d*x + 9*c)/d + 5/1792*a^3*cos(7*d*x + 7*c)/ d - 1/64*a^3*cos(5*d*x + 5*c)/d - 5/64*a^3*cos(3*d*x + 3*c)/d - 19/128*a^3 *cos(d*x + c)/d + 1/5120*a^3*sin(10*d*x + 10*c)/d - 5/2048*a^3*sin(8*d*x + 8*c)/d - 17/1024*a^3*sin(6*d*x + 6*c)/d - 7/256*a^3*sin(4*d*x + 4*c)/d + 25/512*a^3*sin(2*d*x + 2*c)/d
Time = 35.58 (sec) , antiderivative size = 572, normalized size of antiderivative = 3.16 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^6*sin(c + d*x)*(a + a*sin(c + d*x))^3,x)
Output:
(33*a^3*x)/256 - ((333*a^3*tan(c/2 + (d*x)/2)^7)/32 - (577*a^3*tan(c/2 + ( d*x)/2)^5)/160 - (705*a^3*tan(c/2 + (d*x)/2)^3)/128 - (2749*a^3*tan(c/2 + (d*x)/2)^9)/64 + (2749*a^3*tan(c/2 + (d*x)/2)^11)/64 - (333*a^3*tan(c/2 + (d*x)/2)^13)/32 + (577*a^3*tan(c/2 + (d*x)/2)^15)/160 + (705*a^3*tan(c/2 + (d*x)/2)^17)/128 - (33*a^3*tan(c/2 + (d*x)/2)^19)/128 + a^3*((33*c)/256 + (33*d*x)/256) - a^3*((33*c)/256 + (33*d*x)/256 - 10/21) + tan(c/2 + (d*x) /2)^18*(10*a^3*((33*c)/256 + (33*d*x)/256) - a^3*((165*c)/128 + (165*d*x)/ 128 - 2)) + tan(c/2 + (d*x)/2)^2*(10*a^3*((33*c)/256 + (33*d*x)/256) - a^3 *((165*c)/128 + (165*d*x)/128 - 58/21)) + tan(c/2 + (d*x)/2)^14*(120*a^3*( (33*c)/256 + (33*d*x)/256) - a^3*((495*c)/32 + (495*d*x)/32 - 8)) + tan(c/ 2 + (d*x)/2)^6*(120*a^3*((33*c)/256 + (33*d*x)/256) - a^3*((495*c)/32 + (4 95*d*x)/32 - 344/7)) + tan(c/2 + (d*x)/2)^16*(45*a^3*((33*c)/256 + (33*d*x )/256) - a^3*((1485*c)/256 + (1485*d*x)/256 - 18)) + tan(c/2 + (d*x)/2)^4* (45*a^3*((33*c)/256 + (33*d*x)/256) - a^3*((1485*c)/256 + (1485*d*x)/256 - 24/7)) + tan(c/2 + (d*x)/2)^10*(252*a^3*((33*c)/256 + (33*d*x)/256) - a^3 *((2079*c)/64 + (2079*d*x)/64 - 60)) + tan(c/2 + (d*x)/2)^8*(210*a^3*((33* c)/256 + (33*d*x)/256) - a^3*((3465*c)/128 + (3465*d*x)/128 - 28)) + tan(c /2 + (d*x)/2)^12*(210*a^3*((33*c)/256 + (33*d*x)/256) - a^3*((3465*c)/128 + (3465*d*x)/128 - 72)) + (33*a^3*tan(c/2 + (d*x)/2))/128)/(d*(tan(c/2 + ( d*x)/2)^2 + 1)^10)
Time = 0.16 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.91 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (2688 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}+8960 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}+3024 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-20480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-23352 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+7680 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+24570 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+10240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-3465 \cos \left (d x +c \right ) \sin \left (d x +c \right )-6400 \cos \left (d x +c \right )+3465 d x +6400\right )}{26880 d} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(2688*cos(c + d*x)*sin(c + d*x)**9 + 8960*cos(c + d*x)*sin(c + d*x)* *8 + 3024*cos(c + d*x)*sin(c + d*x)**7 - 20480*cos(c + d*x)*sin(c + d*x)** 6 - 23352*cos(c + d*x)*sin(c + d*x)**5 + 7680*cos(c + d*x)*sin(c + d*x)**4 + 24570*cos(c + d*x)*sin(c + d*x)**3 + 10240*cos(c + d*x)*sin(c + d*x)**2 - 3465*cos(c + d*x)*sin(c + d*x) - 6400*cos(c + d*x) + 3465*d*x + 6400))/ (26880*d)