Integrand size = 27, antiderivative size = 153 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 a^2 x}{64}-\frac {a^2 \cos ^7(c+d x)}{28 d}+\frac {5 a^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {5 a^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d} \] Output:
5/64*a^2*x-1/28*a^2*cos(d*x+c)^7/d+5/64*a^2*cos(d*x+c)*sin(d*x+c)/d+5/96*a ^2*cos(d*x+c)^3*sin(d*x+c)/d+1/24*a^2*cos(d*x+c)^5*sin(d*x+c)/d-1/9*cos(d* x+c)^7*(a+a*sin(d*x+c))^2/d-1/36*cos(d*x+c)^7*(a^2+a^2*sin(d*x+c))/d
Time = 0.48 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.69 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (2520 c+2520 d x-3276 \cos (c+d x)-1848 \cos (3 (c+d x))-504 \cos (5 (c+d x))-18 \cos (7 (c+d x))+14 \cos (9 (c+d x))+1008 \sin (2 (c+d x))-504 \sin (4 (c+d x))-336 \sin (6 (c+d x))-63 \sin (8 (c+d x)))}{32256 d} \] Input:
Integrate[Cos[c + d*x]^6*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]
Output:
(a^2*(2520*c + 2520*d*x - 3276*Cos[c + d*x] - 1848*Cos[3*(c + d*x)] - 504* Cos[5*(c + d*x)] - 18*Cos[7*(c + d*x)] + 14*Cos[9*(c + d*x)] + 1008*Sin[2* (c + d*x)] - 504*Sin[4*(c + d*x)] - 336*Sin[6*(c + d*x)] - 63*Sin[8*(c + d *x)]))/(32256*d)
Time = 0.75 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3339, 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.
| \(\displaystyle \int \sin (c+d x) \cos ^6(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x) \cos (c+d x)^6 (a \sin (c+d x)+a)^2dx\) |
| \(\Big \downarrow \) 3339 |
| \(\displaystyle \frac {2}{9} \int \cos ^6(c+d x) (\sin (c+d x) a+a)^2dx-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \int \cos (c+d x)^6 (\sin (c+d x) a+a)^2dx-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {2}{9} \left (\frac {9}{8} a \int \cos ^6(c+d x) (\sin (c+d x) a+a)dx-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \left (\frac {9}{8} a \int \cos (c+d x)^6 (\sin (c+d x) a+a)dx-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {2}{9} \left (\frac {9}{8} a \left (a \int \cos ^6(c+d x)dx-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \left (\frac {9}{8} a \left (a \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {2}{9} \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {2}{9} \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {2}{9} \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {2}{9} \left (\frac {9}{8} a \left (a \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
Input:
Int[Cos[c + d*x]^6*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]
Output:
-1/9*(Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/d + (2*(-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
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 = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.76
\[\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a^{2} \cos \left (d x +c \right )^{7}}{7}}{d}\]
Input:
int(cos(d*x+c)^6*sin(d*x+c)*(a+a*sin(d*x+c))^2,x)
Output:
1/d*(a^2*(-1/9*cos(d*x+c)^7*sin(d*x+c)^2-2/63*cos(d*x+c)^7)+2*a^2*(-1/8*co s(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^2*cos(d*x+c)^7)
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.64 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {448 \, a^{2} \cos \left (d x + c\right )^{9} - 1152 \, a^{2} \cos \left (d x + c\right )^{7} + 315 \, a^{2} d x - 21 \, {\left (48 \, a^{2} \cos \left (d x + c\right )^{7} - 8 \, a^{2} \cos \left (d x + c\right )^{5} - 10 \, a^{2} \cos \left (d x + c\right )^{3} - 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4032 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)*(a+a*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
1/4032*(448*a^2*cos(d*x + c)^9 - 1152*a^2*cos(d*x + c)^7 + 315*a^2*d*x - 2 1*(48*a^2*cos(d*x + c)^7 - 8*a^2*cos(d*x + c)^5 - 10*a^2*cos(d*x + c)^3 - 15*a^2*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (139) = 278\).
Time = 0.96 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.84 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\begin {cases} \frac {5 a^{2} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {5 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {5 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a^{2} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {5 a^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} + \frac {55 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{192 d} + \frac {73 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{192 d} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {5 a^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {2 a^{2} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {a^{2} \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \sin {\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)*(a+a*sin(d*x+c))**2,x)
Output:
Piecewise((5*a**2*x*sin(c + d*x)**8/64 + 5*a**2*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + 15*a**2*x*sin(c + d*x)**4*cos(c + d*x)**4/32 + 5*a**2*x*sin(c + d*x)**2*cos(c + d*x)**6/16 + 5*a**2*x*cos(c + d*x)**8/64 + 5*a**2*sin(c + d*x)**7*cos(c + d*x)/(64*d) + 55*a**2*sin(c + d*x)**5*cos(c + d*x)**3/( 192*d) + 73*a**2*sin(c + d*x)**3*cos(c + d*x)**5/(192*d) - a**2*sin(c + d* x)**2*cos(c + d*x)**7/(7*d) - 5*a**2*sin(c + d*x)*cos(c + d*x)**7/(64*d) - 2*a**2*cos(c + d*x)**9/(63*d) - a**2*cos(c + d*x)**7/(7*d), Ne(d, 0)), (x *(a*sin(c) + a)**2*sin(c)*cos(c)**6, True))
Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {4608 \, a^{2} \cos \left (d x + c\right )^{7} - 512 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{2} - 21 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{32256 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)*(a+a*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
-1/32256*(4608*a^2*cos(d*x + c)^7 - 512*(7*cos(d*x + c)^9 - 9*cos(d*x + c) ^7)*a^2 - 21*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 120*c - 3*sin(8*d*x + 8*c) - 24*sin(4*d*x + 4*c))*a^2)/d
Time = 0.17 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5}{64} \, a^{2} x + \frac {a^{2} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {a^{2} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {11 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {13 \, a^{2} \cos \left (d x + c\right )}{128 \, d} - \frac {a^{2} \sin \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {a^{2} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)*(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
5/64*a^2*x + 1/2304*a^2*cos(9*d*x + 9*c)/d - 1/1792*a^2*cos(7*d*x + 7*c)/d - 1/64*a^2*cos(5*d*x + 5*c)/d - 11/192*a^2*cos(3*d*x + 3*c)/d - 13/128*a^ 2*cos(d*x + c)/d - 1/512*a^2*sin(8*d*x + 8*c)/d - 1/96*a^2*sin(6*d*x + 6*c )/d - 1/64*a^2*sin(4*d*x + 4*c)/d + 1/32*a^2*sin(2*d*x + 2*c)/d
Time = 34.74 (sec) , antiderivative size = 501, normalized size of antiderivative = 3.27 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^6*sin(c + d*x)*(a + a*sin(c + d*x))^2,x)
Output:
(5*a^2*x)/64 - ((83*a^2*tan(c/2 + (d*x)/2)^5)/16 - (191*a^2*tan(c/2 + (d*x )/2)^3)/48 - (145*a^2*tan(c/2 + (d*x)/2)^7)/16 + (145*a^2*tan(c/2 + (d*x)/ 2)^11)/16 - (83*a^2*tan(c/2 + (d*x)/2)^13)/16 + (191*a^2*tan(c/2 + (d*x)/2 )^15)/48 - (5*a^2*tan(c/2 + (d*x)/2)^17)/32 + (a^2*(315*c + 315*d*x))/4032 - (a^2*(315*c + 315*d*x - 1408))/4032 + tan(c/2 + (d*x)/2)^2*((a^2*(315*c + 315*d*x))/448 - (a^2*(2835*c + 2835*d*x - 4608))/4032) + tan(c/2 + (d*x )/2)^16*((a^2*(315*c + 315*d*x))/448 - (a^2*(2835*c + 2835*d*x - 8064))/40 32) + tan(c/2 + (d*x)/2)^4*((a^2*(315*c + 315*d*x))/112 - (a^2*(11340*c + 11340*d*x - 18432))/4032) + tan(c/2 + (d*x)/2)^14*((a^2*(315*c + 315*d*x)) /112 - (a^2*(11340*c + 11340*d*x - 32256))/4032) + tan(c/2 + (d*x)/2)^12*( (a^2*(315*c + 315*d*x))/48 - (a^2*(26460*c + 26460*d*x - 21504))/4032) + t an(c/2 + (d*x)/2)^8*((a^2*(315*c + 315*d*x))/32 - (a^2*(39690*c + 39690*d* x - 16128))/4032) + tan(c/2 + (d*x)/2)^6*((a^2*(315*c + 315*d*x))/48 - (a^ 2*(26460*c + 26460*d*x - 96768))/4032) + tan(c/2 + (d*x)/2)^10*((a^2*(315* c + 315*d*x))/32 - (a^2*(39690*c + 39690*d*x - 161280))/4032) + (5*a^2*tan (c/2 + (d*x)/2))/32)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^9)
Time = 0.16 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (448 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}+1008 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-2856 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-768 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+2478 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+1664 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-315 \cos \left (d x +c \right ) \sin \left (d x +c \right )-704 \cos \left (d x +c \right )+315 d x +704\right )}{4032 d} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*(448*cos(c + d*x)*sin(c + d*x)**8 + 1008*cos(c + d*x)*sin(c + d*x)** 7 - 640*cos(c + d*x)*sin(c + d*x)**6 - 2856*cos(c + d*x)*sin(c + d*x)**5 - 768*cos(c + d*x)*sin(c + d*x)**4 + 2478*cos(c + d*x)*sin(c + d*x)**3 + 16 64*cos(c + d*x)*sin(c + d*x)**2 - 315*cos(c + d*x)*sin(c + d*x) - 704*cos( c + d*x) + 315*d*x + 704))/(4032*d)