\(\int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\) [605]

Optimal result

Integrand size = 29, antiderivative size = 224 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27 a^3 x}{1024}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^9(c+d x)}{d}-\frac {6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^{13}(c+d x)}{13 d}+\frac {27 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {9 a^3 \cos ^3(c+d x) \sin (c+d x)}{512 d}+\frac {9 a^3 \cos ^5(c+d x) \sin (c+d x)}{640 d}-\frac {27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d} \] Output:

27/1024*a^3*x-4/7*a^3*cos(d*x+c)^7/d+a^3*cos(d*x+c)^9/d-6/11*a^3*cos(d*x+c 
)^11/d+1/13*a^3*cos(d*x+c)^13/d+27/1024*a^3*cos(d*x+c)*sin(d*x+c)/d+9/512* 
a^3*cos(d*x+c)^3*sin(d*x+c)/d+9/640*a^3*cos(d*x+c)^5*sin(d*x+c)/d-27/320*a 
^3*cos(d*x+c)^7*sin(d*x+c)/d-9/40*a^3*cos(d*x+c)^7*sin(d*x+c)^3/d-1/4*a^3* 
cos(d*x+c)^7*sin(d*x+c)^5/d
 

Mathematica [A] (verified)

Time = 1.98 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.65 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (720720 c+1081080 d x-1401400 \cos (c+d x)-450450 \cos (3 (c+d x))+150150 \cos (5 (c+d x))+94380 \cos (7 (c+d x))-20020 \cos (9 (c+d x))-11830 \cos (11 (c+d x))+770 \cos (13 (c+d x))+80080 \sin (2 (c+d x))-385385 \sin (4 (c+d x))-40040 \sin (6 (c+d x))+65065 \sin (8 (c+d x))+8008 \sin (10 (c+d x))-5005 \sin (12 (c+d x)))}{41000960 d} \] Input:

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
 

Output:

(a^3*(720720*c + 1081080*d*x - 1401400*Cos[c + d*x] - 450450*Cos[3*(c + d* 
x)] + 150150*Cos[5*(c + d*x)] + 94380*Cos[7*(c + d*x)] - 20020*Cos[9*(c + 
d*x)] - 11830*Cos[11*(c + d*x)] + 770*Cos[13*(c + d*x)] + 80080*Sin[2*(c + 
 d*x)] - 385385*Sin[4*(c + d*x)] - 40040*Sin[6*(c + d*x)] + 65065*Sin[8*(c 
 + d*x)] + 8008*Sin[10*(c + d*x)] - 5005*Sin[12*(c + d*x)]))/(41000960*d)
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 224, 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.

\(\displaystyle \int \sin ^4(c+d x) \cos ^6(c+d x) (a \sin (c+d x)+a)^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin (c+d x)^4 \cos (c+d x)^6 (a \sin (c+d x)+a)^3dx\)

\(\Big \downarrow \) 3352

\(\displaystyle \int \left (a^3 \sin ^7(c+d x) \cos ^6(c+d x)+3 a^3 \sin ^6(c+d x) \cos ^6(c+d x)+3 a^3 \sin ^5(c+d x) \cos ^6(c+d x)+a^3 \sin ^4(c+d x) \cos ^6(c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 \cos ^{13}(c+d x)}{13 d}-\frac {6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^9(c+d x)}{d}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{4 d}-\frac {9 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{40 d}-\frac {27 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac {9 a^3 \sin (c+d x) \cos ^5(c+d x)}{640 d}+\frac {9 a^3 \sin (c+d x) \cos ^3(c+d x)}{512 d}+\frac {27 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {27 a^3 x}{1024}\)

Input:

Int[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
 

Output:

(27*a^3*x)/1024 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (a^3*Cos[c + d*x]^9)/d - 
(6*a^3*Cos[c + d*x]^11)/(11*d) + (a^3*Cos[c + d*x]^13)/(13*d) + (27*a^3*Co 
s[c + d*x]*Sin[c + d*x])/(1024*d) + (9*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(5 
12*d) + (9*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(640*d) - (27*a^3*Cos[c + d*x] 
^7*Sin[c + d*x])/(320*d) - (9*a^3*Cos[c + d*x]^7*Sin[c + d*x]^3)/(40*d) - 
(a^3*Cos[c + d*x]^7*Sin[c + d*x]^5)/(4*d)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3352

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]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.38

\[\frac {a^{3} \left (-\frac {\sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{7}}{13}-\frac {6 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{7}}{143}-\frac {8 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{429}-\frac {16 \cos \left (d x +c \right )^{7}}{3003}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{5}}{12}-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{7}}{24}-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{64}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{7}}{11}-\frac {4 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{99}-\frac {8 \cos \left (d x +c \right )^{7}}{693}\right )+a^{3} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{7}}{10}-\frac {3 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{80}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d}\]

Input:

int(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x)
                                                                                    
                                                                                    
 

Output:

1/d*(a^3*(-1/13*sin(d*x+c)^6*cos(d*x+c)^7-6/143*sin(d*x+c)^4*cos(d*x+c)^7- 
8/429*cos(d*x+c)^7*sin(d*x+c)^2-16/3003*cos(d*x+c)^7)+3*a^3*(-1/12*cos(d*x 
+c)^7*sin(d*x+c)^5-1/24*sin(d*x+c)^3*cos(d*x+c)^7-1/64*cos(d*x+c)^7*sin(d* 
x+c)+1/384*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/10 
24*d*x+5/1024*c)+3*a^3*(-1/11*sin(d*x+c)^4*cos(d*x+c)^7-4/99*cos(d*x+c)^7* 
sin(d*x+c)^2-8/693*cos(d*x+c)^7)+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)+1/160*(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))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.67 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {394240 \, a^{3} \cos \left (d x + c\right )^{13} - 2795520 \, a^{3} \cos \left (d x + c\right )^{11} + 5125120 \, a^{3} \cos \left (d x + c\right )^{9} - 2928640 \, a^{3} \cos \left (d x + c\right )^{7} + 135135 \, a^{3} d x - 1001 \, {\left (1280 \, a^{3} \cos \left (d x + c\right )^{11} - 3712 \, a^{3} \cos \left (d x + c\right )^{9} + 2864 \, a^{3} \cos \left (d x + c\right )^{7} - 72 \, a^{3} \cos \left (d x + c\right )^{5} - 90 \, a^{3} \cos \left (d x + c\right )^{3} - 135 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{5125120 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="frica 
s")
 

Output:

1/5125120*(394240*a^3*cos(d*x + c)^13 - 2795520*a^3*cos(d*x + c)^11 + 5125 
120*a^3*cos(d*x + c)^9 - 2928640*a^3*cos(d*x + c)^7 + 135135*a^3*d*x - 100 
1*(1280*a^3*cos(d*x + c)^11 - 3712*a^3*cos(d*x + c)^9 + 2864*a^3*cos(d*x + 
 c)^7 - 72*a^3*cos(d*x + c)^5 - 90*a^3*cos(d*x + c)^3 - 135*a^3*cos(d*x + 
c))*sin(d*x + c))/d
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 748 vs. \(2 (212) = 424\).

Time = 3.58 (sec) , antiderivative size = 748, normalized size of antiderivative = 3.34 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:

integrate(cos(d*x+c)**6*sin(d*x+c)**4*(a+a*sin(d*x+c))**3,x)
 

Output:

Piecewise((15*a**3*x*sin(c + d*x)**12/1024 + 45*a**3*x*sin(c + d*x)**10*co 
s(c + d*x)**2/512 + 3*a**3*x*sin(c + d*x)**10/256 + 225*a**3*x*sin(c + d*x 
)**8*cos(c + d*x)**4/1024 + 15*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256 
+ 75*a**3*x*sin(c + d*x)**6*cos(c + d*x)**6/256 + 15*a**3*x*sin(c + d*x)** 
6*cos(c + d*x)**4/128 + 225*a**3*x*sin(c + d*x)**4*cos(c + d*x)**8/1024 + 
15*a**3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 45*a**3*x*sin(c + d*x)**2* 
cos(c + d*x)**10/512 + 15*a**3*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 15* 
a**3*x*cos(c + d*x)**12/1024 + 3*a**3*x*cos(c + d*x)**10/256 + 15*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/(1024*d) + 3*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 99*a**3*sin(c 
 + d*x)**7*cos(c + d*x)**5/(512*d) + 7*a**3*sin(c + d*x)**7*cos(c + d*x)** 
3/(128*d) - a**3*sin(c + d*x)**6*cos(c + d*x)**7/(7*d) - 99*a**3*sin(c + d 
*x)**5*cos(c + d*x)**7/(512*d) + a**3*sin(c + d*x)**5*cos(c + d*x)**5/(10* 
d) - 2*a**3*sin(c + d*x)**4*cos(c + d*x)**9/(21*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/(1024*d) 
 - 7*a**3*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - 8*a**3*sin(c + d*x)**2 
*cos(c + d*x)**11/(231*d) - 4*a**3*sin(c + d*x)**2*cos(c + d*x)**9/(21*d) 
- 15*a**3*sin(c + d*x)*cos(c + d*x)**11/(1024*d) - 3*a**3*sin(c + d*x)*cos 
(c + d*x)**9/(256*d) - 16*a**3*cos(c + d*x)**13/(3003*d) - 8*a**3*cos(c + 
d*x)**11/(231*d), Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)**4*cos(c)**6, ...
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.82 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {40960 \, {\left (231 \, \cos \left (d x + c\right )^{13} - 819 \, \cos \left (d x + c\right )^{11} + 1001 \, \cos \left (d x + c\right )^{9} - 429 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 532480 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{3} + 12012 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 15015 \, {\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{123002880 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="maxim 
a")
 

Output:

1/123002880*(40960*(231*cos(d*x + c)^13 - 819*cos(d*x + c)^11 + 1001*cos(d 
*x + c)^9 - 429*cos(d*x + c)^7)*a^3 - 532480*(63*cos(d*x + c)^11 - 154*cos 
(d*x + c)^9 + 99*cos(d*x + c)^7)*a^3 + 12012*(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 + 15015*(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
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27}{1024} \, a^{3} x + \frac {a^{3} \cos \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac {13 \, a^{3} \cos \left (11 \, d x + 11 \, c\right )}{45056 \, d} - \frac {a^{3} \cos \left (9 \, d x + 9 \, c\right )}{2048 \, d} + \frac {33 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{14336 \, d} + \frac {15 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{4096 \, d} - \frac {45 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{4096 \, d} - \frac {35 \, a^{3} \cos \left (d x + c\right )}{1024 \, d} - \frac {a^{3} \sin \left (12 \, d x + 12 \, c\right )}{8192 \, d} + \frac {a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {13 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {77 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="giac" 
)
 

Output:

27/1024*a^3*x + 1/53248*a^3*cos(13*d*x + 13*c)/d - 13/45056*a^3*cos(11*d*x 
 + 11*c)/d - 1/2048*a^3*cos(9*d*x + 9*c)/d + 33/14336*a^3*cos(7*d*x + 7*c) 
/d + 15/4096*a^3*cos(5*d*x + 5*c)/d - 45/4096*a^3*cos(3*d*x + 3*c)/d - 35/ 
1024*a^3*cos(d*x + c)/d - 1/8192*a^3*sin(12*d*x + 12*c)/d + 1/5120*a^3*sin 
(10*d*x + 10*c)/d + 13/8192*a^3*sin(8*d*x + 8*c)/d - 1/1024*a^3*sin(6*d*x 
+ 6*c)/d - 77/8192*a^3*sin(4*d*x + 4*c)/d + 1/512*a^3*sin(2*d*x + 2*c)/d
 

Mupad [B] (verification not implemented)

Time = 36.94 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.73 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Too large to display} \] Input:

int(cos(c + d*x)^6*sin(c + d*x)^4*(a + a*sin(c + d*x))^3,x)
 

Output:

(27*a^3*x)/1024 - ((27*a^3*(c + d*x))/1024 + (171*a^3*tan(c/2 + (d*x)/2)^3 
)/256 - (1603*a^3*tan(c/2 + (d*x)/2)^5)/640 - (59523*a^3*tan(c/2 + (d*x)/2 
)^7)/1280 + (305539*a^3*tan(c/2 + (d*x)/2)^9)/2560 - (93973*a^3*tan(c/2 + 
(d*x)/2)^11)/640 + (93973*a^3*tan(c/2 + (d*x)/2)^15)/640 - (305539*a^3*tan 
(c/2 + (d*x)/2)^17)/2560 + (59523*a^3*tan(c/2 + (d*x)/2)^19)/1280 + (1603* 
a^3*tan(c/2 + (d*x)/2)^21)/640 - (171*a^3*tan(c/2 + (d*x)/2)^23)/256 - (27 
*a^3*tan(c/2 + (d*x)/2)^25)/512 - a^3*((27*c)/1024 + (27*d*x)/1024 - 80/10 
01) + tan(c/2 + (d*x)/2)^2*((351*a^3*(c + d*x))/1024 - a^3*((351*c)/1024 + 
 (351*d*x)/1024 - 80/77)) + tan(c/2 + (d*x)/2)^4*((1053*a^3*(c + d*x))/512 
 - a^3*((1053*c)/512 + (1053*d*x)/512 - 480/77)) + tan(c/2 + (d*x)/2)^20*( 
(3861*a^3*(c + d*x))/512 - a^3*((3861*c)/512 + (3861*d*x)/512 - 32)) + tan 
(c/2 + (d*x)/2)^6*((3861*a^3*(c + d*x))/512 - a^3*((3861*c)/512 + (3861*d* 
x)/512 + 64/7)) + tan(c/2 + (d*x)/2)^14*((11583*a^3*(c + d*x))/256 - a^3*( 
(11583*c)/256 + (11583*d*x)/256 - 320)) + tan(c/2 + (d*x)/2)^12*((11583*a^ 
3*(c + d*x))/256 - a^3*((11583*c)/256 + (11583*d*x)/256 + 1280/7)) + tan(c 
/2 + (d*x)/2)^18*((19305*a^3*(c + d*x))/1024 - a^3*((19305*c)/1024 + (1930 
5*d*x)/1024 - 16)) + tan(c/2 + (d*x)/2)^8*((19305*a^3*(c + d*x))/1024 - a^ 
3*((19305*c)/1024 + (19305*d*x)/1024 - 288/7)) + tan(c/2 + (d*x)/2)^16*((3 
4749*a^3*(c + d*x))/1024 - a^3*((34749*c)/1024 + (34749*d*x)/1024 + 48)) + 
 tan(c/2 + (d*x)/2)^10*((34749*a^3*(c + d*x))/1024 - a^3*((34749*c)/102...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.95 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (394240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{12}+1281280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{11}+430080 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10}-2690688 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}-2938880 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}+816816 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+2498560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+952952 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-76800 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-90090 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-102400 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-135135 \cos \left (d x +c \right ) \sin \left (d x +c \right )-204800 \cos \left (d x +c \right )+135135 d x +204800\right )}{5125120 d} \] Input:

int(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x)
 

Output:

(a**3*(394240*cos(c + d*x)*sin(c + d*x)**12 + 1281280*cos(c + d*x)*sin(c + 
 d*x)**11 + 430080*cos(c + d*x)*sin(c + d*x)**10 - 2690688*cos(c + d*x)*si 
n(c + d*x)**9 - 2938880*cos(c + d*x)*sin(c + d*x)**8 + 816816*cos(c + d*x) 
*sin(c + d*x)**7 + 2498560*cos(c + d*x)*sin(c + d*x)**6 + 952952*cos(c + d 
*x)*sin(c + d*x)**5 - 76800*cos(c + d*x)*sin(c + d*x)**4 - 90090*cos(c + d 
*x)*sin(c + d*x)**3 - 102400*cos(c + d*x)*sin(c + d*x)**2 - 135135*cos(c + 
 d*x)*sin(c + d*x) - 204800*cos(c + d*x) + 135135*d*x + 204800))/(5125120* 
d)