Integrand size = 29, antiderivative size = 183 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {19 a^3 x}{256}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d} \] Output:
19/256*a^3*x-4/7*a^3*cos(d*x+c)^7/d+5/9*a^3*cos(d*x+c)^9/d-1/11*a^3*cos(d* x+c)^11/d+19/256*a^3*cos(d*x+c)*sin(d*x+c)/d+19/384*a^3*cos(d*x+c)^3*sin(d *x+c)/d+19/480*a^3*cos(d*x+c)^5*sin(d*x+c)/d-19/80*a^3*cos(d*x+c)^7*sin(d* x+c)/d-3/10*a^3*cos(d*x+c)^7*sin(d*x+c)^3/d
Time = 0.92 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.69 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (415800 c+526680 d x-568260 \cos (c+d x)-244860 \cos (3 (c+d x))+6930 \cos (5 (c+d x))+40590 \cos (7 (c+d x))+8470 \cos (9 (c+d x))-630 \cos (11 (c+d x))+152460 \sin (2 (c+d x))-138600 \sin (4 (c+d x))-57750 \sin (6 (c+d x))+3465 \sin (8 (c+d x))+4158 \sin (10 (c+d x)))}{7096320 d} \] Input:
Integrate[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(415800*c + 526680*d*x - 568260*Cos[c + d*x] - 244860*Cos[3*(c + d*x) ] + 6930*Cos[5*(c + d*x)] + 40590*Cos[7*(c + d*x)] + 8470*Cos[9*(c + d*x)] - 630*Cos[11*(c + d*x)] + 152460*Sin[2*(c + d*x)] - 138600*Sin[4*(c + d*x )] - 57750*Sin[6*(c + d*x)] + 3465*Sin[8*(c + d*x)] + 4158*Sin[10*(c + d*x )]))/(7096320*d)
Time = 0.56 (sec) , antiderivative size = 183, 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.
| \(\displaystyle \int \sin ^2(c+d x) \cos ^6(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^2 \cos (c+d x)^6 (a \sin (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \sin ^5(c+d x) \cos ^6(c+d x)+3 a^3 \sin ^4(c+d x) \cos ^6(c+d x)+3 a^3 \sin ^3(c+d x) \cos ^6(c+d x)+a^3 \sin ^2(c+d x) \cos ^6(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}-\frac {3 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {19 a^3 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {19 a^3 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac {19 a^3 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac {19 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {19 a^3 x}{256}\) |
Input:
Int[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
(19*a^3*x)/256 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (5*a^3*Cos[c + d*x]^9)/(9* d) - (a^3*Cos[c + d*x]^11)/(11*d) + (19*a^3*Cos[c + d*x]*Sin[c + d*x])/(25 6*d) + (19*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (19*a^3*Cos[c + d*x] ^5*Sin[c + d*x])/(480*d) - (19*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - ( 3*a^3*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*d)
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 = 0.27 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.29
\[\frac {a^{3} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{7}}{11}-\frac {4 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{99}-\frac {8 \cos \left (d x +c \right )^{7}}{693}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{7}}{10}-\frac {3 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{80}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\]
Input:
int(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x)
Output:
1/d*(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)+3*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)+3*a^3*(-1/9*cos(d*x+c)^7*sin(d*x+c)^2-2/63*cos(d*x+ c)^7)+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))
Time = 0.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.68 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {80640 \, a^{3} \cos \left (d x + c\right )^{11} - 492800 \, a^{3} \cos \left (d x + c\right )^{9} + 506880 \, a^{3} \cos \left (d x + c\right )^{7} - 65835 \, a^{3} d x - 231 \, {\left (1152 \, a^{3} \cos \left (d x + c\right )^{9} - 2064 \, a^{3} \cos \left (d x + c\right )^{7} + 152 \, a^{3} \cos \left (d x + c\right )^{5} + 190 \, a^{3} \cos \left (d x + c\right )^{3} + 285 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/887040*(80640*a^3*cos(d*x + c)^11 - 492800*a^3*cos(d*x + c)^9 + 506880* a^3*cos(d*x + c)^7 - 65835*a^3*d*x - 231*(1152*a^3*cos(d*x + c)^9 - 2064*a ^3*cos(d*x + c)^7 + 152*a^3*cos(d*x + c)^5 + 190*a^3*cos(d*x + c)^3 + 285* a^3*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (175) = 350\).
Time = 1.93 (sec) , antiderivative size = 597, normalized size of antiderivative = 3.26 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)**2*(a+a*sin(d*x+c))**3,x)
Output:
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 + 5*a**3*x*sin(c + d*x)**8/128 + 45*a**3*x*sin(c + d*x)**6* cos(c + d*x)**4/128 + 5*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 + 15*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 + 5*a**3*x*si n(c + d*x)**2*cos(c + d*x)**6/32 + 9*a**3*x*cos(c + d*x)**10/256 + 5*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) + 5*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 3*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/(384*d) - a**3*sin(c + d*x)**4*cos(c + d*x )**7/(7*d) - 21*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/(384*d) - 4*a**3*sin(c + d*x)**2*cos(c + d*x) **9/(63*d) - 3*a**3*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 9*a**3*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 5*a**3*sin(c + d*x)*cos(c + d*x)**7/(128*d ) - 8*a**3*cos(c + d*x)**11/(693*d) - 2*a**3*cos(c + d*x)**9/(21*d), Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)**2*cos(c)**6, True))
Time = 0.05 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.90 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {10240 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 337920 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 2079 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 2310 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{7096320 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
-1/7096320*(10240*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^3 - 337920*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^3 - 2079*(32*si n(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*(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^3)/d
Time = 0.25 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.04 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {19}{256} \, a^{3} x - \frac {a^{3} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {11 \, a^{3} \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {41 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {a^{3} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac {53 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac {41 \, a^{3} \cos \left (d x + c\right )}{512 \, d} + \frac {3 \, a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {25 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {5 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {11 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
19/256*a^3*x - 1/11264*a^3*cos(11*d*x + 11*c)/d + 11/9216*a^3*cos(9*d*x + 9*c)/d + 41/7168*a^3*cos(7*d*x + 7*c)/d + 1/1024*a^3*cos(5*d*x + 5*c)/d - 53/1536*a^3*cos(3*d*x + 3*c)/d - 41/512*a^3*cos(d*x + c)/d + 3/5120*a^3*si n(10*d*x + 10*c)/d + 1/2048*a^3*sin(8*d*x + 8*c)/d - 25/3072*a^3*sin(6*d*x + 6*c)/d - 5/256*a^3*sin(4*d*x + 4*c)/d + 11/512*a^3*sin(2*d*x + 2*c)/d
Time = 36.96 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.97 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^6*sin(c + d*x)^2*(a + a*sin(c + d*x))^3,x)
Output:
(19*a^3*x)/256 - ((19*a^3*(c + d*x))/256 - (13*a^3*tan(c/2 + (d*x)/2)^3)/1 2 - (32417*a^3*tan(c/2 + (d*x)/2)^5)/1920 + (466*a^3*tan(c/2 + (d*x)/2)^7) /15 - (2937*a^3*tan(c/2 + (d*x)/2)^9)/64 + (2937*a^3*tan(c/2 + (d*x)/2)^13 )/64 - (466*a^3*tan(c/2 + (d*x)/2)^15)/15 + (32417*a^3*tan(c/2 + (d*x)/2)^ 17)/1920 + (13*a^3*tan(c/2 + (d*x)/2)^19)/12 - (19*a^3*tan(c/2 + (d*x)/2)^ 21)/128 - a^3*((19*c)/256 + (19*d*x)/256 - 148/693) + tan(c/2 + (d*x)/2)^2 *((209*a^3*(c + d*x))/256 - a^3*((209*c)/256 + (209*d*x)/256 - 148/63)) + tan(c/2 + (d*x)/2)^18*((1045*a^3*(c + d*x))/256 - a^3*((1045*c)/256 + (104 5*d*x)/256 - 12)) + tan(c/2 + (d*x)/2)^4*((1045*a^3*(c + d*x))/256 - a^3*( (1045*c)/256 + (1045*d*x)/256 + 16/63)) + tan(c/2 + (d*x)/2)^14*((3135*a^3 *(c + d*x))/128 - a^3*((3135*c)/128 + (3135*d*x)/128 - 16/3)) + tan(c/2 + (d*x)/2)^16*((3135*a^3*(c + d*x))/256 - a^3*((3135*c)/256 + (3135*d*x)/256 - 44/3)) + tan(c/2 + (d*x)/2)^8*((3135*a^3*(c + d*x))/128 - a^3*((3135*c) /128 + (3135*d*x)/128 - 456/7)) + tan(c/2 + (d*x)/2)^6*((3135*a^3*(c + d*x ))/256 - a^3*((3135*c)/256 + (3135*d*x)/256 - 144/7)) + tan(c/2 + (d*x)/2) ^10*((4389*a^3*(c + d*x))/128 - a^3*((4389*c)/128 + (4389*d*x)/128 + 24)) + tan(c/2 + (d*x)/2)^12*((4389*a^3*(c + d*x))/128 - a^3*((4389*c)/128 + (4 389*d*x)/128 - 368/3)) + (19*a^3*tan(c/2 + (d*x)/2))/128)/(d*(tan(c/2 + (d *x)/2)^2 + 1)^11)
Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.98 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (80640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10}+266112 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}+89600 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-587664 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-657920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+201432 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+629760 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+251790 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-47360 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-65835 \cos \left (d x +c \right ) \sin \left (d x +c \right )-94720 \cos \left (d x +c \right )+65835 d x +94720\right )}{887040 d} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(80640*cos(c + d*x)*sin(c + d*x)**10 + 266112*cos(c + d*x)*sin(c + d *x)**9 + 89600*cos(c + d*x)*sin(c + d*x)**8 - 587664*cos(c + d*x)*sin(c + d*x)**7 - 657920*cos(c + d*x)*sin(c + d*x)**6 + 201432*cos(c + d*x)*sin(c + d*x)**5 + 629760*cos(c + d*x)*sin(c + d*x)**4 + 251790*cos(c + d*x)*sin( c + d*x)**3 - 47360*cos(c + d*x)*sin(c + d*x)**2 - 65835*cos(c + d*x)*sin( c + d*x) - 94720*cos(c + d*x) + 65835*d*x + 94720))/(887040*d)