Integrand size = 29, antiderivative size = 184 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {3 a^3 \sin ^{2+n}(c+d x)}{d (2+n)}-\frac {8 a^3 \sin ^{4+n}(c+d x)}{d (4+n)}-\frac {6 a^3 \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {6 a^3 \sin ^{6+n}(c+d x)}{d (6+n)}+\frac {8 a^3 \sin ^{7+n}(c+d x)}{d (7+n)}-\frac {3 a^3 \sin ^{9+n}(c+d x)}{d (9+n)}-\frac {a^3 \sin ^{10+n}(c+d x)}{d (10+n)} \]
a^3*sin(d*x+c)^(1+n)/d/(1+n)+3*a^3*sin(d*x+c)^(2+n)/d/(2+n)-8*a^3*sin(d*x+ c)^(4+n)/d/(4+n)-6*a^3*sin(d*x+c)^(5+n)/d/(5+n)+6*a^3*sin(d*x+c)^(6+n)/d/( 6+n)+8*a^3*sin(d*x+c)^(7+n)/d/(7+n)-3*a^3*sin(d*x+c)^(9+n)/d/(9+n)-a^3*sin (d*x+c)^(10+n)/d/(10+n)
Time = 0.63 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.68 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^{1+n}(c+d x) \left (\frac {1}{1+n}+\frac {3 \sin (c+d x)}{2+n}-\frac {8 \sin ^3(c+d x)}{4+n}-\frac {6 \sin ^4(c+d x)}{5+n}+\frac {6 \sin ^5(c+d x)}{6+n}+\frac {8 \sin ^6(c+d x)}{7+n}-\frac {3 \sin ^8(c+d x)}{9+n}-\frac {\sin ^9(c+d x)}{10+n}\right )}{d} \]
(a^3*Sin[c + d*x]^(1 + n)*((1 + n)^(-1) + (3*Sin[c + d*x])/(2 + n) - (8*Si n[c + d*x]^3)/(4 + n) - (6*Sin[c + d*x]^4)/(5 + n) + (6*Sin[c + d*x]^5)/(6 + n) + (8*Sin[c + d*x]^6)/(7 + n) - (3*Sin[c + d*x]^8)/(9 + n) - Sin[c + d*x]^9/(10 + n)))/d
Time = 0.40 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3042, 3315, 99, 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.
| \(\displaystyle \int \cos ^7(c+d x) (a \sin (c+d x)+a)^3 \sin ^n(c+d x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^7 (a \sin (c+d x)+a)^3 \sin (c+d x)^ndx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {\int \sin ^n(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^6d(a \sin (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (a^9 \sin ^n(c+d x)+3 a^9 \sin ^{n+1}(c+d x)-8 a^9 \sin ^{n+3}(c+d x)-6 a^9 \sin ^{n+4}(c+d x)+6 a^9 \sin ^{n+5}(c+d x)+8 a^9 \sin ^{n+6}(c+d x)-3 a^9 \sin ^{n+8}(c+d x)-a^9 \sin ^{n+9}(c+d x)\right )d(a \sin (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^{10} \sin ^{n+1}(c+d x)}{n+1}+\frac {3 a^{10} \sin ^{n+2}(c+d x)}{n+2}-\frac {8 a^{10} \sin ^{n+4}(c+d x)}{n+4}-\frac {6 a^{10} \sin ^{n+5}(c+d x)}{n+5}+\frac {6 a^{10} \sin ^{n+6}(c+d x)}{n+6}+\frac {8 a^{10} \sin ^{n+7}(c+d x)}{n+7}-\frac {3 a^{10} \sin ^{n+9}(c+d x)}{n+9}-\frac {a^{10} \sin ^{n+10}(c+d x)}{n+10}}{a^7 d}\) |
((a^10*Sin[c + d*x]^(1 + n))/(1 + n) + (3*a^10*Sin[c + d*x]^(2 + n))/(2 + n) - (8*a^10*Sin[c + d*x]^(4 + n))/(4 + n) - (6*a^10*Sin[c + d*x]^(5 + n)) /(5 + n) + (6*a^10*Sin[c + d*x]^(6 + n))/(6 + n) + (8*a^10*Sin[c + d*x]^(7 + n))/(7 + n) - (3*a^10*Sin[c + d*x]^(9 + n))/(9 + n) - (a^10*Sin[c + d*x ]^(10 + n))/(10 + n))/(a^7*d)
3.7.97.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(414\) vs. \(2(184)=368\).
Time = 28.82 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.26
| method | result | size |
| parallelrisch | \(\frac {19 \left (\frac {\left (9+n \right ) \left (7+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (10 d x +10 c \right )}{76}+\frac {25 \left (n^{3}+\frac {492}{25} n^{2}+\frac {2252}{25} n -\frac {4368}{5}\right ) \left (7+n \right ) \left (9+n \right ) \left (5+n \right ) \left (1+n \right ) \cos \left (2 d x +2 c \right )}{38}-\frac {14 \left (7+n \right ) \left (n^{2}+\frac {166}{7} n +\frac {1320}{7}\right ) \left (9+n \right ) \left (5+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (4 d x +4 c \right )}{19}-\frac {51 \left (n +\frac {230}{17}\right ) \left (4+n \right ) \left (7+n \right ) \left (9+n \right ) \left (5+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (6 d x +6 c \right )}{76}-\frac {5 \left (9+n \right ) \left (7+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (8 d x +8 c \right )}{38}+\frac {30 \left (4+n \right ) \left (10+n \right ) \left (6+n \right ) \left (2+n \right ) \left (n^{2}+\frac {108}{5} n +147\right ) \left (1+n \right ) \sin \left (3 d x +3 c \right )}{19}+\frac {10 \left (4+n \right ) \left (10+n \right ) \left (6+n \right ) \left (n^{2}+\frac {76}{5} n +\frac {63}{5}\right ) \left (2+n \right ) \left (1+n \right ) \sin \left (5 d x +5 c \right )}{19}-\frac {5 \left (4+n \right ) \left (n +\frac {99}{5}\right ) \left (10+n \right ) \left (6+n \right ) \left (5+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (7 d x +7 c \right )}{38}-\frac {3 \left (10+n \right ) \left (7+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (9 d x +9 c \right )}{38}+\left (4+n \right ) \left (10+n \right ) \left (6+n \right ) \left (2+n \right ) \left (n^{3}+\frac {471}{19} n^{2}+239 n +\frac {28665}{19}\right ) \sin \left (d x +c \right )+\frac {33 \left (n^{3}+\frac {260}{11} n^{2}+\frac {2276}{11} n +1072\right ) \left (7+n \right ) \left (9+n \right ) \left (5+n \right ) \left (1+n \right )}{38}\right ) \left (\sin ^{n}\left (d x +c \right )\right ) a^{3}}{128 \left (2+n \right ) \left (1+n \right ) \left (10+n \right ) \left (4+n \right ) \left (6+n \right ) \left (5+n \right ) \left (9+n \right ) d \left (7+n \right )}\) | \(415\) |
19/128*(1/76*(9+n)*(7+n)*(6+n)*(5+n)*(4+n)*(2+n)*(1+n)*cos(10*d*x+10*c)+25 /38*(n^3+492/25*n^2+2252/25*n-4368/5)*(7+n)*(9+n)*(5+n)*(1+n)*cos(2*d*x+2* c)-14/19*(7+n)*(n^2+166/7*n+1320/7)*(9+n)*(5+n)*(2+n)*(1+n)*cos(4*d*x+4*c) -51/76*(n+230/17)*(4+n)*(7+n)*(9+n)*(5+n)*(2+n)*(1+n)*cos(6*d*x+6*c)-5/38* (9+n)*(7+n)*(6+n)*(5+n)*(4+n)*(2+n)*(1+n)*cos(8*d*x+8*c)+30/19*(4+n)*(10+n )*(6+n)*(2+n)*(n^2+108/5*n+147)*(1+n)*sin(3*d*x+3*c)+10/19*(4+n)*(10+n)*(6 +n)*(n^2+76/5*n+63/5)*(2+n)*(1+n)*sin(5*d*x+5*c)-5/38*(4+n)*(n+99/5)*(10+n )*(6+n)*(5+n)*(2+n)*(1+n)*sin(7*d*x+7*c)-3/38*(10+n)*(7+n)*(6+n)*(5+n)*(4+ n)*(2+n)*(1+n)*sin(9*d*x+9*c)+(4+n)*(10+n)*(6+n)*(2+n)*(n^3+471/19*n^2+239 *n+28665/19)*sin(d*x+c)+33/38*(n^3+260/11*n^2+2276/11*n+1072)*(7+n)*(9+n)* (5+n)*(1+n))*sin(d*x+c)^n*a^3/(2+n)/(1+n)/(10+n)/(4+n)/(6+n)/(5+n)/(9+n)/d /(7+n)
Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (184) = 368\).
Time = 0.34 (sec) , antiderivative size = 697, normalized size of antiderivative = 3.79 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {{\left ({\left (a^{3} n^{7} + 34 \, a^{3} n^{6} + 472 \, a^{3} n^{5} + 3442 \, a^{3} n^{4} + 14083 \, a^{3} n^{3} + 31804 \, a^{3} n^{2} + 35844 \, a^{3} n + 15120 \, a^{3}\right )} \cos \left (d x + c\right )^{10} - 5 \, {\left (a^{3} n^{7} + 34 \, a^{3} n^{6} + 472 \, a^{3} n^{5} + 3442 \, a^{3} n^{4} + 14083 \, a^{3} n^{3} + 31804 \, a^{3} n^{2} + 35844 \, a^{3} n + 15120 \, a^{3}\right )} \cos \left (d x + c\right )^{8} + 192 \, a^{3} n^{4} + 4 \, {\left (a^{3} n^{7} + 28 \, a^{3} n^{6} + 304 \, a^{3} n^{5} + 1618 \, a^{3} n^{4} + 4375 \, a^{3} n^{3} + 5554 \, a^{3} n^{2} + 2520 \, a^{3} n\right )} \cos \left (d x + c\right )^{6} + 4224 \, a^{3} n^{3} + 31488 \, a^{3} n^{2} + 24 \, {\left (a^{3} n^{6} + 24 \, a^{3} n^{5} + 208 \, a^{3} n^{4} + 786 \, a^{3} n^{3} + 1231 \, a^{3} n^{2} + 630 \, a^{3} n\right )} \cos \left (d x + c\right )^{4} + 87936 \, a^{3} n + 60480 \, a^{3} + 96 \, {\left (a^{3} n^{5} + 22 \, a^{3} n^{4} + 164 \, a^{3} n^{3} + 458 \, a^{3} n^{2} + 315 \, a^{3} n\right )} \cos \left (d x + c\right )^{2} - {\left (3 \, {\left (a^{3} n^{7} + 35 \, a^{3} n^{6} + 497 \, a^{3} n^{5} + 3689 \, a^{3} n^{4} + 15302 \, a^{3} n^{3} + 34916 \, a^{3} n^{2} + 39640 \, a^{3} n + 16800 \, a^{3}\right )} \cos \left (d x + c\right )^{8} - 192 \, a^{3} n^{4} - 4 \, {\left (a^{3} n^{7} + 31 \, a^{3} n^{6} + 385 \, a^{3} n^{5} + 2485 \, a^{3} n^{4} + 8974 \, a^{3} n^{3} + 18004 \, a^{3} n^{2} + 18360 \, a^{3} n + 7200 \, a^{3}\right )} \cos \left (d x + c\right )^{6} - 4224 \, a^{3} n^{3} - 31488 \, a^{3} n^{2} - 24 \, {\left (a^{3} n^{6} + 26 \, a^{3} n^{5} + 255 \, a^{3} n^{4} + 1210 \, a^{3} n^{3} + 2924 \, a^{3} n^{2} + 3384 \, a^{3} n + 1440 \, a^{3}\right )} \cos \left (d x + c\right )^{4} - 93696 \, a^{3} n - 92160 \, a^{3} - 96 \, {\left (a^{3} n^{5} + 23 \, a^{3} n^{4} + 186 \, a^{3} n^{3} + 652 \, a^{3} n^{2} + 968 \, a^{3} n + 480 \, a^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 44 \, d n^{7} + 812 \, d n^{6} + 8162 \, d n^{5} + 48503 \, d n^{4} + 172634 \, d n^{3} + 353884 \, d n^{2} + 373560 \, d n + 151200 \, d} \]
((a^3*n^7 + 34*a^3*n^6 + 472*a^3*n^5 + 3442*a^3*n^4 + 14083*a^3*n^3 + 3180 4*a^3*n^2 + 35844*a^3*n + 15120*a^3)*cos(d*x + c)^10 - 5*(a^3*n^7 + 34*a^3 *n^6 + 472*a^3*n^5 + 3442*a^3*n^4 + 14083*a^3*n^3 + 31804*a^3*n^2 + 35844* a^3*n + 15120*a^3)*cos(d*x + c)^8 + 192*a^3*n^4 + 4*(a^3*n^7 + 28*a^3*n^6 + 304*a^3*n^5 + 1618*a^3*n^4 + 4375*a^3*n^3 + 5554*a^3*n^2 + 2520*a^3*n)*c os(d*x + c)^6 + 4224*a^3*n^3 + 31488*a^3*n^2 + 24*(a^3*n^6 + 24*a^3*n^5 + 208*a^3*n^4 + 786*a^3*n^3 + 1231*a^3*n^2 + 630*a^3*n)*cos(d*x + c)^4 + 879 36*a^3*n + 60480*a^3 + 96*(a^3*n^5 + 22*a^3*n^4 + 164*a^3*n^3 + 458*a^3*n^ 2 + 315*a^3*n)*cos(d*x + c)^2 - (3*(a^3*n^7 + 35*a^3*n^6 + 497*a^3*n^5 + 3 689*a^3*n^4 + 15302*a^3*n^3 + 34916*a^3*n^2 + 39640*a^3*n + 16800*a^3)*cos (d*x + c)^8 - 192*a^3*n^4 - 4*(a^3*n^7 + 31*a^3*n^6 + 385*a^3*n^5 + 2485*a ^3*n^4 + 8974*a^3*n^3 + 18004*a^3*n^2 + 18360*a^3*n + 7200*a^3)*cos(d*x + c)^6 - 4224*a^3*n^3 - 31488*a^3*n^2 - 24*(a^3*n^6 + 26*a^3*n^5 + 255*a^3*n ^4 + 1210*a^3*n^3 + 2924*a^3*n^2 + 3384*a^3*n + 1440*a^3)*cos(d*x + c)^4 - 93696*a^3*n - 92160*a^3 - 96*(a^3*n^5 + 23*a^3*n^4 + 186*a^3*n^3 + 652*a^ 3*n^2 + 968*a^3*n + 480*a^3)*cos(d*x + c)^2)*sin(d*x + c))*sin(d*x + c)^n/ (d*n^8 + 44*d*n^7 + 812*d*n^6 + 8162*d*n^5 + 48503*d*n^4 + 172634*d*n^3 + 353884*d*n^2 + 373560*d*n + 151200*d)
Leaf count of result is larger than twice the leaf count of optimal. 41196 vs. \(2 (158) = 316\).
Time = 83.34 (sec) , antiderivative size = 41196, normalized size of antiderivative = 223.89 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Too large to display} \]
Piecewise((x*(a*sin(c) + a)**3*sin(c)**n*cos(c)**7, Eq(d, 0)), (-a**3*log( sin(c + d*x))/d + 48*a**3/(35*d*sin(c + d*x)) - a**3*cos(c + d*x)**2/(2*d* sin(c + d*x)**2) + 3*a**3/(8*d*sin(c + d*x)**2) - 24*a**3*cos(c + d*x)**2/ (35*d*sin(c + d*x)**3) + 16*a**3/(315*d*sin(c + d*x)**3) + a**3*cos(c + d* x)**4/(4*d*sin(c + d*x)**4) - 3*a**3*cos(c + d*x)**2/(8*d*sin(c + d*x)**4) + 18*a**3*cos(c + d*x)**4/(35*d*sin(c + d*x)**5) - 8*a**3*cos(c + d*x)**2 /(105*d*sin(c + d*x)**5) - a**3*cos(c + d*x)**6/(6*d*sin(c + d*x)**6) + 3* a**3*cos(c + d*x)**4/(8*d*sin(c + d*x)**6) - 3*a**3*cos(c + d*x)**6/(7*d*s in(c + d*x)**7) + 2*a**3*cos(c + d*x)**4/(21*d*sin(c + d*x)**7) - 3*a**3*c os(c + d*x)**6/(8*d*sin(c + d*x)**8) - a**3*cos(c + d*x)**6/(9*d*sin(c + d *x)**9), Eq(n, -10)), (-3*a**3*log(sin(c + d*x))/d - 16*a**3*sin(c + d*x)/ (5*d) - 8*a**3*cos(c + d*x)**2/(5*d*sin(c + d*x)) + 48*a**3/(35*d*sin(c + d*x)) - 3*a**3*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) + a**3/(8*d*sin(c + d *x)**2) + 2*a**3*cos(c + d*x)**4/(5*d*sin(c + d*x)**3) - 24*a**3*cos(c + d *x)**2/(35*d*sin(c + d*x)**3) + 3*a**3*cos(c + d*x)**4/(4*d*sin(c + d*x)** 4) - a**3*cos(c + d*x)**2/(8*d*sin(c + d*x)**4) - a**3*cos(c + d*x)**6/(5* d*sin(c + d*x)**5) + 18*a**3*cos(c + d*x)**4/(35*d*sin(c + d*x)**5) - a**3 *cos(c + d*x)**6/(2*d*sin(c + d*x)**6) + a**3*cos(c + d*x)**4/(8*d*sin(c + d*x)**6) - 3*a**3*cos(c + d*x)**6/(7*d*sin(c + d*x)**7) - a**3*cos(c + d* x)**6/(8*d*sin(c + d*x)**8), Eq(n, -9)), (-a**3*tan(c/2 + d*x/2)**18/(8...
Time = 0.21 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.90 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {\frac {a^{3} \sin \left (d x + c\right )^{n + 10}}{n + 10} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 9}}{n + 9} - \frac {8 \, a^{3} \sin \left (d x + c\right )^{n + 7}}{n + 7} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{n + 6}}{n + 6} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{n + 5}}{n + 5} + \frac {8 \, a^{3} \sin \left (d x + c\right )^{n + 4}}{n + 4} - \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a^{3} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]
-(a^3*sin(d*x + c)^(n + 10)/(n + 10) + 3*a^3*sin(d*x + c)^(n + 9)/(n + 9) - 8*a^3*sin(d*x + c)^(n + 7)/(n + 7) - 6*a^3*sin(d*x + c)^(n + 6)/(n + 6) + 6*a^3*sin(d*x + c)^(n + 5)/(n + 5) + 8*a^3*sin(d*x + c)^(n + 4)/(n + 4) - 3*a^3*sin(d*x + c)^(n + 2)/(n + 2) - a^3*sin(d*x + c)^(n + 1)/(n + 1))/d
Leaf count of result is larger than twice the leaf count of optimal. 1360 vs. \(2 (184) = 368\).
Time = 0.57 (sec) , antiderivative size = 1360, normalized size of antiderivative = 7.39 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Too large to display} \]
-((n^3*sin(d*x + c)^n*sin(d*x + c)^10 + 18*n^2*sin(d*x + c)^n*sin(d*x + c) ^10 - 3*n^3*sin(d*x + c)^n*sin(d*x + c)^8 + 104*n*sin(d*x + c)^n*sin(d*x + c)^10 - 60*n^2*sin(d*x + c)^n*sin(d*x + c)^8 + 192*sin(d*x + c)^n*sin(d*x + c)^10 + 3*n^3*sin(d*x + c)^n*sin(d*x + c)^6 - 372*n*sin(d*x + c)^n*sin( d*x + c)^8 + 66*n^2*sin(d*x + c)^n*sin(d*x + c)^6 - 720*sin(d*x + c)^n*sin (d*x + c)^8 - n^3*sin(d*x + c)^n*sin(d*x + c)^4 + 456*n*sin(d*x + c)^n*sin (d*x + c)^6 - 24*n^2*sin(d*x + c)^n*sin(d*x + c)^4 + 960*sin(d*x + c)^n*si n(d*x + c)^6 - 188*n*sin(d*x + c)^n*sin(d*x + c)^4 - 480*sin(d*x + c)^n*si n(d*x + c)^4)*a^3/(n^4 + 28*n^3 + 284*n^2 + 1232*n + 1920) + 3*(n^3*sin(d* x + c)^n*sin(d*x + c)^9 + 15*n^2*sin(d*x + c)^n*sin(d*x + c)^9 - 3*n^3*sin (d*x + c)^n*sin(d*x + c)^7 + 71*n*sin(d*x + c)^n*sin(d*x + c)^9 - 51*n^2*s in(d*x + c)^n*sin(d*x + c)^7 + 105*sin(d*x + c)^n*sin(d*x + c)^9 + 3*n^3*s in(d*x + c)^n*sin(d*x + c)^5 - 261*n*sin(d*x + c)^n*sin(d*x + c)^7 + 57*n^ 2*sin(d*x + c)^n*sin(d*x + c)^5 - 405*sin(d*x + c)^n*sin(d*x + c)^7 - n^3* sin(d*x + c)^n*sin(d*x + c)^3 + 333*n*sin(d*x + c)^n*sin(d*x + c)^5 - 21*n ^2*sin(d*x + c)^n*sin(d*x + c)^3 + 567*sin(d*x + c)^n*sin(d*x + c)^5 - 143 *n*sin(d*x + c)^n*sin(d*x + c)^3 - 315*sin(d*x + c)^n*sin(d*x + c)^3)*a^3/ (n^4 + 24*n^3 + 206*n^2 + 744*n + 945) + 3*(n^3*sin(d*x + c)^n*sin(d*x + c )^8 + 12*n^2*sin(d*x + c)^n*sin(d*x + c)^8 - 3*n^3*sin(d*x + c)^n*sin(d*x + c)^6 + 44*n*sin(d*x + c)^n*sin(d*x + c)^8 - 42*n^2*sin(d*x + c)^n*sin...
Time = 17.99 (sec) , antiderivative size = 1130, normalized size of antiderivative = 6.14 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Too large to display} \]
(3*a^3*sin(c + d*x)^n*(6117676*n + 3058196*n^2 + 755233*n^3 + 109542*n^4 + 9800*n^5 + 502*n^6 + 11*n^7 + 3714480))/(256*d*(373560*n + 353884*n^2 + 1 72634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) - (5* a^3*sin(c + d*x)^n*cos(8*c + 8*d*x)*(35844*n + 31804*n^2 + 14083*n^3 + 344 2*n^4 + 472*n^5 + 34*n^6 + n^7 + 15120))/(256*d*(373560*n + 353884*n^2 + 1 72634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) + (a^ 3*sin(c + d*x)^n*cos(10*c + 10*d*x)*(35844*n + 31804*n^2 + 14083*n^3 + 344 2*n^4 + 472*n^5 + 34*n^6 + n^7 + 15120))/(512*d*(373560*n + 353884*n^2 + 1 72634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) - (a^ 3*sin(c + d*x)*sin(c + d*x)^n*(n*16168200i + n^2*7143148i + n^3*1614322i + n^4*215083i + n^5*18019i + n^6*889i + n^7*19i + 13759200i)*1i)/(128*d*(37 3560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) - (3*a^3*sin(c + d*x)^n*cos(6*c + 6*d*x)*(1320260*n + 11 00668*n^2 + 446515*n^3 + 97426*n^4 + 11608*n^5 + 706*n^6 + 17*n^7 + 579600 ))/(512*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812 *n^6 + 44*n^7 + n^8 + 151200)) - (a^3*sin(c + d*x)^n*cos(4*c + 4*d*x)*(172 9500*n + 1246276*n^2 + 413653*n^3 + 71710*n^4 + 6760*n^5 + 334*n^6 + 7*n^7 + 831600))/(64*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n ^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) + (a^3*sin(c + d*x)^n*cos(2*c + 2*d *x)*(122059*n^3 - 2395364*n^2 - 9293340*n + 119842*n^4 + 17176*n^5 + 10...