Integrand size = 21, antiderivative size = 77 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^9}{9 b^3 d}+\frac {a (a+b \sin (c+d x))^{10}}{5 b^3 d}-\frac {(a+b \sin (c+d x))^{11}}{11 b^3 d} \]
-1/9*(a^2-b^2)*(a+b*sin(d*x+c))^9/b^3/d+1/5*a*(a+b*sin(d*x+c))^10/b^3/d-1/ 11*(a+b*sin(d*x+c))^11/b^3/d
Time = 0.55 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.73 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {(a+b \sin (c+d x))^9 \left (-2 a^2+65 b^2+45 b^2 \cos (2 (c+d x))+18 a b \sin (c+d x)\right )}{990 b^3 d} \]
((a + b*Sin[c + d*x])^9*(-2*a^2 + 65*b^2 + 45*b^2*Cos[2*(c + d*x)] + 18*a* b*Sin[c + d*x]))/(990*b^3*d)
Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3147, 476, 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.
| \(\displaystyle \int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^3 (a+b \sin (c+d x))^8dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {\int (a+b \sin (c+d x))^8 \left (b^2-b^2 \sin ^2(c+d x)\right )d(b \sin (c+d x))}{b^3 d}\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \frac {\int \left (-(a+b \sin (c+d x))^{10}+2 a (a+b \sin (c+d x))^9+\left (b^2-a^2\right ) (a+b \sin (c+d x))^8\right )d(b \sin (c+d x))}{b^3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{9} \left (a^2-b^2\right ) (a+b \sin (c+d x))^9-\frac {1}{11} (a+b \sin (c+d x))^{11}+\frac {1}{5} a (a+b \sin (c+d x))^{10}}{b^3 d}\) |
(-1/9*((a^2 - b^2)*(a + b*Sin[c + d*x])^9) + (a*(a + b*Sin[c + d*x])^10)/5 - (a + b*Sin[c + d*x])^11/11)/(b^3*d)
3.5.14.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs. \(2(71)=142\).
Time = 2.99 (sec) , antiderivative size = 236, normalized size of antiderivative = 3.06
| method | result | size |
| derivativedivides | \(-\frac {\frac {b^{8} \left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {4 a \,b^{7} \left (\sin ^{10}\left (d x +c \right )\right )}{5}+\frac {\left (28 a^{2} b^{6}-b^{8}\right ) \left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (56 a^{3} b^{5}-8 a \,b^{7}\right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (70 a^{4} b^{4}-28 a^{2} b^{6}\right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (56 a^{5} b^{3}-56 a^{3} b^{5}\right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (28 a^{6} b^{2}-70 a^{4} b^{4}\right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (8 a^{7} b -56 a^{5} b^{3}\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (a^{8}-28 a^{6} b^{2}\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}-4 \left (\sin ^{2}\left (d x +c \right )\right ) a^{7} b -a^{8} \sin \left (d x +c \right )}{d}\) | \(236\) |
| default | \(-\frac {\frac {b^{8} \left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {4 a \,b^{7} \left (\sin ^{10}\left (d x +c \right )\right )}{5}+\frac {\left (28 a^{2} b^{6}-b^{8}\right ) \left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (56 a^{3} b^{5}-8 a \,b^{7}\right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (70 a^{4} b^{4}-28 a^{2} b^{6}\right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (56 a^{5} b^{3}-56 a^{3} b^{5}\right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (28 a^{6} b^{2}-70 a^{4} b^{4}\right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (8 a^{7} b -56 a^{5} b^{3}\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (a^{8}-28 a^{6} b^{2}\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}-4 \left (\sin ^{2}\left (d x +c \right )\right ) a^{7} b -a^{8} \sin \left (d x +c \right )}{d}\) | \(236\) |
| parallelrisch | \(\frac {\left (42240 a^{8}-295680 a^{6} b^{2}-554400 a^{4} b^{4}-147840 a^{2} b^{6}-3630 b^{8}\right ) \sin \left (3 d x +3 c \right )+\left (-506880 a^{7} b -1330560 a^{5} b^{3}-665280 a^{3} b^{5}-55440 a \,b^{7}\right ) \cos \left (2 d x +2 c \right )+\left (-126720 a^{7} b +110880 a^{3} b^{5}+15840 a \,b^{7}\right ) \cos \left (4 d x +4 c \right )+\left (147840 a^{5} b^{3}+73920 a^{3} b^{5}+3960 a \,b^{7}\right ) \cos \left (6 d x +6 c \right )+\left (-177408 a^{6} b^{2}-110880 a^{4} b^{4}+495 b^{8}\right ) \sin \left (5 d x +5 c \right )+\left (79200 a^{4} b^{4}+23760 a^{2} b^{6}+495 b^{8}\right ) \sin \left (7 d x +7 c \right )+\left (-27720 a^{3} b^{5}-3960 a \,b^{7}\right ) \cos \left (8 d x +8 c \right )+\left (-6160 a^{2} b^{6}-275 b^{8}\right ) \sin \left (9 d x +9 c \right )+792 \cos \left (10 d x +10 c \right ) a \,b^{7}+45 \sin \left (11 d x +11 c \right ) b^{8}+\left (380160 a^{8}+1774080 a^{6} b^{2}+1663200 a^{4} b^{4}+332640 a^{2} b^{6}+6930 b^{8}\right ) \sin \left (d x +c \right )+633600 a^{7} b +1182720 a^{5} b^{3}+508200 a^{3} b^{5}+38808 a \,b^{7}}{506880 d}\) | \(367\) |
| risch | \(\frac {3 a^{8} \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (11 d x +11 c \right ) b^{8}}{11264 d}+\frac {7 \sin \left (d x +c \right ) b^{8}}{512 d}-\frac {5 \sin \left (9 d x +9 c \right ) b^{8}}{9216 d}-\frac {7 a^{3} b^{5} \cos \left (8 d x +8 c \right )}{128 d}-\frac {a^{7} b \cos \left (2 d x +2 c \right )}{d}-\frac {21 a^{5} b^{3} \cos \left (2 d x +2 c \right )}{8 d}+\frac {7 \sin \left (d x +c \right ) a^{6} b^{2}}{2 d}+\frac {7 a^{3} b^{5} \cos \left (4 d x +4 c \right )}{32 d}+\frac {7 a^{5} b^{3} \cos \left (6 d x +6 c \right )}{24 d}+\frac {7 a^{3} b^{5} \cos \left (6 d x +6 c \right )}{48 d}+\frac {a \,b^{7} \cos \left (6 d x +6 c \right )}{128 d}-\frac {21 a^{3} b^{5} \cos \left (2 d x +2 c \right )}{16 d}-\frac {7 a \,b^{7} \cos \left (2 d x +2 c \right )}{64 d}+\frac {a \,b^{7} \cos \left (10 d x +10 c \right )}{640 d}+\frac {5 \sin \left (7 d x +7 c \right ) a^{4} b^{4}}{32 d}+\frac {3 \sin \left (7 d x +7 c \right ) a^{2} b^{6}}{64 d}+\frac {\sin \left (7 d x +7 c \right ) b^{8}}{1024 d}+\frac {\sin \left (5 d x +5 c \right ) b^{8}}{1024 d}+\frac {\sin \left (3 d x +3 c \right ) a^{8}}{12 d}-\frac {11 \sin \left (3 d x +3 c \right ) b^{8}}{1536 d}-\frac {a^{7} b \cos \left (4 d x +4 c \right )}{4 d}+\frac {a \,b^{7} \cos \left (4 d x +4 c \right )}{32 d}+\frac {105 \sin \left (d x +c \right ) a^{4} b^{4}}{32 d}+\frac {21 \sin \left (d x +c \right ) a^{2} b^{6}}{32 d}-\frac {7 \sin \left (3 d x +3 c \right ) a^{6} b^{2}}{12 d}-\frac {35 \sin \left (3 d x +3 c \right ) a^{4} b^{4}}{32 d}-\frac {7 \sin \left (3 d x +3 c \right ) a^{2} b^{6}}{24 d}-\frac {7 \sin \left (5 d x +5 c \right ) a^{6} b^{2}}{20 d}-\frac {7 \sin \left (5 d x +5 c \right ) a^{4} b^{4}}{32 d}-\frac {7 \sin \left (9 d x +9 c \right ) a^{2} b^{6}}{576 d}-\frac {a \,b^{7} \cos \left (8 d x +8 c \right )}{128 d}\) | \(589\) |
-1/d*(1/11*b^8*sin(d*x+c)^11+4/5*a*b^7*sin(d*x+c)^10+1/9*(28*a^2*b^6-b^8)* sin(d*x+c)^9+1/8*(56*a^3*b^5-8*a*b^7)*sin(d*x+c)^8+1/7*(70*a^4*b^4-28*a^2* b^6)*sin(d*x+c)^7+1/6*(56*a^5*b^3-56*a^3*b^5)*sin(d*x+c)^6+1/5*(28*a^6*b^2 -70*a^4*b^4)*sin(d*x+c)^5+1/4*(8*a^7*b-56*a^5*b^3)*sin(d*x+c)^4+1/3*(a^8-2 8*a^6*b^2)*sin(d*x+c)^3-4*sin(d*x+c)^2*a^7*b-a^8*sin(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (71) = 142\).
Time = 0.31 (sec) , antiderivative size = 310, normalized size of antiderivative = 4.03 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {396 \, a b^{7} \cos \left (d x + c\right )^{10} - 495 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{8} + 660 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{6} - 990 \, {\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (45 \, b^{8} \cos \left (d x + c\right )^{10} - 10 \, {\left (154 \, a^{2} b^{6} + 17 \, b^{8}\right )} \cos \left (d x + c\right )^{8} + 330 \, a^{8} + 1848 \, a^{6} b^{2} + 1980 \, a^{4} b^{4} + 440 \, a^{2} b^{6} + 10 \, b^{8} + 10 \, {\left (495 \, a^{4} b^{4} + 418 \, a^{2} b^{6} + 23 \, b^{8}\right )} \cos \left (d x + c\right )^{6} - 12 \, {\left (231 \, a^{6} b^{2} + 660 \, a^{4} b^{4} + 275 \, a^{2} b^{6} + 10 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + {\left (165 \, a^{8} + 924 \, a^{6} b^{2} + 990 \, a^{4} b^{4} + 220 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{495 \, d} \]
1/495*(396*a*b^7*cos(d*x + c)^10 - 495*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^ 8 + 660*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^6 - 990*(a^7*b + 7 *a^5*b^3 + 7*a^3*b^5 + a*b^7)*cos(d*x + c)^4 + (45*b^8*cos(d*x + c)^10 - 1 0*(154*a^2*b^6 + 17*b^8)*cos(d*x + c)^8 + 330*a^8 + 1848*a^6*b^2 + 1980*a^ 4*b^4 + 440*a^2*b^6 + 10*b^8 + 10*(495*a^4*b^4 + 418*a^2*b^6 + 23*b^8)*cos (d*x + c)^6 - 12*(231*a^6*b^2 + 660*a^4*b^4 + 275*a^2*b^6 + 10*b^8)*cos(d* x + c)^4 + (165*a^8 + 924*a^6*b^2 + 990*a^4*b^4 + 220*a^2*b^6 + 5*b^8)*cos (d*x + c)^2)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (65) = 130\).
Time = 1.91 (sec) , antiderivative size = 468, normalized size of antiderivative = 6.08 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=\begin {cases} \frac {2 a^{8} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{8} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {2 a^{7} b \cos ^{4}{\left (c + d x \right )}}{d} + \frac {56 a^{6} b^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {28 a^{6} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} - \frac {14 a^{5} b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {14 a^{5} b^{3} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {4 a^{4} b^{4} \sin ^{7}{\left (c + d x \right )}}{d} + \frac {14 a^{4} b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {14 a^{3} b^{5} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {28 a^{3} b^{5} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {7 a^{3} b^{5} \cos ^{8}{\left (c + d x \right )}}{3 d} + \frac {8 a^{2} b^{6} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac {4 a^{2} b^{6} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {2 a b^{7} \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {2 a b^{7} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {a b^{7} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {a b^{7} \cos ^{10}{\left (c + d x \right )}}{5 d} + \frac {2 b^{8} \sin ^{11}{\left (c + d x \right )}}{99 d} + \frac {b^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{9 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{8} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((2*a**8*sin(c + d*x)**3/(3*d) + a**8*sin(c + d*x)*cos(c + d*x)** 2/d - 2*a**7*b*cos(c + d*x)**4/d + 56*a**6*b**2*sin(c + d*x)**5/(15*d) + 2 8*a**6*b**2*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) - 14*a**5*b**3*sin(c + d *x)**2*cos(c + d*x)**4/d - 14*a**5*b**3*cos(c + d*x)**6/(3*d) + 4*a**4*b** 4*sin(c + d*x)**7/d + 14*a**4*b**4*sin(c + d*x)**5*cos(c + d*x)**2/d - 14* a**3*b**5*sin(c + d*x)**4*cos(c + d*x)**4/d - 28*a**3*b**5*sin(c + d*x)**2 *cos(c + d*x)**6/(3*d) - 7*a**3*b**5*cos(c + d*x)**8/(3*d) + 8*a**2*b**6*s in(c + d*x)**9/(9*d) + 4*a**2*b**6*sin(c + d*x)**7*cos(c + d*x)**2/d - 2*a *b**7*sin(c + d*x)**6*cos(c + d*x)**4/d - 2*a*b**7*sin(c + d*x)**4*cos(c + d*x)**6/d - a*b**7*sin(c + d*x)**2*cos(c + d*x)**8/d - a*b**7*cos(c + d*x )**10/(5*d) + 2*b**8*sin(c + d*x)**11/(99*d) + b**8*sin(c + d*x)**9*cos(c + d*x)**2/(9*d), Ne(d, 0)), (x*(a + b*sin(c))**8*cos(c)**3, True))
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (71) = 142\).
Time = 0.19 (sec) , antiderivative size = 233, normalized size of antiderivative = 3.03 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {45 \, b^{8} \sin \left (d x + c\right )^{11} + 396 \, a b^{7} \sin \left (d x + c\right )^{10} - 1980 \, a^{7} b \sin \left (d x + c\right )^{2} + 55 \, {\left (28 \, a^{2} b^{6} - b^{8}\right )} \sin \left (d x + c\right )^{9} - 495 \, a^{8} \sin \left (d x + c\right ) + 495 \, {\left (7 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (d x + c\right )^{8} + 990 \, {\left (5 \, a^{4} b^{4} - 2 \, a^{2} b^{6}\right )} \sin \left (d x + c\right )^{7} + 4620 \, {\left (a^{5} b^{3} - a^{3} b^{5}\right )} \sin \left (d x + c\right )^{6} + 1386 \, {\left (2 \, a^{6} b^{2} - 5 \, a^{4} b^{4}\right )} \sin \left (d x + c\right )^{5} + 990 \, {\left (a^{7} b - 7 \, a^{5} b^{3}\right )} \sin \left (d x + c\right )^{4} + 165 \, {\left (a^{8} - 28 \, a^{6} b^{2}\right )} \sin \left (d x + c\right )^{3}}{495 \, d} \]
-1/495*(45*b^8*sin(d*x + c)^11 + 396*a*b^7*sin(d*x + c)^10 - 1980*a^7*b*si n(d*x + c)^2 + 55*(28*a^2*b^6 - b^8)*sin(d*x + c)^9 - 495*a^8*sin(d*x + c) + 495*(7*a^3*b^5 - a*b^7)*sin(d*x + c)^8 + 990*(5*a^4*b^4 - 2*a^2*b^6)*si n(d*x + c)^7 + 4620*(a^5*b^3 - a^3*b^5)*sin(d*x + c)^6 + 1386*(2*a^6*b^2 - 5*a^4*b^4)*sin(d*x + c)^5 + 990*(a^7*b - 7*a^5*b^3)*sin(d*x + c)^4 + 165* (a^8 - 28*a^6*b^2)*sin(d*x + c)^3)/d
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (71) = 142\).
Time = 0.40 (sec) , antiderivative size = 272, normalized size of antiderivative = 3.53 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {45 \, b^{8} \sin \left (d x + c\right )^{11} + 396 \, a b^{7} \sin \left (d x + c\right )^{10} + 1540 \, a^{2} b^{6} \sin \left (d x + c\right )^{9} - 55 \, b^{8} \sin \left (d x + c\right )^{9} + 3465 \, a^{3} b^{5} \sin \left (d x + c\right )^{8} - 495 \, a b^{7} \sin \left (d x + c\right )^{8} + 4950 \, a^{4} b^{4} \sin \left (d x + c\right )^{7} - 1980 \, a^{2} b^{6} \sin \left (d x + c\right )^{7} + 4620 \, a^{5} b^{3} \sin \left (d x + c\right )^{6} - 4620 \, a^{3} b^{5} \sin \left (d x + c\right )^{6} + 2772 \, a^{6} b^{2} \sin \left (d x + c\right )^{5} - 6930 \, a^{4} b^{4} \sin \left (d x + c\right )^{5} + 990 \, a^{7} b \sin \left (d x + c\right )^{4} - 6930 \, a^{5} b^{3} \sin \left (d x + c\right )^{4} + 165 \, a^{8} \sin \left (d x + c\right )^{3} - 4620 \, a^{6} b^{2} \sin \left (d x + c\right )^{3} - 1980 \, a^{7} b \sin \left (d x + c\right )^{2} - 495 \, a^{8} \sin \left (d x + c\right )}{495 \, d} \]
-1/495*(45*b^8*sin(d*x + c)^11 + 396*a*b^7*sin(d*x + c)^10 + 1540*a^2*b^6* sin(d*x + c)^9 - 55*b^8*sin(d*x + c)^9 + 3465*a^3*b^5*sin(d*x + c)^8 - 495 *a*b^7*sin(d*x + c)^8 + 4950*a^4*b^4*sin(d*x + c)^7 - 1980*a^2*b^6*sin(d*x + c)^7 + 4620*a^5*b^3*sin(d*x + c)^6 - 4620*a^3*b^5*sin(d*x + c)^6 + 2772 *a^6*b^2*sin(d*x + c)^5 - 6930*a^4*b^4*sin(d*x + c)^5 + 990*a^7*b*sin(d*x + c)^4 - 6930*a^5*b^3*sin(d*x + c)^4 + 165*a^8*sin(d*x + c)^3 - 4620*a^6*b ^2*sin(d*x + c)^3 - 1980*a^7*b*sin(d*x + c)^2 - 495*a^8*sin(d*x + c))/d
Time = 4.65 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.00 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {a^8}{3}-\frac {28\,a^6\,b^2}{3}\right )-{\sin \left (c+d\,x\right )}^5\,\left (14\,a^4\,b^4-\frac {28\,a^6\,b^2}{5}\right )-{\sin \left (c+d\,x\right )}^7\,\left (4\,a^2\,b^6-10\,a^4\,b^4\right )-a^8\,\sin \left (c+d\,x\right )-{\sin \left (c+d\,x\right )}^9\,\left (\frac {b^8}{9}-\frac {28\,a^2\,b^6}{9}\right )+\frac {b^8\,{\sin \left (c+d\,x\right )}^{11}}{11}-4\,a^7\,b\,{\sin \left (c+d\,x\right )}^2+\frac {4\,a\,b^7\,{\sin \left (c+d\,x\right )}^{10}}{5}+2\,a^5\,b\,{\sin \left (c+d\,x\right )}^4\,\left (a^2-7\,b^2\right )+a\,b^5\,{\sin \left (c+d\,x\right )}^8\,\left (7\,a^2-b^2\right )+\frac {28\,a^3\,b^3\,{\sin \left (c+d\,x\right )}^6\,\left (a^2-b^2\right )}{3}}{d} \]
-(sin(c + d*x)^3*(a^8/3 - (28*a^6*b^2)/3) - sin(c + d*x)^5*(14*a^4*b^4 - ( 28*a^6*b^2)/5) - sin(c + d*x)^7*(4*a^2*b^6 - 10*a^4*b^4) - a^8*sin(c + d*x ) - sin(c + d*x)^9*(b^8/9 - (28*a^2*b^6)/9) + (b^8*sin(c + d*x)^11)/11 - 4 *a^7*b*sin(c + d*x)^2 + (4*a*b^7*sin(c + d*x)^10)/5 + 2*a^5*b*sin(c + d*x) ^4*(a^2 - 7*b^2) + a*b^5*sin(c + d*x)^8*(7*a^2 - b^2) + (28*a^3*b^3*sin(c + d*x)^6*(a^2 - b^2))/3)/d