Integrand size = 21, antiderivative size = 203 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=\frac {11 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{d}-\frac {14 a^3 \cos ^3(c+d x)}{3 d}-\frac {7 a^3 \cos ^4(c+d x)}{2 d}+\frac {6 a^3 \cos ^5(c+d x)}{5 d}+\frac {11 a^3 \cos ^6(c+d x)}{6 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {3 a^3 \cos ^8(c+d x)}{8 d}-\frac {a^3 \cos ^9(c+d x)}{9 d}+\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
11*a^3*cos(d*x+c)/d+3*a^3*cos(d*x+c)^2/d-14/3*a^3*cos(d*x+c)^3/d-7/2*a^3*c os(d*x+c)^4/d+6/5*a^3*cos(d*x+c)^5/d+11/6*a^3*cos(d*x+c)^6/d+1/7*a^3*cos(d *x+c)^7/d-3/8*a^3*cos(d*x+c)^8/d-1/9*a^3*cos(d*x+c)^9/d+a^3*ln(cos(d*x+c)) /d+3*a^3*sec(d*x+c)/d+1/2*a^3*sec(d*x+c)^2/d
Time = 1.93 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.73 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=\frac {a^3 (471450+11624760 \cos (c+d x)+2188872 \cos (3 (c+d x))+41160 \cos (4 (c+d x))-204156 \cos (5 (c+d x))-35805 \cos (6 (c+d x))+22972 \cos (7 (c+d x))+9030 \cos (8 (c+d x))-820 \cos (9 (c+d x))-945 \cos (10 (c+d x))-140 \cos (11 (c+d x))+645120 \log (\cos (c+d x))+210 \cos (2 (c+d x)) (-413+3072 \log (\cos (c+d x)))) \sec ^2(c+d x)}{1290240 d} \]
(a^3*(471450 + 11624760*Cos[c + d*x] + 2188872*Cos[3*(c + d*x)] + 41160*Co s[4*(c + d*x)] - 204156*Cos[5*(c + d*x)] - 35805*Cos[6*(c + d*x)] + 22972* Cos[7*(c + d*x)] + 9030*Cos[8*(c + d*x)] - 820*Cos[9*(c + d*x)] - 945*Cos[ 10*(c + d*x)] - 140*Cos[11*(c + d*x)] + 645120*Log[Cos[c + d*x]] + 210*Cos [2*(c + d*x)]*(-413 + 3072*Log[Cos[c + d*x]]))*Sec[c + d*x]^2)/(1290240*d)
Time = 0.47 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 4360, 25, 25, 3042, 25, 3315, 27, 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.
| \(\displaystyle \int \sin ^9(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^9 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \sin ^6(c+d x) \tan ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \sin ^6(c+d x) \tan ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \sin ^6(c+d x) \tan ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )^9 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^9 \left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle -\frac {\int (a-a \cos (c+d x))^4 (\cos (c+d x) a+a)^7 \sec ^3(c+d x)d(a \cos (c+d x))}{a^9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(a-a \cos (c+d x))^4 (\cos (c+d x) a+a)^7 \sec ^3(c+d x)}{a^3}d(a \cos (c+d x))}{a^6 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {\int \left (\cos ^8(c+d x) a^8+3 \cos ^7(c+d x) a^8-\cos ^6(c+d x) a^8-11 \cos ^5(c+d x) a^8-6 \cos ^4(c+d x) a^8+14 \cos ^3(c+d x) a^8+\sec ^3(c+d x) a^8+14 \cos ^2(c+d x) a^8+3 \sec ^2(c+d x) a^8-6 \cos (c+d x) a^8-\sec (c+d x) a^8-11 a^8\right )d(a \cos (c+d x))}{a^6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{9} a^9 \cos ^9(c+d x)+\frac {3}{8} a^9 \cos ^8(c+d x)-\frac {1}{7} a^9 \cos ^7(c+d x)-\frac {11}{6} a^9 \cos ^6(c+d x)-\frac {6}{5} a^9 \cos ^5(c+d x)+\frac {7}{2} a^9 \cos ^4(c+d x)+\frac {14}{3} a^9 \cos ^3(c+d x)-3 a^9 \cos ^2(c+d x)-11 a^9 \cos (c+d x)-\frac {1}{2} a^9 \sec ^2(c+d x)-3 a^9 \sec (c+d x)-a^9 \log (a \cos (c+d x))}{a^6 d}\) |
-((-11*a^9*Cos[c + d*x] - 3*a^9*Cos[c + d*x]^2 + (14*a^9*Cos[c + d*x]^3)/3 + (7*a^9*Cos[c + d*x]^4)/2 - (6*a^9*Cos[c + d*x]^5)/5 - (11*a^9*Cos[c + d *x]^6)/6 - (a^9*Cos[c + d*x]^7)/7 + (3*a^9*Cos[c + d*x]^8)/8 + (a^9*Cos[c + d*x]^9)/9 - a^9*Log[a*Cos[c + d*x]] - 3*a^9*Sec[c + d*x] - (a^9*Sec[c + d*x]^2)/2)/(a^6*d))
3.1.38.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 3.02 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.07
| method | result | size |
| parallelrisch | \(\frac {a^{3} \left (-645120 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+645120 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+645120 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-35805 \cos \left (6 d x +6 c \right )+22972 \cos \left (7 d x +7 c \right )+9030 \cos \left (8 d x +8 c \right )-820 \cos \left (9 d x +9 c \right )-945 \cos \left (10 d x +10 c \right )-140 \cos \left (11 d x +11 c \right )+11624760 \cos \left (d x +c \right )+6529934 \cos \left (2 d x +2 c \right )+2188872 \cos \left (3 d x +3 c \right )+41160 \cos \left (4 d x +4 c \right )-204156 \cos \left (5 d x +5 c \right )+7088114\right )}{645120 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(217\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{8}}{2}+\frac {2 \sin \left (d x +c \right )^{6}}{3}+\sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{2}+4 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(252\) |
| default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{8}}{2}+\frac {2 \sin \left (d x +c \right )^{6}}{3}+\sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{2}+4 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(252\) |
| parts | \(-\frac {a^{3} \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9 d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{8}}{2}+\frac {2 \sin \left (d x +c \right )^{6}}{3}+\sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{2}+4 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(260\) |
| risch | \(-\frac {2 i a^{3} c}{d}-i a^{3} x -\frac {a^{3} \cos \left (9 d x +9 c \right )}{2304 d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {25 a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{64 d}+\frac {57 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{256 d}+\frac {1059 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{256 d}+\frac {1059 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{256 d}+\frac {57 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{256 d}-\frac {25 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{64 d}+\frac {2 a^{3} \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {3 a^{3} \cos \left (8 d x +8 c \right )}{1024 d}-\frac {3 a^{3} \cos \left (7 d x +7 c \right )}{1792 d}+\frac {13 a^{3} \cos \left (6 d x +6 c \right )}{384 d}+\frac {3 a^{3} \cos \left (5 d x +5 c \right )}{40 d}-\frac {45 a^{3} \cos \left (4 d x +4 c \right )}{256 d}\) | \(295\) |
1/645120*a^3*(-645120*(1+cos(2*d*x+2*c))*ln(sec(1/2*d*x+1/2*c)^2)+645120*( 1+cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c)-1)+645120*(1+cos(2*d*x+2*c))*ln(ta n(1/2*d*x+1/2*c)+1)-35805*cos(6*d*x+6*c)+22972*cos(7*d*x+7*c)+9030*cos(8*d *x+8*c)-820*cos(9*d*x+9*c)-945*cos(10*d*x+10*c)-140*cos(11*d*x+11*c)+11624 760*cos(d*x+c)+6529934*cos(2*d*x+2*c)+2188872*cos(3*d*x+3*c)+41160*cos(4*d *x+4*c)-204156*cos(5*d*x+5*c)+7088114)/d/(1+cos(2*d*x+2*c))
Time = 0.31 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.90 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=-\frac {35840 \, a^{3} \cos \left (d x + c\right )^{11} + 120960 \, a^{3} \cos \left (d x + c\right )^{10} - 46080 \, a^{3} \cos \left (d x + c\right )^{9} - 591360 \, a^{3} \cos \left (d x + c\right )^{8} - 387072 \, a^{3} \cos \left (d x + c\right )^{7} + 1128960 \, a^{3} \cos \left (d x + c\right )^{6} + 1505280 \, a^{3} \cos \left (d x + c\right )^{5} - 967680 \, a^{3} \cos \left (d x + c\right )^{4} - 3548160 \, a^{3} \cos \left (d x + c\right )^{3} - 322560 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + 212205 \, a^{3} \cos \left (d x + c\right )^{2} - 967680 \, a^{3} \cos \left (d x + c\right ) - 161280 \, a^{3}}{322560 \, d \cos \left (d x + c\right )^{2}} \]
-1/322560*(35840*a^3*cos(d*x + c)^11 + 120960*a^3*cos(d*x + c)^10 - 46080* a^3*cos(d*x + c)^9 - 591360*a^3*cos(d*x + c)^8 - 387072*a^3*cos(d*x + c)^7 + 1128960*a^3*cos(d*x + c)^6 + 1505280*a^3*cos(d*x + c)^5 - 967680*a^3*co s(d*x + c)^4 - 3548160*a^3*cos(d*x + c)^3 - 322560*a^3*cos(d*x + c)^2*log( -cos(d*x + c)) + 212205*a^3*cos(d*x + c)^2 - 967680*a^3*cos(d*x + c) - 161 280*a^3)/(d*cos(d*x + c)^2)
Timed out. \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.78 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=-\frac {280 \, a^{3} \cos \left (d x + c\right )^{9} + 945 \, a^{3} \cos \left (d x + c\right )^{8} - 360 \, a^{3} \cos \left (d x + c\right )^{7} - 4620 \, a^{3} \cos \left (d x + c\right )^{6} - 3024 \, a^{3} \cos \left (d x + c\right )^{5} + 8820 \, a^{3} \cos \left (d x + c\right )^{4} + 11760 \, a^{3} \cos \left (d x + c\right )^{3} - 7560 \, a^{3} \cos \left (d x + c\right )^{2} - 27720 \, a^{3} \cos \left (d x + c\right ) - 2520 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {1260 \, {\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{2520 \, d} \]
-1/2520*(280*a^3*cos(d*x + c)^9 + 945*a^3*cos(d*x + c)^8 - 360*a^3*cos(d*x + c)^7 - 4620*a^3*cos(d*x + c)^6 - 3024*a^3*cos(d*x + c)^5 + 8820*a^3*cos (d*x + c)^4 + 11760*a^3*cos(d*x + c)^3 - 7560*a^3*cos(d*x + c)^2 - 27720*a ^3*cos(d*x + c) - 2520*a^3*log(cos(d*x + c)) - 1260*(6*a^3*cos(d*x + c) + a^3)/cos(d*x + c)^2)/d
Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (187) = 374\).
Time = 0.49 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.95 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=-\frac {2520 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {1260 \, {\left (9 \, a^{3} + \frac {2 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}} + \frac {45257 \, a^{3} - \frac {392193 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {1467972 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3001908 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3232782 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2359854 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1190196 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {397764 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {79281 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {7129 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}}}{2520 \, d} \]
-1/2520*(2520*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 2 520*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) - 1260*(9*a^3 + 2*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 3*a^3*(cos(d*x + c) - 1)^ 2/(cos(d*x + c) + 1)^2)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^2 + (4 5257*a^3 - 392193*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1467972*a^3* (cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 3001908*a^3*(cos(d*x + c) - 1) ^3/(cos(d*x + c) + 1)^3 + 3232782*a^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 2359854*a^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 1190196*a^ 3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 397764*a^3*(cos(d*x + c) - 1 )^7/(cos(d*x + c) + 1)^7 + 79281*a^3*(cos(d*x + c) - 1)^8/(cos(d*x + c) + 1)^8 - 7129*a^3*(cos(d*x + c) - 1)^9/(cos(d*x + c) + 1)^9)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^9)/d
Time = 12.63 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.77 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=\frac {\frac {3\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{{\cos \left (c+d\,x\right )}^2}+11\,a^3\,\cos \left (c+d\,x\right )+3\,a^3\,{\cos \left (c+d\,x\right )}^2-\frac {14\,a^3\,{\cos \left (c+d\,x\right )}^3}{3}-\frac {7\,a^3\,{\cos \left (c+d\,x\right )}^4}{2}+\frac {6\,a^3\,{\cos \left (c+d\,x\right )}^5}{5}+\frac {11\,a^3\,{\cos \left (c+d\,x\right )}^6}{6}+\frac {a^3\,{\cos \left (c+d\,x\right )}^7}{7}-\frac {3\,a^3\,{\cos \left (c+d\,x\right )}^8}{8}-\frac {a^3\,{\cos \left (c+d\,x\right )}^9}{9}+a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
((3*a^3*cos(c + d*x) + a^3/2)/cos(c + d*x)^2 + 11*a^3*cos(c + d*x) + 3*a^3 *cos(c + d*x)^2 - (14*a^3*cos(c + d*x)^3)/3 - (7*a^3*cos(c + d*x)^4)/2 + ( 6*a^3*cos(c + d*x)^5)/5 + (11*a^3*cos(c + d*x)^6)/6 + (a^3*cos(c + d*x)^7) /7 - (3*a^3*cos(c + d*x)^8)/8 - (a^3*cos(c + d*x)^9)/9 + a^3*log(cos(c + d *x)))/d