Integrand size = 21, antiderivative size = 210 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=-\frac {805 a^3 x}{128}-\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {603 a^3 \cos (c+d x) \sin (c+d x)}{128 d}-\frac {293 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \sin ^3(c+d x)}{3 d}-\frac {2 a^3 \sin ^5(c+d x)}{5 d}-\frac {3 a^3 \sin ^7(c+d x)}{7 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d} \]
-805/128*a^3*x-1/2*a^3*arctanh(sin(d*x+c))/d+603/128*a^3*cos(d*x+c)*sin(d* x+c)/d-293/192*a^3*cos(d*x+c)^3*sin(d*x+c)/d-1/48*a^3*cos(d*x+c)^5*sin(d*x +c)/d+1/8*a^3*cos(d*x+c)^7*sin(d*x+c)/d-1/3*a^3*sin(d*x+c)^3/d-2/5*a^3*sin (d*x+c)^5/d-3/7*a^3*sin(d*x+c)^7/d+3*a^3*tan(d*x+c)/d+1/2*a^3*sec(d*x+c)*t an(d*x+c)/d
Time = 1.79 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.74 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=\frac {a^3 \sec ^2(c+d x) \left (-1352400 c-1352400 d x-215040 \text {arctanh}(\sin (c+d x)) \cos ^2(c+d x)-1352400 (c+d x) \cos (2 (c+d x))+173600 \sin (c+d x)+1052520 \sin (2 (c+d x))-11648 \sin (3 (c+d x))+175280 \sin (4 (c+d x))+22784 \sin (5 (c+d x))-18095 \sin (6 (c+d x))-6288 \sin (7 (c+d x))+770 \sin (8 (c+d x))+720 \sin (9 (c+d x))+105 \sin (10 (c+d x))\right )}{430080 d} \]
(a^3*Sec[c + d*x]^2*(-1352400*c - 1352400*d*x - 215040*ArcTanh[Sin[c + d*x ]]*Cos[c + d*x]^2 - 1352400*(c + d*x)*Cos[2*(c + d*x)] + 173600*Sin[c + d* x] + 1052520*Sin[2*(c + d*x)] - 11648*Sin[3*(c + d*x)] + 175280*Sin[4*(c + d*x)] + 22784*Sin[5*(c + d*x)] - 18095*Sin[6*(c + d*x)] - 6288*Sin[7*(c + d*x)] + 770*Sin[8*(c + d*x)] + 720*Sin[9*(c + d*x)] + 105*Sin[10*(c + d*x )]))/(430080*d)
Time = 0.66 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4360, 25, 25, 3042, 3351, 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.
| \(\displaystyle \int \sin ^8(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^8 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \sin ^5(c+d x) \tan ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \sin ^5(c+d x) \tan ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \sin ^5(c+d x) \tan ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x+\frac {\pi }{2}\right )^8 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\cos ^8(c+d x) a^{11}+3 \cos ^7(c+d x) a^{11}-\cos ^6(c+d x) a^{11}-11 \cos ^5(c+d x) a^{11}-6 \cos ^4(c+d x) a^{11}+14 \cos ^3(c+d x) a^{11}+\sec ^3(c+d x) a^{11}+14 \cos ^2(c+d x) a^{11}+3 \sec ^2(c+d x) a^{11}-6 \cos (c+d x) a^{11}-\sec (c+d x) a^{11}-11 a^{11}\right )dx}{a^8}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {a^{11} \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^{11} \sin ^7(c+d x)}{7 d}-\frac {2 a^{11} \sin ^5(c+d x)}{5 d}-\frac {a^{11} \sin ^3(c+d x)}{3 d}+\frac {3 a^{11} \tan (c+d x)}{d}+\frac {a^{11} \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac {a^{11} \sin (c+d x) \cos ^5(c+d x)}{48 d}-\frac {293 a^{11} \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {603 a^{11} \sin (c+d x) \cos (c+d x)}{128 d}+\frac {a^{11} \tan (c+d x) \sec (c+d x)}{2 d}-\frac {805 a^{11} x}{128}}{a^8}\) |
((-805*a^11*x)/128 - (a^11*ArcTanh[Sin[c + d*x]])/(2*d) + (603*a^11*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (293*a^11*Cos[c + d*x]^3*Sin[c + d*x])/(192 *d) - (a^11*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a^11*Cos[c + d*x]^7*Sin [c + d*x])/(8*d) - (a^11*Sin[c + d*x]^3)/(3*d) - (2*a^11*Sin[c + d*x]^5)/( 5*d) - (3*a^11*Sin[c + d*x]^7)/(7*d) + (3*a^11*Tan[c + d*x])/d + (a^11*Sec [c + d*x]*Tan[c + d*x])/(2*d))/a^8
3.1.48.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/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 = 2.48 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.94
| method | result | size |
| parallelrisch | \(\frac {a^{3} \left (107520 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-107520 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-1352400 d x \cos \left (2 d x +2 c \right )-1352400 d x +22784 \sin \left (5 d x +5 c \right )-18095 \sin \left (6 d x +6 c \right )-6288 \sin \left (7 d x +7 c \right )+770 \sin \left (8 d x +8 c \right )+720 \sin \left (9 d x +9 c \right )+105 \sin \left (10 d x +10 c \right )+173600 \sin \left (d x +c \right )+1052520 \sin \left (2 d x +2 c \right )-11648 \sin \left (3 d x +3 c \right )+175280 \sin \left (4 d x +4 c \right )\right )}{215040 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(197\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{7}}{2}+\frac {7 \sin \left (d x +c \right )^{5}}{10}+\frac {7 \sin \left (d x +c \right )^{3}}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(272\) |
| default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{7}}{2}+\frac {7 \sin \left (d x +c \right )^{5}}{10}+\frac {7 \sin \left (d x +c \right )^{3}}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(272\) |
| parts | \(\frac {a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{7}}{2}+\frac {7 \sin \left (d x +c \right )^{5}}{10}+\frac {7 \sin \left (d x +c \right )^{3}}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )}{d}\) | \(280\) |
| risch | \(-\frac {805 a^{3} x}{128}+\frac {127 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {127 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}-\frac {i a^{3} \left ({\mathrm e}^{3 i \left (d x +c \right )}-6 \,{\mathrm e}^{2 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}-6\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {67 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{384 d}+\frac {47 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {67 i a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{384 d}-\frac {47 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{128 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {a^{3} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {3 a^{3} \sin \left (7 d x +7 c \right )}{448 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}-\frac {23 a^{3} \sin \left (5 d x +5 c \right )}{320 d}-\frac {23 a^{3} \sin \left (4 d x +4 c \right )}{128 d}\) | \(298\) |
1/215040*a^3*(107520*(1+cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c)-1)-107520*(1 +cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c)+1)-1352400*d*x*cos(2*d*x+2*c)-13524 00*d*x+22784*sin(5*d*x+5*c)-18095*sin(6*d*x+6*c)-6288*sin(7*d*x+7*c)+770*s in(8*d*x+8*c)+720*sin(9*d*x+9*c)+105*sin(10*d*x+10*c)+173600*sin(d*x+c)+10 52520*sin(2*d*x+2*c)-11648*sin(3*d*x+3*c)+175280*sin(4*d*x+4*c))/d/(1+cos( 2*d*x+2*c))
Time = 0.30 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.97 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=-\frac {84525 \, a^{3} d x \cos \left (d x + c\right )^{2} + 3360 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3360 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (1680 \, a^{3} \cos \left (d x + c\right )^{9} + 5760 \, a^{3} \cos \left (d x + c\right )^{8} - 280 \, a^{3} \cos \left (d x + c\right )^{7} - 22656 \, a^{3} \cos \left (d x + c\right )^{6} - 20510 \, a^{3} \cos \left (d x + c\right )^{5} + 32512 \, a^{3} \cos \left (d x + c\right )^{4} + 63315 \, a^{3} \cos \left (d x + c\right )^{3} - 15616 \, a^{3} \cos \left (d x + c\right )^{2} + 40320 \, a^{3} \cos \left (d x + c\right ) + 6720 \, a^{3}\right )} \sin \left (d x + c\right )}{13440 \, d \cos \left (d x + c\right )^{2}} \]
-1/13440*(84525*a^3*d*x*cos(d*x + c)^2 + 3360*a^3*cos(d*x + c)^2*log(sin(d *x + c) + 1) - 3360*a^3*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - (1680*a^3* cos(d*x + c)^9 + 5760*a^3*cos(d*x + c)^8 - 280*a^3*cos(d*x + c)^7 - 22656* a^3*cos(d*x + c)^6 - 20510*a^3*cos(d*x + c)^5 + 32512*a^3*cos(d*x + c)^4 + 63315*a^3*cos(d*x + c)^3 - 15616*a^3*cos(d*x + c)^2 + 40320*a^3*cos(d*x + c) + 6720*a^3)*sin(d*x + c))/(d*cos(d*x + c)^2)
Timed out. \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.39 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=-\frac {1536 \, {\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a^{3} - 1792 \, {\left (12 \, \sin \left (d x + c\right )^{5} + 40 \, \sin \left (d x + c\right )^{3} - \frac {30 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 180 \, \sin \left (d x + c\right )\right )} a^{3} - 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 6720 \, {\left (105 \, d x + 105 \, c - \frac {87 \, \tan \left (d x + c\right )^{5} + 136 \, \tan \left (d x + c\right )^{3} + 57 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1} - 48 \, \tan \left (d x + c\right )\right )} a^{3}}{107520 \, d} \]
-1/107520*(1536*(30*sin(d*x + c)^7 + 42*sin(d*x + c)^5 + 70*sin(d*x + c)^3 - 105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1) + 210*sin(d*x + c ))*a^3 - 1792*(12*sin(d*x + c)^5 + 40*sin(d*x + c)^3 - 30*sin(d*x + c)/(si n(d*x + c)^2 - 1) - 105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1) + 180*sin(d*x + c))*a^3 - 35*(128*sin(2*d*x + 2*c)^3 + 840*d*x + 840*c + 3 *sin(8*d*x + 8*c) + 168*sin(4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*a^3 + 672 0*(105*d*x + 105*c - (87*tan(d*x + c)^5 + 136*tan(d*x + c)^3 + 57*tan(d*x + c))/(tan(d*x + c)^6 + 3*tan(d*x + c)^4 + 3*tan(d*x + c)^2 + 1) - 48*tan( d*x + c))*a^3)/d
Time = 0.47 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.16 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=-\frac {84525 \, {\left (d x + c\right )} a^{3} + 6720 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 6720 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {13440 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {2 \, {\left (44205 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 303065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 841981 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1123793 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 487983 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 490749 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 267225 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 44205 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8}}}{13440 \, d} \]
-1/13440*(84525*(d*x + c)*a^3 + 6720*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1) ) - 6720*a^3*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 13440*(5*a^3*tan(1/2*d*x + 1/2*c)^3 - 7*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2 + 2*(44205*a^3*tan(1/2*d*x + 1/2*c)^15 + 303065*a^3*tan(1/2*d*x + 1/2*c)^13 + 841981*a^3*tan(1/2*d*x + 1/2*c)^11 + 1123793*a^3*tan(1/2*d*x + 1/2*c)^9 + 487983*a^3*tan(1/2*d*x + 1/2*c)^7 - 490749*a^3*tan(1/2*d*x + 1/2*c)^5 - 267225*a^3*tan(1/2*d*x + 1/2*c)^3 - 44205*a^3*tan(1/2*d*x + 1/2*c))/(tan( 1/2*d*x + 1/2*c)^2 + 1)^8)/d
Time = 14.84 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.52 \[ \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx=\frac {-\frac {741\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{64}-\frac {12469\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{192}-\frac {5027\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{40}-\frac {19211\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{420}+\frac {199977\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{1120}+\frac {877061\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3360}+\frac {10233\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{140}+\frac {6243\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40}+\frac {4967\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}+\frac {869\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {805\,a^3\,x}{128} \]
((4967*a^3*tan(c/2 + (d*x)/2)^3)/64 + (6243*a^3*tan(c/2 + (d*x)/2)^5)/40 + (10233*a^3*tan(c/2 + (d*x)/2)^7)/140 + (877061*a^3*tan(c/2 + (d*x)/2)^9)/ 3360 + (199977*a^3*tan(c/2 + (d*x)/2)^11)/1120 - (19211*a^3*tan(c/2 + (d*x )/2)^13)/420 - (5027*a^3*tan(c/2 + (d*x)/2)^15)/40 - (12469*a^3*tan(c/2 + (d*x)/2)^17)/192 - (741*a^3*tan(c/2 + (d*x)/2)^19)/64 + (869*a^3*tan(c/2 + (d*x)/2))/64)/(d*(6*tan(c/2 + (d*x)/2)^2 + 13*tan(c/2 + (d*x)/2)^4 + 8*ta n(c/2 + (d*x)/2)^6 - 14*tan(c/2 + (d*x)/2)^8 - 28*tan(c/2 + (d*x)/2)^10 - 14*tan(c/2 + (d*x)/2)^12 + 8*tan(c/2 + (d*x)/2)^14 + 13*tan(c/2 + (d*x)/2) ^16 + 6*tan(c/2 + (d*x)/2)^18 + tan(c/2 + (d*x)/2)^20 + 1)) - (a^3*atanh(t an(c/2 + (d*x)/2)))/d - (805*a^3*x)/128