Integrand size = 19, antiderivative size = 165 \[ \int (a+a \sec (c+d x)) \sin ^8(c+d x) \, dx=\frac {35 a x}{128}+\frac {a \text {arctanh}(\sin (c+d x))}{d}-\frac {a \sin (c+d x)}{d}-\frac {35 a \cos (c+d x) \sin (c+d x)}{128 d}-\frac {a \sin ^3(c+d x)}{3 d}-\frac {35 a \cos (c+d x) \sin ^3(c+d x)}{192 d}-\frac {a \sin ^5(c+d x)}{5 d}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{48 d}-\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \cos (c+d x) \sin ^7(c+d x)}{8 d} \]
35/128*a*x+a*arctanh(sin(d*x+c))/d-a*sin(d*x+c)/d-35/128*a*cos(d*x+c)*sin( d*x+c)/d-1/3*a*sin(d*x+c)^3/d-35/192*a*cos(d*x+c)*sin(d*x+c)^3/d-1/5*a*sin (d*x+c)^5/d-7/48*a*cos(d*x+c)*sin(d*x+c)^5/d-1/7*a*sin(d*x+c)^7/d-1/8*a*co s(d*x+c)*sin(d*x+c)^7/d
Time = 1.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.64 \[ \int (a+a \sec (c+d x)) \sin ^8(c+d x) \, dx=\frac {a \left (107520 \text {arctanh}(\sin (c+d x))-107520 \sin (c+d x)-35840 \sin ^3(c+d x)-21504 \sin ^5(c+d x)-15360 \sin ^7(c+d x)+35 (840 c+840 d x-672 \sin (2 (c+d x))+168 \sin (4 (c+d x))-32 \sin (6 (c+d x))+3 \sin (8 (c+d x)))\right )}{107520 d} \]
(a*(107520*ArcTanh[Sin[c + d*x]] - 107520*Sin[c + d*x] - 35840*Sin[c + d*x ]^3 - 21504*Sin[c + d*x]^5 - 15360*Sin[c + d*x]^7 + 35*(840*c + 840*d*x - 672*Sin[2*(c + d*x)] + 168*Sin[4*(c + d*x)] - 32*Sin[6*(c + d*x)] + 3*Sin[ 8*(c + d*x)])))/(107520*d)
Time = 0.70 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.99, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.947, Rules used = {3042, 4360, 25, 25, 3042, 3317, 3042, 3072, 254, 2009, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \sin ^8(c+d x) (a \sec (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^8 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\left (\sin ^7(c+d x) \tan (c+d x) (a (-\cos (c+d x))-a)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\left ((\cos (c+d x) a+a) \sin ^7(c+d x) \tan (c+d x)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \sin ^7(c+d x) \tan (c+d x) (a \cos (c+d x)+a)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x+\frac {\pi }{2}\right )^8 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \sin ^8(c+d x)dx+a \int \sin ^7(c+d x) \tan (c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \sin (c+d x)^8dx+a \int \sin (c+d x)^7 \tan (c+d x)dx\) |
| \(\Big \downarrow \) 3072 |
| \(\displaystyle \frac {a \int \frac {\sin ^8(c+d x)}{1-\sin ^2(c+d x)}d\sin (c+d x)}{d}+a \int \sin (c+d x)^8dx\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {a \int \left (-\sin ^6(c+d x)-\sin ^4(c+d x)-\sin ^2(c+d x)+\frac {1}{1-\sin ^2(c+d x)}-1\right )d\sin (c+d x)}{d}+a \int \sin (c+d x)^8dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \int \sin (c+d x)^8dx+\frac {a \left (\text {arctanh}(\sin (c+d x))-\frac {1}{7} \sin ^7(c+d x)-\frac {1}{5} \sin ^5(c+d x)-\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {7}{8} \int \sin ^6(c+d x)dx-\frac {\sin ^7(c+d x) \cos (c+d x)}{8 d}\right )+\frac {a \left (\text {arctanh}(\sin (c+d x))-\frac {1}{7} \sin ^7(c+d x)-\frac {1}{5} \sin ^5(c+d x)-\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {7}{8} \int \sin (c+d x)^6dx-\frac {\sin ^7(c+d x) \cos (c+d x)}{8 d}\right )+\frac {a \left (\text {arctanh}(\sin (c+d x))-\frac {1}{7} \sin ^7(c+d x)-\frac {1}{5} \sin ^5(c+d x)-\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {7}{8} \left (\frac {5}{6} \int \sin ^4(c+d x)dx-\frac {\sin ^5(c+d x) \cos (c+d x)}{6 d}\right )-\frac {\sin ^7(c+d x) \cos (c+d x)}{8 d}\right )+\frac {a \left (\text {arctanh}(\sin (c+d x))-\frac {1}{7} \sin ^7(c+d x)-\frac {1}{5} \sin ^5(c+d x)-\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {7}{8} \left (\frac {5}{6} \int \sin (c+d x)^4dx-\frac {\sin ^5(c+d x) \cos (c+d x)}{6 d}\right )-\frac {\sin ^7(c+d x) \cos (c+d x)}{8 d}\right )+\frac {a \left (\text {arctanh}(\sin (c+d x))-\frac {1}{7} \sin ^7(c+d x)-\frac {1}{5} \sin ^5(c+d x)-\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 d}\right )-\frac {\sin ^5(c+d x) \cos (c+d x)}{6 d}\right )-\frac {\sin ^7(c+d x) \cos (c+d x)}{8 d}\right )+\frac {a \left (\text {arctanh}(\sin (c+d x))-\frac {1}{7} \sin ^7(c+d x)-\frac {1}{5} \sin ^5(c+d x)-\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 d}\right )-\frac {\sin ^5(c+d x) \cos (c+d x)}{6 d}\right )-\frac {\sin ^7(c+d x) \cos (c+d x)}{8 d}\right )+\frac {a \left (\text {arctanh}(\sin (c+d x))-\frac {1}{7} \sin ^7(c+d x)-\frac {1}{5} \sin ^5(c+d x)-\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 d}\right )-\frac {\sin ^5(c+d x) \cos (c+d x)}{6 d}\right )-\frac {\sin ^7(c+d x) \cos (c+d x)}{8 d}\right )+\frac {a \left (\text {arctanh}(\sin (c+d x))-\frac {1}{7} \sin ^7(c+d x)-\frac {1}{5} \sin ^5(c+d x)-\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {a \left (\text {arctanh}(\sin (c+d x))-\frac {1}{7} \sin ^7(c+d x)-\frac {1}{5} \sin ^5(c+d x)-\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}+a \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 d}\right )-\frac {\sin ^5(c+d x) \cos (c+d x)}{6 d}\right )-\frac {\sin ^7(c+d x) \cos (c+d x)}{8 d}\right )\) |
(a*(ArcTanh[Sin[c + d*x]] - Sin[c + d*x] - Sin[c + d*x]^3/3 - Sin[c + d*x] ^5/5 - Sin[c + d*x]^7/7))/d + a*(-1/8*(Cos[c + d*x]*Sin[c + d*x]^7)/d + (7 *(-1/6*(Cos[c + d*x]*Sin[c + d*x]^5)/d + (5*(-1/4*(Cos[c + d*x]*Sin[c + d* x]^3)/d + (3*(x/2 - (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/6))/8)
3.1.10.3.1 Defintions of rubi rules used
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ (ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)], x ]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
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 = 1.86 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(116\) |
| default | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(116\) |
| parts | \(\frac {a \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {a \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(118\) |
| parallelrisch | \(-\frac {a \left (-\frac {105 d x}{4}+\sin \left (6 d x +6 c \right )-\frac {3 \sin \left (7 d x +7 c \right )}{14}-\frac {3 \sin \left (8 d x +8 c \right )}{32}+96 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-96 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {279 \sin \left (d x +c \right )}{2}+21 \sin \left (2 d x +2 c \right )-\frac {37 \sin \left (3 d x +3 c \right )}{2}-\frac {21 \sin \left (4 d x +4 c \right )}{4}+\frac {27 \sin \left (5 d x +5 c \right )}{10}\right )}{96 d}\) | \(123\) |
| risch | \(\frac {35 a x}{128}-\frac {93 i a \,{\mathrm e}^{-i \left (d x +c \right )}}{128 d}+\frac {93 i a \,{\mathrm e}^{i \left (d x +c \right )}}{128 d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {a \sin \left (8 d x +8 c \right )}{1024 d}+\frac {a \sin \left (7 d x +7 c \right )}{448 d}-\frac {a \sin \left (6 d x +6 c \right )}{96 d}-\frac {9 a \sin \left (5 d x +5 c \right )}{320 d}+\frac {7 a \sin \left (4 d x +4 c \right )}{128 d}+\frac {37 a \sin \left (3 d x +3 c \right )}{192 d}-\frac {7 a \sin \left (2 d x +2 c \right )}{32 d}\) | \(180\) |
| norman | \(\frac {\frac {35 a x}{128}-\frac {163 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {1335 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{64 d}-\frac {24223 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{320 d}-\frac {359453 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2240 d}-\frac {724649 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{6720 d}-\frac {45859 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{960 d}-\frac {2395 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{192 d}-\frac {93 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 d}+\frac {35 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16}+\frac {245 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32}+\frac {245 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16}+\frac {1225 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64}+\frac {245 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16}+\frac {245 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{32}+\frac {35 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16}+\frac {35 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{128}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(312\) |
1/d*(a*(-1/7*sin(d*x+c)^7-1/5*sin(d*x+c)^5-1/3*sin(d*x+c)^3-sin(d*x+c)+ln( sec(d*x+c)+tan(d*x+c)))+a*(-1/8*(sin(d*x+c)^7+7/6*sin(d*x+c)^5+35/24*sin(d *x+c)^3+35/16*sin(d*x+c))*cos(d*x+c)+35/128*d*x+35/128*c))
Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.75 \[ \int (a+a \sec (c+d x)) \sin ^8(c+d x) \, dx=\frac {3675 \, a d x + 6720 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) - 6720 \, a \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (1680 \, a \cos \left (d x + c\right )^{7} + 1920 \, a \cos \left (d x + c\right )^{6} - 7000 \, a \cos \left (d x + c\right )^{5} - 8448 \, a \cos \left (d x + c\right )^{4} + 11410 \, a \cos \left (d x + c\right )^{3} + 15616 \, a \cos \left (d x + c\right )^{2} - 9765 \, a \cos \left (d x + c\right ) - 22528 \, a\right )} \sin \left (d x + c\right )}{13440 \, d} \]
1/13440*(3675*a*d*x + 6720*a*log(sin(d*x + c) + 1) - 6720*a*log(-sin(d*x + c) + 1) + (1680*a*cos(d*x + c)^7 + 1920*a*cos(d*x + c)^6 - 7000*a*cos(d*x + c)^5 - 8448*a*cos(d*x + c)^4 + 11410*a*cos(d*x + c)^3 + 15616*a*cos(d*x + c)^2 - 9765*a*cos(d*x + c) - 22528*a)*sin(d*x + c))/d
Timed out. \[ \int (a+a \sec (c+d x)) \sin ^8(c+d x) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.77 \[ \int (a+a \sec (c+d x)) \sin ^8(c+d x) \, dx=-\frac {512 \, {\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a - 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a}{107520 \, d} \]
-1/107520*(512*(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 - 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)/d
Time = 0.31 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.05 \[ \int (a+a \sec (c+d x)) \sin ^8(c+d x) \, dx=\frac {3675 \, {\left (d x + c\right )} a + 13440 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 13440 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (9765 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 83825 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 321013 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 724649 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1078359 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 508683 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 140175 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 17115 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8}}}{13440 \, d} \]
1/13440*(3675*(d*x + c)*a + 13440*a*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 1 3440*a*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(9765*a*tan(1/2*d*x + 1/2*c) ^15 + 83825*a*tan(1/2*d*x + 1/2*c)^13 + 321013*a*tan(1/2*d*x + 1/2*c)^11 + 724649*a*tan(1/2*d*x + 1/2*c)^9 + 1078359*a*tan(1/2*d*x + 1/2*c)^7 + 5086 83*a*tan(1/2*d*x + 1/2*c)^5 + 140175*a*tan(1/2*d*x + 1/2*c)^3 + 17115*a*ta n(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^8)/d
Time = 13.77 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.91 \[ \int (a+a \sec (c+d x)) \sin ^8(c+d x) \, dx=\frac {35\,a\,x}{128}+\frac {2\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {7\,a\,\sin \left (2\,c+2\,d\,x\right )}{32\,d}+\frac {37\,a\,\sin \left (3\,c+3\,d\,x\right )}{192\,d}+\frac {7\,a\,\sin \left (4\,c+4\,d\,x\right )}{128\,d}-\frac {9\,a\,\sin \left (5\,c+5\,d\,x\right )}{320\,d}-\frac {a\,\sin \left (6\,c+6\,d\,x\right )}{96\,d}+\frac {a\,\sin \left (7\,c+7\,d\,x\right )}{448\,d}+\frac {a\,\sin \left (8\,c+8\,d\,x\right )}{1024\,d}-\frac {93\,a\,\sin \left (c+d\,x\right )}{64\,d} \]
(35*a*x)/128 + (2*a*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d - (7*a *sin(2*c + 2*d*x))/(32*d) + (37*a*sin(3*c + 3*d*x))/(192*d) + (7*a*sin(4*c + 4*d*x))/(128*d) - (9*a*sin(5*c + 5*d*x))/(320*d) - (a*sin(6*c + 6*d*x)) /(96*d) + (a*sin(7*c + 7*d*x))/(448*d) + (a*sin(8*c + 8*d*x))/(1024*d) - ( 93*a*sin(c + d*x))/(64*d)