Integrand size = 21, antiderivative size = 125 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {5 x}{128 a}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac {5 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d} \]
-5/128*x/a-5/128*cos(d*x+c)*sin(d*x+c)/a/d+5/64*cos(d*x+c)^3*sin(d*x+c)/a/ d+5/48*cos(d*x+c)^3*sin(d*x+c)^3/a/d+1/8*cos(d*x+c)^3*sin(d*x+c)^5/a/d+1/7 *sin(d*x+c)^7/a/d
Time = 1.61 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (1176 c-840 d x+1680 \sin (c+d x)+336 \sin (2 (c+d x))-1008 \sin (3 (c+d x))+168 \sin (4 (c+d x))+336 \sin (5 (c+d x))-112 \sin (6 (c+d x))-48 \sin (7 (c+d x))+21 \sin (8 (c+d x))-1176 \tan \left (\frac {c}{2}\right )\right )}{10752 a d (1+\sec (c+d x))} \]
(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(1176*c - 840*d*x + 1680*Sin[c + d*x] + 3 36*Sin[2*(c + d*x)] - 1008*Sin[3*(c + d*x)] + 168*Sin[4*(c + d*x)] + 336*S in[5*(c + d*x)] - 112*Sin[6*(c + d*x)] - 48*Sin[7*(c + d*x)] + 21*Sin[8*(c + d*x)] - 1176*Tan[c/2]))/(10752*a*d*(1 + Sec[c + d*x]))
Time = 0.76 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 4360, 25, 25, 3042, 3318, 3042, 3044, 15, 3048, 3042, 3048, 3042, 3048, 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.
| \(\displaystyle \int \frac {\sin ^8(c+d x)}{a \sec (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^8}{a-a \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\sin ^8(c+d x) \cos (c+d x)}{a (-\cos (c+d x))-a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos (c+d x) \sin ^8(c+d x)}{\cos (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sin ^8(c+d x) \cos (c+d x)}{a \cos (c+d x)+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \cos \left (c+d x+\frac {\pi }{2}\right )^8}{a \sin \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cos (c+d x) \sin ^6(c+d x)dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^6(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos (c+d x) \sin (c+d x)^6dx}{a}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^6dx}{a}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {\int \sin ^6(c+d x)d\sin (c+d x)}{a d}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^6dx}{a}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\sin ^7(c+d x)}{7 a d}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^6dx}{a}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\sin ^7(c+d x)}{7 a d}-\frac {\frac {5}{8} \int \cos ^2(c+d x) \sin ^4(c+d x)dx-\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^7(c+d x)}{7 a d}-\frac {\frac {5}{8} \int \cos (c+d x)^2 \sin (c+d x)^4dx-\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\sin ^7(c+d x)}{7 a d}-\frac {\frac {5}{8} \left (\frac {1}{2} \int \cos ^2(c+d x) \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^7(c+d x)}{7 a d}-\frac {\frac {5}{8} \left (\frac {1}{2} \int \cos (c+d x)^2 \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\sin ^7(c+d x)}{7 a d}-\frac {\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \int \cos ^2(c+d x)dx-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^7(c+d x)}{7 a d}-\frac {\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\sin ^7(c+d x)}{7 a d}-\frac {\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\sin ^7(c+d x)}{7 a d}-\frac {\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 d}}{a}\) |
Sin[c + d*x]^7/(7*a*d) - (-1/8*(Cos[c + d*x]^3*Sin[c + d*x]^5)/d + (5*(-1/ 6*(Cos[c + d*x]^3*Sin[c + d*x]^3)/d + (-1/4*(Cos[c + d*x]^3*Sin[c + d*x])/ d + (x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d))/4)/2))/8)/a
3.1.65.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && 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 = 0.67 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.79
| method | result | size |
| parallelrisch | \(\frac {-840 d x +1680 \sin \left (d x +c \right )-48 \sin \left (7 d x +7 c \right )+336 \sin \left (5 d x +5 c \right )-1008 \sin \left (3 d x +3 c \right )+21 \sin \left (8 d x +8 c \right )-112 \sin \left (6 d x +6 c \right )+168 \sin \left (4 d x +4 c \right )+336 \sin \left (2 d x +2 c \right )}{21504 d a}\) | \(99\) |
| risch | \(-\frac {5 x}{128 a}+\frac {5 \sin \left (d x +c \right )}{64 a d}+\frac {\sin \left (8 d x +8 c \right )}{1024 d a}-\frac {\sin \left (7 d x +7 c \right )}{448 d a}-\frac {\sin \left (6 d x +6 c \right )}{192 d a}+\frac {\sin \left (5 d x +5 c \right )}{64 d a}+\frac {\sin \left (4 d x +4 c \right )}{128 d a}-\frac {3 \sin \left (3 d x +3 c \right )}{64 d a}+\frac {\sin \left (2 d x +2 c \right )}{64 d a}\) | \(141\) |
| derivativedivides | \(\frac {-\frac {256 \left (-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16384}-\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{49152}-\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{49152}-\frac {5053 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{344064}-\frac {44099 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{344064}+\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{49152}+\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{49152}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{16384}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(142\) |
| default | \(\frac {-\frac {256 \left (-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16384}-\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{49152}-\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{49152}-\frac {5053 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{344064}-\frac {44099 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{344064}+\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{49152}+\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{49152}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{16384}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(142\) |
| norman | \(\frac {-\frac {5 x}{128 a}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}+\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192 a d}+\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192 a d}+\frac {5053 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{1344 a d}+\frac {44099 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{1344 a d}-\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{192 a d}-\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{192 a d}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 a d}-\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16 a}-\frac {35 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32 a}-\frac {35 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16 a}-\frac {175 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 a}-\frac {35 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16 a}-\frac {35 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{32 a}-\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16 a}-\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{128 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}\) | \(310\) |
1/21504*(-840*d*x+1680*sin(d*x+c)-48*sin(7*d*x+7*c)+336*sin(5*d*x+5*c)-100 8*sin(3*d*x+3*c)+21*sin(8*d*x+8*c)-112*sin(6*d*x+6*c)+168*sin(4*d*x+4*c)+3 36*sin(2*d*x+2*c))/d/a
Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {105 \, d x - {\left (336 \, \cos \left (d x + c\right )^{7} - 384 \, \cos \left (d x + c\right )^{6} - 952 \, \cos \left (d x + c\right )^{5} + 1152 \, \cos \left (d x + c\right )^{4} + 826 \, \cos \left (d x + c\right )^{3} - 1152 \, \cos \left (d x + c\right )^{2} - 105 \, \cos \left (d x + c\right ) + 384\right )} \sin \left (d x + c\right )}{2688 \, a d} \]
-1/2688*(105*d*x - (336*cos(d*x + c)^7 - 384*cos(d*x + c)^6 - 952*cos(d*x + c)^5 + 1152*cos(d*x + c)^4 + 826*cos(d*x + c)^3 - 1152*cos(d*x + c)^2 - 105*cos(d*x + c) + 384)*sin(d*x + c))/(a*d)
\[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\sin ^{8}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (113) = 226\).
Time = 0.29 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.88 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2681 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5053 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {44099 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {2681 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {805 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {105 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}}}{a + \frac {8 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {56 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {56 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {8 \, a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{1344 \, d} \]
1/1344*((105*sin(d*x + c)/(cos(d*x + c) + 1) + 805*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2681*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5053*sin(d*x + c )^7/(cos(d*x + c) + 1)^7 + 44099*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 268 1*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 805*sin(d*x + c)^13/(cos(d*x + c ) + 1)^13 - 105*sin(d*x + c)^15/(cos(d*x + c) + 1)^15)/(a + 8*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 28*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 56* a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 70*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 56*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 28*a*sin(d*x + c)^12 /(cos(d*x + c) + 1)^12 + 8*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + a*sin (d*x + c)^16/(cos(d*x + c) + 1)^16) - 105*arctan(sin(d*x + c)/(cos(d*x + c ) + 1))/a)/d
Time = 0.34 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.11 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 805 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2681 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 44099 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 5053 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2681 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 805 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8} a}}{2688 \, d} \]
-1/2688*(105*(d*x + c)/a + 2*(105*tan(1/2*d*x + 1/2*c)^15 + 805*tan(1/2*d* x + 1/2*c)^13 + 2681*tan(1/2*d*x + 1/2*c)^11 - 44099*tan(1/2*d*x + 1/2*c)^ 9 - 5053*tan(1/2*d*x + 1/2*c)^7 - 2681*tan(1/2*d*x + 1/2*c)^5 - 805*tan(1/ 2*d*x + 1/2*c)^3 - 105*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1) ^8*a))/d
Time = 16.33 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}-\frac {115\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}-\frac {383\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+\frac {44099\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1344}+\frac {5053\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1344}+\frac {383\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {115\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8}-\frac {5\,x}{128\,a} \]
((5*tan(c/2 + (d*x)/2))/64 + (115*tan(c/2 + (d*x)/2)^3)/192 + (383*tan(c/2 + (d*x)/2)^5)/192 + (5053*tan(c/2 + (d*x)/2)^7)/1344 + (44099*tan(c/2 + ( d*x)/2)^9)/1344 - (383*tan(c/2 + (d*x)/2)^11)/192 - (115*tan(c/2 + (d*x)/2 )^13)/192 - (5*tan(c/2 + (d*x)/2)^15)/64)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^ 8) - (5*x)/(128*a)