Integrand size = 21, antiderivative size = 238 \[ \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {5 \text {arctanh}(\sin (c+d x))}{512 a^8 d}-\frac {a}{36 d (a+a \sin (c+d x))^9}-\frac {1}{32 d (a+a \sin (c+d x))^8}-\frac {3}{112 a d (a+a \sin (c+d x))^7}-\frac {1}{48 a^2 d (a+a \sin (c+d x))^6}-\frac {1}{64 a^3 d (a+a \sin (c+d x))^5}-\frac {7}{768 a^5 d (a+a \sin (c+d x))^3}-\frac {3}{256 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {1}{128 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac {1}{1024 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac {9}{1024 d \left (a^8+a^8 \sin (c+d x)\right )} \]
5/512*arctanh(sin(d*x+c))/a^8/d-1/36*a/d/(a+a*sin(d*x+c))^9-1/32/d/(a+a*si n(d*x+c))^8-3/112/a/d/(a+a*sin(d*x+c))^7-1/48/a^2/d/(a+a*sin(d*x+c))^6-1/6 4/a^3/d/(a+a*sin(d*x+c))^5-7/768/a^5/d/(a+a*sin(d*x+c))^3-3/256/d/(a^2+a^2 *sin(d*x+c))^4-1/128/d/(a^4+a^4*sin(d*x+c))^2+1/1024/d/(a^8-a^8*sin(d*x+c) )-9/1024/d/(a^8+a^8*sin(d*x+c))
Time = 1.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.74 \[ \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {\sec ^2(c+d x) \left (5120-315 \text {arctanh}(\sin (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{18}+9019 \sin (c+d x)+11736 \sin ^2(c+d x)+7074 \sin ^3(c+d x)-5544 \sin ^4(c+d x)-16128 \sin ^5(c+d x)-15960 \sin ^6(c+d x)-8610 \sin ^7(c+d x)-2520 \sin ^8(c+d x)-315 \sin ^9(c+d x)\right )}{32256 a^8 d (1+\sin (c+d x))^8} \]
-1/32256*(Sec[c + d*x]^2*(5120 - 315*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/ 2] - Sin[(c + d*x)/2])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^18 + 9019*S in[c + d*x] + 11736*Sin[c + d*x]^2 + 7074*Sin[c + d*x]^3 - 5544*Sin[c + d* x]^4 - 16128*Sin[c + d*x]^5 - 15960*Sin[c + d*x]^6 - 8610*Sin[c + d*x]^7 - 2520*Sin[c + d*x]^8 - 315*Sin[c + d*x]^9))/(a^8*d*(1 + Sin[c + d*x])^8)
Time = 0.38 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3146, 54, 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.
| \(\displaystyle \int \frac {\sec ^3(c+d x)}{(a \sin (c+d x)+a)^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^3 (a \sin (c+d x)+a)^8}dx\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {a^3 \int \frac {1}{(a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^{10}}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {a^3 \int \left (\frac {1}{1024 a^{10} (a-a \sin (c+d x))^2}+\frac {9}{1024 a^{10} (\sin (c+d x) a+a)^2}+\frac {1}{64 a^9 (\sin (c+d x) a+a)^3}+\frac {7}{256 a^8 (\sin (c+d x) a+a)^4}+\frac {3}{64 a^7 (\sin (c+d x) a+a)^5}+\frac {5}{64 a^6 (\sin (c+d x) a+a)^6}+\frac {1}{8 a^5 (\sin (c+d x) a+a)^7}+\frac {3}{16 a^4 (\sin (c+d x) a+a)^8}+\frac {1}{4 a^3 (\sin (c+d x) a+a)^9}+\frac {1}{4 a^2 (\sin (c+d x) a+a)^{10}}+\frac {5}{512 a^{10} \left (a^2-a^2 \sin ^2(c+d x)\right )}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \left (\frac {5 \text {arctanh}(\sin (c+d x))}{512 a^{11}}+\frac {1}{1024 a^{10} (a-a \sin (c+d x))}-\frac {9}{1024 a^{10} (a \sin (c+d x)+a)}-\frac {1}{128 a^9 (a \sin (c+d x)+a)^2}-\frac {7}{768 a^8 (a \sin (c+d x)+a)^3}-\frac {3}{256 a^7 (a \sin (c+d x)+a)^4}-\frac {1}{64 a^6 (a \sin (c+d x)+a)^5}-\frac {1}{48 a^5 (a \sin (c+d x)+a)^6}-\frac {3}{112 a^4 (a \sin (c+d x)+a)^7}-\frac {1}{32 a^3 (a \sin (c+d x)+a)^8}-\frac {1}{36 a^2 (a \sin (c+d x)+a)^9}\right )}{d}\) |
(a^3*((5*ArcTanh[Sin[c + d*x]])/(512*a^11) + 1/(1024*a^10*(a - a*Sin[c + d *x])) - 1/(36*a^2*(a + a*Sin[c + d*x])^9) - 1/(32*a^3*(a + a*Sin[c + d*x]) ^8) - 3/(112*a^4*(a + a*Sin[c + d*x])^7) - 1/(48*a^5*(a + a*Sin[c + d*x])^ 6) - 1/(64*a^6*(a + a*Sin[c + d*x])^5) - 3/(256*a^7*(a + a*Sin[c + d*x])^4 ) - 7/(768*a^8*(a + a*Sin[c + d*x])^3) - 1/(128*a^9*(a + a*Sin[c + d*x])^2 ) - 9/(1024*a^10*(a + a*Sin[c + d*x]))))/d
3.1.98.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 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 + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Time = 5.51 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.63
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{1024 \left (\sin \left (d x +c \right )-1\right )}-\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{1024}-\frac {1}{36 \left (1+\sin \left (d x +c \right )\right )^{9}}-\frac {1}{32 \left (1+\sin \left (d x +c \right )\right )^{8}}-\frac {3}{112 \left (1+\sin \left (d x +c \right )\right )^{7}}-\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{6}}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {3}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {7}{768 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {1}{128 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {9}{1024 \left (1+\sin \left (d x +c \right )\right )}+\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{1024}}{d \,a^{8}}\) | \(151\) |
| default | \(\frac {-\frac {1}{1024 \left (\sin \left (d x +c \right )-1\right )}-\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{1024}-\frac {1}{36 \left (1+\sin \left (d x +c \right )\right )^{9}}-\frac {1}{32 \left (1+\sin \left (d x +c \right )\right )^{8}}-\frac {3}{112 \left (1+\sin \left (d x +c \right )\right )^{7}}-\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{6}}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {3}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {7}{768 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {1}{128 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {9}{1024 \left (1+\sin \left (d x +c \right )\right )}+\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{1024}}{d \,a^{8}}\) | \(151\) |
| risch | \(-\frac {i \left (-1404864 i {\mathrm e}^{8 i \left (d x +c \right )}+315 \,{\mathrm e}^{19 i \left (d x +c \right )}-1404864 i {\mathrm e}^{12 i \left (d x +c \right )}-37275 \,{\mathrm e}^{17 i \left (d x +c \right )}+1084608 i {\mathrm e}^{6 i \left (d x +c \right )}+510468 \,{\mathrm e}^{15 i \left (d x +c \right )}-1655008 i {\mathrm e}^{10 i \left (d x +c \right )}-1587204 \,{\mathrm e}^{13 i \left (d x +c \right )}+5040 i {\mathrm e}^{2 i \left (d x +c \right )}+158498 \,{\mathrm e}^{11 i \left (d x +c \right )}-168000 i {\mathrm e}^{16 i \left (d x +c \right )}-158498 \,{\mathrm e}^{9 i \left (d x +c \right )}+1084608 i {\mathrm e}^{14 i \left (d x +c \right )}+1587204 \,{\mathrm e}^{7 i \left (d x +c \right )}+5040 i {\mathrm e}^{18 i \left (d x +c \right )}-510468 \,{\mathrm e}^{5 i \left (d x +c \right )}-168000 i {\mathrm e}^{4 i \left (d x +c \right )}+37275 \,{\mathrm e}^{3 i \left (d x +c \right )}-315 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{16128 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{18} \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )^{2} a^{8} d}-\frac {5 \ln \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )}{512 a^{8} d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{512 a^{8} d}\) | \(300\) |
| parallelrisch | \(\frac {\left (-1028160 \sin \left (3 d x +3 c \right )+1028160 \sin \left (5 d x +5 c \right )-166320 \sin \left (7 d x +7 c \right )+5040 \sin \left (9 d x +9 c \right )-315 \cos \left (10 d x +10 c \right )+417690 \cos \left (2 d x +2 c \right )+1413720 \cos \left (4 d x +4 c \right )-498015 \cos \left (6 d x +6 c \right )+37170 \cos \left (8 d x +8 c \right )-2227680 \sin \left (d x +c \right )-1531530\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (1028160 \sin \left (3 d x +3 c \right )-1028160 \sin \left (5 d x +5 c \right )+166320 \sin \left (7 d x +7 c \right )-5040 \sin \left (9 d x +9 c \right )+315 \cos \left (10 d x +10 c \right )-417690 \cos \left (2 d x +2 c \right )-1413720 \cos \left (4 d x +4 c \right )+498015 \cos \left (6 d x +6 c \right )-37170 \cos \left (8 d x +8 c \right )+2227680 \sin \left (d x +c \right )+1531530\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+13537272 \sin \left (3 d x +3 c \right )-15690744 \sin \left (5 d x +5 c \right )+2628810 \sin \left (7 d x +7 c \right )-81290 \sin \left (9 d x +9 c \right )+5120 \cos \left (10 d x +10 c \right )-9598848 \cos \left (2 d x +2 c \right )-20809344 \cos \left (4 d x +4 c \right )+7758720 \cos \left (6 d x +6 c \right )-594080 \cos \left (8 d x +8 c \right )+36525636 \sin \left (d x +c \right )+23238432}{32256 a^{8} d \left (4862-16 \sin \left (9 d x +9 c \right )-3264 \sin \left (5 d x +5 c \right )+528 \sin \left (7 d x +7 c \right )+3264 \sin \left (3 d x +3 c \right )+7072 \sin \left (d x +c \right )+1581 \cos \left (6 d x +6 c \right )+\cos \left (10 d x +10 c \right )-4488 \cos \left (4 d x +4 c \right )-118 \cos \left (8 d x +8 c \right )-1326 \cos \left (2 d x +2 c \right )\right )}\) | \(471\) |
1/d/a^8*(-1/1024/(sin(d*x+c)-1)-5/1024*ln(sin(d*x+c)-1)-1/36/(1+sin(d*x+c) )^9-1/32/(1+sin(d*x+c))^8-3/112/(1+sin(d*x+c))^7-1/48/(1+sin(d*x+c))^6-1/6 4/(1+sin(d*x+c))^5-3/256/(1+sin(d*x+c))^4-7/768/(1+sin(d*x+c))^3-1/128/(1+ sin(d*x+c))^2-9/1024/(1+sin(d*x+c))+5/1024*ln(1+sin(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (217) = 434\).
Time = 0.36 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.87 \[ \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {5040 \, \cos \left (d x + c\right )^{8} - 52080 \, \cos \left (d x + c\right )^{6} + 137088 \, \cos \left (d x + c\right )^{4} - 114624 \, \cos \left (d x + c\right )^{2} + 315 \, {\left (\cos \left (d x + c\right )^{10} - 32 \, \cos \left (d x + c\right )^{8} + 160 \, \cos \left (d x + c\right )^{6} - 256 \, \cos \left (d x + c\right )^{4} + 128 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (\cos \left (d x + c\right )^{8} - 10 \, \cos \left (d x + c\right )^{6} + 24 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \, {\left (\cos \left (d x + c\right )^{10} - 32 \, \cos \left (d x + c\right )^{8} + 160 \, \cos \left (d x + c\right )^{6} - 256 \, \cos \left (d x + c\right )^{4} + 128 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (\cos \left (d x + c\right )^{8} - 10 \, \cos \left (d x + c\right )^{6} + 24 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (315 \, \cos \left (d x + c\right )^{8} - 9870 \, \cos \left (d x + c\right )^{6} + 43848 \, \cos \left (d x + c\right )^{4} - 52272 \, \cos \left (d x + c\right )^{2} + 8960\right )} \sin \left (d x + c\right ) + 14336}{64512 \, {\left (a^{8} d \cos \left (d x + c\right )^{10} - 32 \, a^{8} d \cos \left (d x + c\right )^{8} + 160 \, a^{8} d \cos \left (d x + c\right )^{6} - 256 \, a^{8} d \cos \left (d x + c\right )^{4} + 128 \, a^{8} d \cos \left (d x + c\right )^{2} - 8 \, {\left (a^{8} d \cos \left (d x + c\right )^{8} - 10 \, a^{8} d \cos \left (d x + c\right )^{6} + 24 \, a^{8} d \cos \left (d x + c\right )^{4} - 16 \, a^{8} d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \]
1/64512*(5040*cos(d*x + c)^8 - 52080*cos(d*x + c)^6 + 137088*cos(d*x + c)^ 4 - 114624*cos(d*x + c)^2 + 315*(cos(d*x + c)^10 - 32*cos(d*x + c)^8 + 160 *cos(d*x + c)^6 - 256*cos(d*x + c)^4 + 128*cos(d*x + c)^2 - 8*(cos(d*x + c )^8 - 10*cos(d*x + c)^6 + 24*cos(d*x + c)^4 - 16*cos(d*x + c)^2)*sin(d*x + c))*log(sin(d*x + c) + 1) - 315*(cos(d*x + c)^10 - 32*cos(d*x + c)^8 + 16 0*cos(d*x + c)^6 - 256*cos(d*x + c)^4 + 128*cos(d*x + c)^2 - 8*(cos(d*x + c)^8 - 10*cos(d*x + c)^6 + 24*cos(d*x + c)^4 - 16*cos(d*x + c)^2)*sin(d*x + c))*log(-sin(d*x + c) + 1) + 2*(315*cos(d*x + c)^8 - 9870*cos(d*x + c)^6 + 43848*cos(d*x + c)^4 - 52272*cos(d*x + c)^2 + 8960)*sin(d*x + c) + 1433 6)/(a^8*d*cos(d*x + c)^10 - 32*a^8*d*cos(d*x + c)^8 + 160*a^8*d*cos(d*x + c)^6 - 256*a^8*d*cos(d*x + c)^4 + 128*a^8*d*cos(d*x + c)^2 - 8*(a^8*d*cos( d*x + c)^8 - 10*a^8*d*cos(d*x + c)^6 + 24*a^8*d*cos(d*x + c)^4 - 16*a^8*d* cos(d*x + c)^2)*sin(d*x + c))
Timed out. \[ \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.04 \[ \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {\frac {2 \, {\left (315 \, \sin \left (d x + c\right )^{9} + 2520 \, \sin \left (d x + c\right )^{8} + 8610 \, \sin \left (d x + c\right )^{7} + 15960 \, \sin \left (d x + c\right )^{6} + 16128 \, \sin \left (d x + c\right )^{5} + 5544 \, \sin \left (d x + c\right )^{4} - 7074 \, \sin \left (d x + c\right )^{3} - 11736 \, \sin \left (d x + c\right )^{2} - 9019 \, \sin \left (d x + c\right ) - 5120\right )}}{a^{8} \sin \left (d x + c\right )^{10} + 8 \, a^{8} \sin \left (d x + c\right )^{9} + 27 \, a^{8} \sin \left (d x + c\right )^{8} + 48 \, a^{8} \sin \left (d x + c\right )^{7} + 42 \, a^{8} \sin \left (d x + c\right )^{6} - 42 \, a^{8} \sin \left (d x + c\right )^{4} - 48 \, a^{8} \sin \left (d x + c\right )^{3} - 27 \, a^{8} \sin \left (d x + c\right )^{2} - 8 \, a^{8} \sin \left (d x + c\right ) - a^{8}} - \frac {315 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{8}} + \frac {315 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{8}}}{64512 \, d} \]
-1/64512*(2*(315*sin(d*x + c)^9 + 2520*sin(d*x + c)^8 + 8610*sin(d*x + c)^ 7 + 15960*sin(d*x + c)^6 + 16128*sin(d*x + c)^5 + 5544*sin(d*x + c)^4 - 70 74*sin(d*x + c)^3 - 11736*sin(d*x + c)^2 - 9019*sin(d*x + c) - 5120)/(a^8* sin(d*x + c)^10 + 8*a^8*sin(d*x + c)^9 + 27*a^8*sin(d*x + c)^8 + 48*a^8*si n(d*x + c)^7 + 42*a^8*sin(d*x + c)^6 - 42*a^8*sin(d*x + c)^4 - 48*a^8*sin( d*x + c)^3 - 27*a^8*sin(d*x + c)^2 - 8*a^8*sin(d*x + c) - a^8) - 315*log(s in(d*x + c) + 1)/a^8 + 315*log(sin(d*x + c) - 1)/a^8)/d
Time = 0.39 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.70 \[ \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {\frac {2520 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{8}} - \frac {2520 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{8}} + \frac {504 \, {\left (5 \, \sin \left (d x + c\right ) - 6\right )}}{a^{8} {\left (\sin \left (d x + c\right ) - 1\right )}} - \frac {7129 \, \sin \left (d x + c\right )^{9} + 68697 \, \sin \left (d x + c\right )^{8} + 296964 \, \sin \left (d x + c\right )^{7} + 758772 \, \sin \left (d x + c\right )^{6} + 1271214 \, \sin \left (d x + c\right )^{5} + 1465758 \, \sin \left (d x + c\right )^{4} + 1191540 \, \sin \left (d x + c\right )^{3} + 693828 \, \sin \left (d x + c\right )^{2} + 295425 \, \sin \left (d x + c\right ) + 89553}{a^{8} {\left (\sin \left (d x + c\right ) + 1\right )}^{9}}}{516096 \, d} \]
1/516096*(2520*log(abs(sin(d*x + c) + 1))/a^8 - 2520*log(abs(sin(d*x + c) - 1))/a^8 + 504*(5*sin(d*x + c) - 6)/(a^8*(sin(d*x + c) - 1)) - (7129*sin( d*x + c)^9 + 68697*sin(d*x + c)^8 + 296964*sin(d*x + c)^7 + 758772*sin(d*x + c)^6 + 1271214*sin(d*x + c)^5 + 1465758*sin(d*x + c)^4 + 1191540*sin(d* x + c)^3 + 693828*sin(d*x + c)^2 + 295425*sin(d*x + c) + 89553)/(a^8*(sin( d*x + c) + 1)^9))/d
Time = 0.60 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.97 \[ \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {\frac {5\,{\sin \left (c+d\,x\right )}^9}{512}+\frac {5\,{\sin \left (c+d\,x\right )}^8}{64}+\frac {205\,{\sin \left (c+d\,x\right )}^7}{768}+\frac {95\,{\sin \left (c+d\,x\right )}^6}{192}+\frac {{\sin \left (c+d\,x\right )}^5}{2}+\frac {11\,{\sin \left (c+d\,x\right )}^4}{64}-\frac {393\,{\sin \left (c+d\,x\right )}^3}{1792}-\frac {163\,{\sin \left (c+d\,x\right )}^2}{448}-\frac {9019\,\sin \left (c+d\,x\right )}{32256}-\frac {10}{63}}{d\,\left (-a^8\,{\sin \left (c+d\,x\right )}^{10}-8\,a^8\,{\sin \left (c+d\,x\right )}^9-27\,a^8\,{\sin \left (c+d\,x\right )}^8-48\,a^8\,{\sin \left (c+d\,x\right )}^7-42\,a^8\,{\sin \left (c+d\,x\right )}^6+42\,a^8\,{\sin \left (c+d\,x\right )}^4+48\,a^8\,{\sin \left (c+d\,x\right )}^3+27\,a^8\,{\sin \left (c+d\,x\right )}^2+8\,a^8\,\sin \left (c+d\,x\right )+a^8\right )}+\frac {5\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{512\,a^8\,d} \]
((11*sin(c + d*x)^4)/64 - (163*sin(c + d*x)^2)/448 - (393*sin(c + d*x)^3)/ 1792 - (9019*sin(c + d*x))/32256 + sin(c + d*x)^5/2 + (95*sin(c + d*x)^6)/ 192 + (205*sin(c + d*x)^7)/768 + (5*sin(c + d*x)^8)/64 + (5*sin(c + d*x)^9 )/512 - 10/63)/(d*(8*a^8*sin(c + d*x) + a^8 + 27*a^8*sin(c + d*x)^2 + 48*a ^8*sin(c + d*x)^3 + 42*a^8*sin(c + d*x)^4 - 42*a^8*sin(c + d*x)^6 - 48*a^8 *sin(c + d*x)^7 - 27*a^8*sin(c + d*x)^8 - 8*a^8*sin(c + d*x)^9 - a^8*sin(c + d*x)^10)) + (5*atanh(sin(c + d*x)))/(512*a^8*d)