Integrand size = 27, antiderivative size = 134 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 \text {arctanh}(\cos (c+d x))}{128 a d}+\frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d} \]
5/128*arctanh(cos(d*x+c))/a/d+1/7*cot(d*x+c)^7/a/d+5/128*cot(d*x+c)*csc(d* x+c)/a/d-5/64*cot(d*x+c)*csc(d*x+c)^3/a/d+5/48*cot(d*x+c)^3*csc(d*x+c)^3/a /d-1/8*cot(d*x+c)^5*csc(d*x+c)^3/a/d
Leaf count is larger than twice the leaf count of optimal. \(291\) vs. \(2(134)=268\).
Time = 1.91 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.17 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^8(c+d x) \left (-24710 \cos (c+d x)-12530 \cos (3 (c+d x))-5558 \cos (5 (c+d x))-210 \cos (7 (c+d x))+3675 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-5880 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2940 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-840 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+105 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3675 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+5880 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2940 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+840 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-105 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+5376 \sin (2 (c+d x))+5376 \sin (4 (c+d x))+2304 \sin (6 (c+d x))+384 \sin (8 (c+d x))\right )}{344064 a d} \]
(Csc[c + d*x]^8*(-24710*Cos[c + d*x] - 12530*Cos[3*(c + d*x)] - 5558*Cos[5 *(c + d*x)] - 210*Cos[7*(c + d*x)] + 3675*Log[Cos[(c + d*x)/2]] - 5880*Cos [2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 2940*Cos[4*(c + d*x)]*Log[Cos[(c + d *x)/2]] - 840*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 105*Cos[8*(c + d*x) ]*Log[Cos[(c + d*x)/2]] - 3675*Log[Sin[(c + d*x)/2]] + 5880*Cos[2*(c + d*x )]*Log[Sin[(c + d*x)/2]] - 2940*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 8 40*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 105*Cos[8*(c + d*x)]*Log[Sin[( c + d*x)/2]] + 5376*Sin[2*(c + d*x)] + 5376*Sin[4*(c + d*x)] + 2304*Sin[6* (c + d*x)] + 384*Sin[8*(c + d*x)]))/(344064*a*d)
Time = 0.82 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {3042, 3318, 3042, 3087, 15, 3091, 3042, 3091, 3042, 3091, 3042, 4255, 3042, 4257}
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.
| \(\displaystyle \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^9 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cot ^6(c+d x) \csc ^3(c+d x)dx}{a}-\frac {\int \cot ^6(c+d x) \csc ^2(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^2 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \cot ^6(c+d x)d(-\cot (c+d x))}{a d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}+\frac {\cot ^7(c+d x)}{7 a d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {5}{8} \int \cot ^4(c+d x) \csc ^3(c+d x)dx-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}+\frac {\cot ^7(c+d x)}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5}{8} \int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}+\frac {\cot ^7(c+d x)}{7 a d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {5}{8} \left (-\frac {1}{2} \int \cot ^2(c+d x) \csc ^3(c+d x)dx-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}+\frac {\cot ^7(c+d x)}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5}{8} \left (-\frac {1}{2} \int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}+\frac {\cot ^7(c+d x)}{7 a d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \int \csc ^3(c+d x)dx+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}+\frac {\cot ^7(c+d x)}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \int \csc (c+d x)^3dx+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}+\frac {\cot ^7(c+d x)}{7 a d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}+\frac {\cot ^7(c+d x)}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}+\frac {\cot ^7(c+d x)}{7 a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}+\frac {\cot ^7(c+d x)}{7 a d}\) |
Cot[c + d*x]^7/(7*a*d) + (-1/8*(Cot[c + d*x]^5*Csc[c + d*x]^3)/d - (5*(-1/ 6*(Cot[c + d*x]^3*Csc[c + d*x]^3)/d + ((Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (-1/2*ArcTanh[Cos[c + d*x]]/d - (Cot[c + d*x]*Csc[c + d*x])/(2*d))/4)/2 ))/8)/a
3.8.18.3.1 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[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.42 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.69
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-6 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {2}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d a}\) | \(226\) |
| default | \(\frac {\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-6 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {2}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d a}\) | \(226\) |
| parallelrisch | \(\frac {21 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-21 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-112 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+112 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+336 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-336 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+168 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-168 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1008 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1008 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+336 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-336 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1680 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1680 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1680 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{43008 d a}\) | \(226\) |
| risch | \(-\frac {13440 i {\mathrm e}^{10 i \left (d x +c \right )}+105 \,{\mathrm e}^{15 i \left (d x +c \right )}+2688 i {\mathrm e}^{14 i \left (d x +c \right )}+2779 \,{\mathrm e}^{13 i \left (d x +c \right )}-8064 i {\mathrm e}^{4 i \left (d x +c \right )}+6265 \,{\mathrm e}^{11 i \left (d x +c \right )}-13440 i {\mathrm e}^{8 i \left (d x +c \right )}+12355 \,{\mathrm e}^{9 i \left (d x +c \right )}+384 i {\mathrm e}^{2 i \left (d x +c \right )}+12355 \,{\mathrm e}^{7 i \left (d x +c \right )}+8064 i {\mathrm e}^{6 i \left (d x +c \right )}+6265 \,{\mathrm e}^{5 i \left (d x +c \right )}-2688 i {\mathrm e}^{12 i \left (d x +c \right )}+2779 \,{\mathrm e}^{3 i \left (d x +c \right )}-384 i+105 \,{\mathrm e}^{i \left (d x +c \right )}}{1344 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d a}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d a}\) | \(238\) |
1/256/d/a*(1/8*tan(1/2*d*x+1/2*c)^8-2/7*tan(1/2*d*x+1/2*c)^7-2/3*tan(1/2*d *x+1/2*c)^6+2*tan(1/2*d*x+1/2*c)^5+tan(1/2*d*x+1/2*c)^4-6*tan(1/2*d*x+1/2* c)^3+2*tan(1/2*d*x+1/2*c)^2+10*tan(1/2*d*x+1/2*c)+2/3/tan(1/2*d*x+1/2*c)^6 -10*ln(tan(1/2*d*x+1/2*c))-1/tan(1/2*d*x+1/2*c)^4-2/tan(1/2*d*x+1/2*c)^2-2 /tan(1/2*d*x+1/2*c)^5-10/tan(1/2*d*x+1/2*c)-1/8/tan(1/2*d*x+1/2*c)^8+2/7/t an(1/2*d*x+1/2*c)^7+6/tan(1/2*d*x+1/2*c)^3)
Time = 0.26 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.61 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {768 \, \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - 210 \, \cos \left (d x + c\right )^{7} - 1022 \, \cos \left (d x + c\right )^{5} + 770 \, \cos \left (d x + c\right )^{3} + 105 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 105 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 210 \, \cos \left (d x + c\right )}{5376 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]
1/5376*(768*cos(d*x + c)^7*sin(d*x + c) - 210*cos(d*x + c)^7 - 1022*cos(d* x + c)^5 + 770*cos(d*x + c)^3 + 105*(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6 *cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2) - 105* (cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2) - 210*cos(d*x + c))/(a*d*cos(d*x + c)^8 - 4*a*d*cos(d*x + c)^6 + 6*a*d*cos(d*x + c)^4 - 4*a*d*cos(d*x + c)^2 + a*d)
Timed out. \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (122) = 244\).
Time = 0.22 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.64 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {1680 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {336 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1008 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {168 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {336 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {112 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {48 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {21 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}{a} - \frac {1680 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {48 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {112 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {336 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {168 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {336 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1680 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 21\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{8}}{a \sin \left (d x + c\right )^{8}}}{43008 \, d} \]
1/43008*((1680*sin(d*x + c)/(cos(d*x + c) + 1) + 336*sin(d*x + c)^2/(cos(d *x + c) + 1)^2 - 1008*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 168*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 336*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 112* sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 48*sin(d*x + c)^7/(cos(d*x + c) + 1) ^7 + 21*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)/a - 1680*log(sin(d*x + c)/(co s(d*x + c) + 1))/a + (48*sin(d*x + c)/(cos(d*x + c) + 1) + 112*sin(d*x + c )^2/(cos(d*x + c) + 1)^2 - 336*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 168*s in(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1008*sin(d*x + c)^5/(cos(d*x + c) + 1 )^5 - 336*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 1680*sin(d*x + c)^7/(cos(d *x + c) + 1)^7 - 21)*(cos(d*x + c) + 1)^8/(a*sin(d*x + c)^8))/d
Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (122) = 244\).
Time = 0.38 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.04 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {1680 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {21 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 48 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 112 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 336 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 168 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1008 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 336 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1680 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{8}} - \frac {4566 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1008 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 168 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{43008 \, d} \]
-1/43008*(1680*log(abs(tan(1/2*d*x + 1/2*c)))/a - (21*a^7*tan(1/2*d*x + 1/ 2*c)^8 - 48*a^7*tan(1/2*d*x + 1/2*c)^7 - 112*a^7*tan(1/2*d*x + 1/2*c)^6 + 336*a^7*tan(1/2*d*x + 1/2*c)^5 + 168*a^7*tan(1/2*d*x + 1/2*c)^4 - 1008*a^7 *tan(1/2*d*x + 1/2*c)^3 + 336*a^7*tan(1/2*d*x + 1/2*c)^2 + 1680*a^7*tan(1/ 2*d*x + 1/2*c))/a^8 - (4566*tan(1/2*d*x + 1/2*c)^8 - 1680*tan(1/2*d*x + 1/ 2*c)^7 - 336*tan(1/2*d*x + 1/2*c)^6 + 1008*tan(1/2*d*x + 1/2*c)^5 - 168*ta n(1/2*d*x + 1/2*c)^4 - 336*tan(1/2*d*x + 1/2*c)^3 + 112*tan(1/2*d*x + 1/2* c)^2 + 48*tan(1/2*d*x + 1/2*c) - 21)/(a*tan(1/2*d*x + 1/2*c)^8))/d
Time = 12.89 (sec) , antiderivative size = 435, normalized size of antiderivative = 3.25 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-21\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+48\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}-48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+112\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-168\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+168\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-112\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1680\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{43008\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8} \]
-(21*cos(c/2 + (d*x)/2)^16 - 21*sin(c/2 + (d*x)/2)^16 + 48*cos(c/2 + (d*x) /2)*sin(c/2 + (d*x)/2)^15 - 48*cos(c/2 + (d*x)/2)^15*sin(c/2 + (d*x)/2) + 112*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^14 - 336*cos(c/2 + (d*x)/2)^3* sin(c/2 + (d*x)/2)^13 - 168*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^12 + 1 008*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^11 - 336*cos(c/2 + (d*x)/2)^6* sin(c/2 + (d*x)/2)^10 - 1680*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^9 + 1 680*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^7 + 336*cos(c/2 + (d*x)/2)^10* sin(c/2 + (d*x)/2)^6 - 1008*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^5 + 1 68*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^4 + 336*cos(c/2 + (d*x)/2)^13* sin(c/2 + (d*x)/2)^3 - 112*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^2 + 16 80*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^8)/(43008*a*d*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^8)