Integrand size = 21, antiderivative size = 106 \[ \int \frac {\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 \text {arctanh}(\cos (c+d x))}{16 a d}-\frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d}-\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d} \]
-5/16*arctanh(cos(d*x+c))/a/d-1/7*cot(d*x+c)^7/a/d+5/16*cot(d*x+c)*csc(d*x +c)/a/d-5/24*cot(d*x+c)^3*csc(d*x+c)/a/d+1/6*cot(d*x+c)^5*csc(d*x+c)/a/d
Leaf count is larger than twice the leaf count of optimal. \(284\) vs. \(2(106)=212\).
Time = 1.60 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.68 \[ \int \frac {\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^5(c+d x) \left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (1680 \cos (c+d x)+1008 \cos (3 (c+d x))+336 \cos (5 (c+d x))+48 \cos (7 (c+d x))+3675 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-3675 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-1190 \sin (2 (c+d x))-2205 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+2205 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+392 \sin (4 (c+d x))+735 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-735 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-462 \sin (6 (c+d x))-105 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+105 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))\right )}{86016 a d (1+\sin (c+d x))} \]
-1/86016*(Csc[c + d*x]^5*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(1680*Cos [c + d*x] + 1008*Cos[3*(c + d*x)] + 336*Cos[5*(c + d*x)] + 48*Cos[7*(c + d *x)] + 3675*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 3675*Log[Sin[(c + d*x)/2] ]*Sin[c + d*x] - 1190*Sin[2*(c + d*x)] - 2205*Log[Cos[(c + d*x)/2]]*Sin[3* (c + d*x)] + 2205*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 392*Sin[4*(c + d*x)] + 735*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 735*Log[Sin[(c + d*x) /2]]*Sin[5*(c + d*x)] - 462*Sin[6*(c + d*x)] - 105*Log[Cos[(c + d*x)/2]]*S in[7*(c + d*x)] + 105*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/(a*d*(1 + S in[c + d*x]))
Time = 0.62 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3185, 3042, 3087, 15, 3091, 3042, 3091, 3042, 3091, 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.
\(\displaystyle \int \frac {\cot ^8(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x)^8 (a \sin (c+d x)+a)}dx\) |
\(\Big \downarrow \) 3185 |
\(\displaystyle \frac {\int \cot ^6(c+d x) \csc ^2(c+d x)dx}{a}-\frac {\int \cot ^6(c+d x) \csc (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^2 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle \frac {\int \cot ^6(c+d x)d(-\cot (c+d x))}{a d}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}-\frac {\cot ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle -\frac {-\frac {5}{6} \int \cot ^4(c+d x) \csc (c+d x)dx-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\cot ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {5}{6} \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\cot ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle -\frac {-\frac {5}{6} \left (-\frac {3}{4} \int \cot ^2(c+d x) \csc (c+d x)dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\cot ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {5}{6} \left (-\frac {3}{4} \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\cot ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle -\frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\cot ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\cot ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {-\frac {5}{6} \left (-\frac {3}{4} \left (\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\cot ^7(c+d x)}{7 a d}\) |
-1/7*Cot[c + d*x]^7/(a*d) - (-1/6*(Cot[c + d*x]^5*Csc[c + d*x])/d - (5*(-1 /4*(Cot[c + d*x]^3*Csc[c + d*x])/d - (3*(ArcTanh[Cos[c + d*x]]/(2*d) - (Co t[c + d*x]*Csc[c + d*x])/(2*d)))/4))/6)/a
3.1.60.3.1 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[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[1/a Int[Sec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x ] - Simp[1/(b*g) Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /; Fre eQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Result contains complex when optimal does not.
Time = 3.67 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.58
method | result | size |
risch | \(-\frac {-336 i {\mathrm e}^{12 i \left (d x +c \right )}+231 \,{\mathrm e}^{13 i \left (d x +c \right )}-196 \,{\mathrm e}^{11 i \left (d x +c \right )}-1680 i {\mathrm e}^{8 i \left (d x +c \right )}+595 \,{\mathrm e}^{9 i \left (d x +c \right )}-1008 i {\mathrm e}^{4 i \left (d x +c \right )}-595 \,{\mathrm e}^{5 i \left (d x +c \right )}+196 \,{\mathrm e}^{3 i \left (d x +c \right )}-48 i-231 \,{\mathrm e}^{i \left (d x +c \right )}}{168 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d a}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d a}\) | \(168\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {15}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+40 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d a}\) | \(200\) |
default | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {15}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+40 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d a}\) | \(200\) |
-1/168*(-336*I*exp(12*I*(d*x+c))+231*exp(13*I*(d*x+c))-196*exp(11*I*(d*x+c ))-1680*I*exp(8*I*(d*x+c))+595*exp(9*I*(d*x+c))-1008*I*exp(4*I*(d*x+c))-59 5*exp(5*I*(d*x+c))+196*exp(3*I*(d*x+c))-48*I-231*exp(I*(d*x+c)))/d/a/(exp( 2*I*(d*x+c))-1)^7-5/16/d/a*ln(exp(I*(d*x+c))+1)+5/16/d/a*ln(exp(I*(d*x+c)) -1)
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (96) = 192\).
Time = 0.30 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.87 \[ \int \frac {\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {96 \, \cos \left (d x + c\right )^{7} - 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 14 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
1/672*(96*cos(d*x + c)^7 - 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos( d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/ 2)*sin(d*x + c) - 14*(33*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c))/((a*d*cos(d*x + c)^6 - 3*a*d*cos(d*x + c)^4 + 3*a*d*cos (d*x + c)^2 - a*d)*sin(d*x + c))
\[ \int \frac {\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\cot ^{8}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (96) = 192\).
Time = 0.21 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.97 \[ \int \frac {\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {315 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {63 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {63 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {7 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a} - \frac {840 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {{\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {63 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {63 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {315 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {105 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a \sin \left (d x + c\right )^{7}}}{2688 \, d} \]
-1/2688*((105*sin(d*x + c)/(cos(d*x + c) + 1) + 315*sin(d*x + c)^2/(cos(d* x + c) + 1)^2 - 63*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 63*sin(d*x + c)^4 /(cos(d*x + c) + 1)^4 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 7*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a - 840*log(sin(d*x + c)/(cos(d*x + c) + 1))/a - (7*sin(d*x + c)/(cos(d*x + c) + 1) + 21*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 63*sin(d*x + c)^3/(cos(d* x + c) + 1)^3 - 63*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 315*sin(d*x + c)^ 5/(cos(d*x + c) + 1)^5 + 105*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 3)*(cos (d*x + c) + 1)^7/(a*sin(d*x + c)^7))/d
Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (96) = 192\).
Time = 0.44 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.30 \[ \int \frac {\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {3 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 63 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 63 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{7}} - \frac {2178 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{2688 \, d} \]
1/2688*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a + (3*a^6*tan(1/2*d*x + 1/2*c) ^7 - 7*a^6*tan(1/2*d*x + 1/2*c)^6 - 21*a^6*tan(1/2*d*x + 1/2*c)^5 + 63*a^6 *tan(1/2*d*x + 1/2*c)^4 + 63*a^6*tan(1/2*d*x + 1/2*c)^3 - 315*a^6*tan(1/2* d*x + 1/2*c)^2 - 105*a^6*tan(1/2*d*x + 1/2*c))/a^7 - (2178*tan(1/2*d*x + 1 /2*c)^7 - 105*tan(1/2*d*x + 1/2*c)^6 - 315*tan(1/2*d*x + 1/2*c)^5 + 63*tan (1/2*d*x + 1/2*c)^4 + 63*tan(1/2*d*x + 1/2*c)^3 - 21*tan(1/2*d*x + 1/2*c)^ 2 - 7*tan(1/2*d*x + 1/2*c) + 3)/(a*tan(1/2*d*x + 1/2*c)^7))/d
Time = 7.85 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.65 \[ \int \frac {\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+7\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2688\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
(3*sin(c/2 + (d*x)/2)^14 - 3*cos(c/2 + (d*x)/2)^14 - 7*cos(c/2 + (d*x)/2)* sin(c/2 + (d*x)/2)^13 + 7*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2) - 21*co s(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 + 63*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^11 + 63*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 - 315*cos(c /2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9 - 105*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 105*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 + 315*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5 - 63*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x )/2)^4 - 63*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^3 + 21*cos(c/2 + (d*x )/2)^12*sin(c/2 + (d*x)/2)^2 + 840*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/ 2))*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7)/(2688*a*d*cos(c/2 + (d*x)/2 )^7*sin(c/2 + (d*x)/2)^7)