Integrand size = 29, antiderivative size = 238 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-a^3 x+\frac {125 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \]
-a^3*x+125/128*a^3*arctanh(cos(d*x+c))/d-a^3*cot(d*x+c)/d+1/3*a^3*cot(d*x+ c)^3/d-1/5*a^3*cot(d*x+c)^5/d-3/7*a^3*cot(d*x+c)^7/d-115/128*a^3*cot(d*x+c )*csc(d*x+c)/d+5/8*a^3*cot(d*x+c)^3*csc(d*x+c)/d-1/2*a^3*cot(d*x+c)^5*csc( d*x+c)/d-5/64*a^3*cot(d*x+c)*csc(d*x+c)^3/d+5/48*a^3*cot(d*x+c)^3*csc(d*x+ c)^3/d-1/8*a^3*cot(d*x+c)^5*csc(d*x+c)^3/d
Time = 1.17 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.17 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (-215040 c-215040 d x-118784 \cot \left (\frac {1}{2} (c+d x)\right )-108780 \csc ^2\left (\frac {1}{2} (c+d x)\right )+210000 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-210000 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+108780 \sec ^2\left (\frac {1}{2} (c+d x)\right )-17010 \sec ^4\left (\frac {1}{2} (c+d x)\right )+700 \sec ^6\left (\frac {1}{2} (c+d x)\right )+105 \sec ^8\left (\frac {1}{2} (c+d x)\right )+71936 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+\csc ^4\left (\frac {1}{2} (c+d x)\right ) (17010-4496 \sin (c+d x))-15 \csc ^8\left (\frac {1}{2} (c+d x)\right ) (7+24 \sin (c+d x))+4 \csc ^6\left (\frac {1}{2} (c+d x)\right ) (-175+732 \sin (c+d x))+118784 \tan \left (\frac {1}{2} (c+d x)\right )-5856 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+720 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{215040 d} \]
(a^3*(-215040*c - 215040*d*x - 118784*Cot[(c + d*x)/2] - 108780*Csc[(c + d *x)/2]^2 + 210000*Log[Cos[(c + d*x)/2]] - 210000*Log[Sin[(c + d*x)/2]] + 1 08780*Sec[(c + d*x)/2]^2 - 17010*Sec[(c + d*x)/2]^4 + 700*Sec[(c + d*x)/2] ^6 + 105*Sec[(c + d*x)/2]^8 + 71936*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + Cs c[(c + d*x)/2]^4*(17010 - 4496*Sin[c + d*x]) - 15*Csc[(c + d*x)/2]^8*(7 + 24*Sin[c + d*x]) + 4*Csc[(c + d*x)/2]^6*(-175 + 732*Sin[c + d*x]) + 118784 *Tan[(c + d*x)/2] - 5856*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] + 720*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2]))/(215040*d)
Time = 0.57 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3352, 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.
| \(\displaystyle \int \cot ^6(c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^3}{\sin (c+d x)^9}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \cot ^6(c+d x)+a^3 \cot ^6(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^6(c+d x) \csc (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {125 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-a^3 x\) |
-(a^3*x) + (125*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x]^7)/(7*d) - (115*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Cot[ c + d*x]^3*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x])/(2*d) - (5*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)
3.7.17.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Time = 0.46 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.64
| method | result | size |
| parallelrisch | \(-\frac {3805 \left (\frac {1228800 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{761}+\left (\sec ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )-\frac {1817 \cos \left (3 d x +3 c \right )}{3805}+\frac {1861 \cos \left (5 d x +5 c \right )}{3805}-\frac {777 \cos \left (7 d x +7 c \right )}{3805}\right ) \left (\csc ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {3072 \left (\sec ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )+\frac {\cos \left (3 d x +3 c \right )}{25}+\frac {29 \cos \left (5 d x +5 c \right )}{75}-\frac {29 \cos \left (7 d x +7 c \right )}{525}\right ) \left (\csc ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{761}+\frac {6291456 d x}{3805}\right ) a^{3}}{6291456 d}\) | \(153\) |
| risch | \(-a^{3} x +\frac {a^{3} \left (27195 \,{\mathrm e}^{15 i \left (d x +c \right )}-65135 \,{\mathrm e}^{13 i \left (d x +c \right )}+63595 \,{\mathrm e}^{11 i \left (d x +c \right )}+161280 i {\mathrm e}^{12 i \left (d x +c \right )}-133175 \,{\mathrm e}^{9 i \left (d x +c \right )}-286720 i {\mathrm e}^{10 i \left (d x +c \right )}-133175 \,{\mathrm e}^{7 i \left (d x +c \right )}+519680 i {\mathrm e}^{8 i \left (d x +c \right )}+63595 \,{\mathrm e}^{5 i \left (d x +c \right )}-544768 i {\mathrm e}^{6 i \left (d x +c \right )}-65135 \,{\mathrm e}^{3 i \left (d x +c \right )}+254464 i {\mathrm e}^{4 i \left (d x +c \right )}+27195 \,{\mathrm e}^{i \left (d x +c \right )}-118784 i {\mathrm e}^{2 i \left (d x +c \right )}+14848 i\right )}{6720 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}-\frac {125 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}+\frac {125 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}\) | \(232\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(296\) |
| default | \(\frac {a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(296\) |
-3805/6291456*(1228800/761*ln(tan(1/2*d*x+1/2*c))+sec(1/2*d*x+1/2*c)^8*(co s(d*x+c)-1817/3805*cos(3*d*x+3*c)+1861/3805*cos(5*d*x+5*c)-777/3805*cos(7* d*x+7*c))*csc(1/2*d*x+1/2*c)^8+3072/761*sec(1/2*d*x+1/2*c)^7*(cos(d*x+c)+1 /25*cos(3*d*x+3*c)+29/75*cos(5*d*x+5*c)-29/525*cos(7*d*x+7*c))*csc(1/2*d*x +1/2*c)^7+6291456/3805*d*x)*a^3/d
Time = 0.28 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.52 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {26880 \, a^{3} d x \cos \left (d x + c\right )^{8} - 107520 \, a^{3} d x \cos \left (d x + c\right )^{6} - 54390 \, a^{3} \cos \left (d x + c\right )^{7} + 161280 \, a^{3} d x \cos \left (d x + c\right )^{4} + 127750 \, a^{3} \cos \left (d x + c\right )^{5} - 107520 \, a^{3} d x \cos \left (d x + c\right )^{2} - 96250 \, a^{3} \cos \left (d x + c\right )^{3} + 26880 \, a^{3} d x + 26250 \, a^{3} \cos \left (d x + c\right ) - 13125 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 13125 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 256 \, {\left (116 \, a^{3} \cos \left (d x + c\right )^{7} - 406 \, a^{3} \cos \left (d x + c\right )^{5} + 350 \, a^{3} \cos \left (d x + c\right )^{3} - 105 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{26880 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
-1/26880*(26880*a^3*d*x*cos(d*x + c)^8 - 107520*a^3*d*x*cos(d*x + c)^6 - 5 4390*a^3*cos(d*x + c)^7 + 161280*a^3*d*x*cos(d*x + c)^4 + 127750*a^3*cos(d *x + c)^5 - 107520*a^3*d*x*cos(d*x + c)^2 - 96250*a^3*cos(d*x + c)^3 + 268 80*a^3*d*x + 26250*a^3*cos(d*x + c) - 13125*(a^3*cos(d*x + c)^8 - 4*a^3*co s(d*x + c)^6 + 6*a^3*cos(d*x + c)^4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(1/2* cos(d*x + c) + 1/2) + 13125*(a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6 + 6 *a^3*cos(d*x + c)^4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2) - 256*(116*a^3*cos(d*x + c)^7 - 406*a^3*cos(d*x + c)^5 + 350*a^3*cos( d*x + c)^3 - 105*a^3*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^8 - 4*d*c os(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)
Timed out. \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.11 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {1792 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} + 35 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 840 \, a^{3} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {11520 \, a^{3}}{\tan \left (d x + c\right )^{7}}}{26880 \, d} \]
-1/26880*(1792*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3) /tan(d*x + c)^5)*a^3 + 35*a^3*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*l og(cos(d*x + c) - 1)) - 840*a^3*(2*(33*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 11520*a^3/tan (d*x + c)^7)/d
Time = 0.50 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.27 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3696 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 14280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 77280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 215040 \, {\left (d x + c\right )} a^{3} - 210000 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 122640 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {570750 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 122640 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 77280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 14280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3696 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{215040 \, d} \]
1/215040*(105*a^3*tan(1/2*d*x + 1/2*c)^8 + 720*a^3*tan(1/2*d*x + 1/2*c)^7 + 1120*a^3*tan(1/2*d*x + 1/2*c)^6 - 3696*a^3*tan(1/2*d*x + 1/2*c)^5 - 1428 0*a^3*tan(1/2*d*x + 1/2*c)^4 - 560*a^3*tan(1/2*d*x + 1/2*c)^3 + 77280*a^3* tan(1/2*d*x + 1/2*c)^2 - 215040*(d*x + c)*a^3 - 210000*a^3*log(abs(tan(1/2 *d*x + 1/2*c))) + 122640*a^3*tan(1/2*d*x + 1/2*c) + (570750*a^3*tan(1/2*d* x + 1/2*c)^8 - 122640*a^3*tan(1/2*d*x + 1/2*c)^7 - 77280*a^3*tan(1/2*d*x + 1/2*c)^6 + 560*a^3*tan(1/2*d*x + 1/2*c)^5 + 14280*a^3*tan(1/2*d*x + 1/2*c )^4 + 3696*a^3*tan(1/2*d*x + 1/2*c)^3 - 1120*a^3*tan(1/2*d*x + 1/2*c)^2 - 720*a^3*tan(1/2*d*x + 1/2*c) - 105*a^3)/tan(1/2*d*x + 1/2*c)^8)/d
Time = 12.65 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.63 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {23\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {17\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {11\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {23\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {2\,a^3\,\mathrm {atan}\left (\frac {128\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+125\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{125\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-128\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {125\,a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{128\,d}-\frac {73\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {73\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]
(a^3*cot(c/2 + (d*x)/2)^3)/(384*d) - (23*a^3*cot(c/2 + (d*x)/2)^2)/(64*d) + (17*a^3*cot(c/2 + (d*x)/2)^4)/(256*d) + (11*a^3*cot(c/2 + (d*x)/2)^5)/(6 40*d) - (a^3*cot(c/2 + (d*x)/2)^6)/(192*d) - (3*a^3*cot(c/2 + (d*x)/2)^7)/ (896*d) - (a^3*cot(c/2 + (d*x)/2)^8)/(2048*d) + (23*a^3*tan(c/2 + (d*x)/2) ^2)/(64*d) - (a^3*tan(c/2 + (d*x)/2)^3)/(384*d) - (17*a^3*tan(c/2 + (d*x)/ 2)^4)/(256*d) - (11*a^3*tan(c/2 + (d*x)/2)^5)/(640*d) + (a^3*tan(c/2 + (d* x)/2)^6)/(192*d) + (3*a^3*tan(c/2 + (d*x)/2)^7)/(896*d) + (a^3*tan(c/2 + ( d*x)/2)^8)/(2048*d) - (2*a^3*atan((128*cos(c/2 + (d*x)/2) + 125*sin(c/2 + (d*x)/2))/(125*cos(c/2 + (d*x)/2) - 128*sin(c/2 + (d*x)/2))))/d - (125*a^3 *log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(128*d) - (73*a^3*cot(c/2 + ( d*x)/2))/(128*d) + (73*a^3*tan(c/2 + (d*x)/2))/(128*d)