Integrand size = 29, antiderivative size = 228 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {33 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{3 d}+\frac {33 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {29 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \]
33/256*a^3*arctanh(cos(d*x+c))/d-4/7*a^3*cot(d*x+c)^7/d-1/3*a^3*cot(d*x+c) ^9/d+33/256*a^3*cot(d*x+c)*csc(d*x+c)/d-29/128*a^3*cot(d*x+c)*csc(d*x+c)^3 /d+5/16*a^3*cot(d*x+c)^3*csc(d*x+c)^3/d-3/8*a^3*cot(d*x+c)^5*csc(d*x+c)^3/ d-1/32*a^3*cot(d*x+c)*csc(d*x+c)^5/d+1/16*a^3*cot(d*x+c)^3*csc(d*x+c)^5/d- 1/10*a^3*cot(d*x+c)^5*csc(d*x+c)^5/d
Time = 7.35 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.60 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (51200 \cot \left (\frac {1}{2} (c+d x)\right )+13860 \csc ^2\left (\frac {1}{2} (c+d x)\right )+55440 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-55440 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-13860 \sec ^2\left (\frac {1}{2} (c+d x)\right )+19320 \sec ^4\left (\frac {1}{2} (c+d x)\right )-5250 \sec ^6\left (\frac {1}{2} (c+d x)\right )+315 \sec ^8\left (\frac {1}{2} (c+d x)\right )+42 \sec ^{10}\left (\frac {1}{2} (c+d x)\right )+164800 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+3840 \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )+\csc ^6\left (\frac {1}{2} (c+d x)\right ) (5250-60 \sin (c+d x))-14 \csc ^{10}\left (\frac {1}{2} (c+d x)\right ) (3+10 \sin (c+d x))+5 \csc ^8\left (\frac {1}{2} (c+d x)\right ) (-63+172 \sin (c+d x))-20 \csc ^4\left (\frac {1}{2} (c+d x)\right ) (966+515 \sin (c+d x))-51200 \tan \left (\frac {1}{2} (c+d x)\right )-1720 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+280 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{430080 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
(a^3*(1 + Sin[c + d*x])^3*(51200*Cot[(c + d*x)/2] + 13860*Csc[(c + d*x)/2] ^2 + 55440*Log[Cos[(c + d*x)/2]] - 55440*Log[Sin[(c + d*x)/2]] - 13860*Sec [(c + d*x)/2]^2 + 19320*Sec[(c + d*x)/2]^4 - 5250*Sec[(c + d*x)/2]^6 + 315 *Sec[(c + d*x)/2]^8 + 42*Sec[(c + d*x)/2]^10 + 164800*Csc[c + d*x]^3*Sin[( c + d*x)/2]^4 + 3840*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 + Csc[(c + d*x)/2]^ 6*(5250 - 60*Sin[c + d*x]) - 14*Csc[(c + d*x)/2]^10*(3 + 10*Sin[c + d*x]) + 5*Csc[(c + d*x)/2]^8*(-63 + 172*Sin[c + d*x]) - 20*Csc[(c + d*x)/2]^4*(9 66 + 515*Sin[c + d*x]) - 51200*Tan[(c + d*x)/2] - 1720*Sec[(c + d*x)/2]^6* Tan[(c + d*x)/2] + 280*Sec[(c + d*x)/2]^8*Tan[(c + d*x)/2]))/(430080*d*(Co s[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.61 (sec) , antiderivative size = 228, 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.
\(\displaystyle \int \cot ^6(c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^3}{\sin (c+d x)^{11}}dx\) |
\(\Big \downarrow \) 3352 |
\(\displaystyle \int \left (a^3 \cot ^6(c+d x) \csc ^5(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^3(c+d x)+a^3 \cot ^6(c+d x) \csc ^2(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {33 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac {29 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {33 a^3 \cot (c+d x) \csc (c+d x)}{256 d}\) |
(33*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (a ^3*Cot[c + d*x]^9)/(3*d) + (33*a^3*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (2 9*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(16*d) - (3*a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d ) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)
3.7.19.3.1 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 = 0.58 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.74
method | result | size |
parallelrisch | \(-\frac {2749 \left (\frac {4325376 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2749}+\left (\csc ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {666 \cos \left (3 d x +3 c \right )}{2749}+\frac {1154 \cos \left (5 d x +5 c \right )}{13745}-\frac {705 \cos \left (7 d x +7 c \right )}{5498}-\frac {33 \cos \left (9 d x +9 c \right )}{5498}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {12800 \cos \left (d x +c \right )}{2749}+\frac {7168 \cos \left (3 d x +3 c \right )}{2749}+\frac {13312 \cos \left (5 d x +5 c \right )}{19243}+\frac {256 \cos \left (7 d x +7 c \right )}{19243}-\frac {1280 \cos \left (9 d x +9 c \right )}{57729}\right ) \left (\sec ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{3}}{33554432 d}\) | \(168\) |
risch | \(-\frac {a^{3} \left (3465 \,{\mathrm e}^{19 i \left (d x +c \right )}+74025 \,{\mathrm e}^{17 i \left (d x +c \right )}-107520 i {\mathrm e}^{14 i \left (d x +c \right )}-48468 \,{\mathrm e}^{15 i \left (d x +c \right )}+241920 i {\mathrm e}^{16 i \left (d x +c \right )}-139860 \,{\mathrm e}^{13 i \left (d x +c \right )}+376320 i {\mathrm e}^{8 i \left (d x +c \right )}-577290 \,{\mathrm e}^{11 i \left (d x +c \right )}-26880 i {\mathrm e}^{18 i \left (d x +c \right )}-577290 \,{\mathrm e}^{9 i \left (d x +c \right )}-660480 i {\mathrm e}^{6 i \left (d x +c \right )}-139860 \,{\mathrm e}^{7 i \left (d x +c \right )}+967680 i {\mathrm e}^{12 i \left (d x +c \right )}-48468 \,{\mathrm e}^{5 i \left (d x +c \right )}-806400 i {\mathrm e}^{10 i \left (d x +c \right )}+74025 \,{\mathrm e}^{3 i \left (d x +c \right )}+46080 i {\mathrm e}^{4 i \left (d x +c \right )}+3465 \,{\mathrm e}^{i \left (d x +c \right )}-37120 i {\mathrm e}^{2 i \left (d x +c \right )}+6400 i\right )}{13440 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}-\frac {33 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}+\frac {33 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}\) | \(284\) |
derivativedivides | \(\frac {-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\) | \(334\) |
default | \(\frac {-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\) | \(334\) |
-2749/33554432*(4325376/2749*ln(tan(1/2*d*x+1/2*c))+csc(1/2*d*x+1/2*c)^9*( sec(1/2*d*x+1/2*c)*(cos(d*x+c)+666/2749*cos(3*d*x+3*c)+1154/13745*cos(5*d* x+5*c)-705/5498*cos(7*d*x+7*c)-33/5498*cos(9*d*x+9*c))*csc(1/2*d*x+1/2*c)+ 12800/2749*cos(d*x+c)+7168/2749*cos(3*d*x+3*c)+13312/19243*cos(5*d*x+5*c)+ 256/19243*cos(7*d*x+7*c)-1280/57729*cos(9*d*x+9*c))*sec(1/2*d*x+1/2*c)^9)* a^3/d
Time = 0.28 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.43 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {6930 \, a^{3} \cos \left (d x + c\right )^{9} + 21420 \, a^{3} \cos \left (d x + c\right )^{7} - 59136 \, a^{3} \cos \left (d x + c\right )^{5} + 32340 \, a^{3} \cos \left (d x + c\right )^{3} - 6930 \, a^{3} \cos \left (d x + c\right ) - 3465 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3465 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2560 \, {\left (5 \, a^{3} \cos \left (d x + c\right )^{9} - 12 \, a^{3} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{53760 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
-1/53760*(6930*a^3*cos(d*x + c)^9 + 21420*a^3*cos(d*x + c)^7 - 59136*a^3*c os(d*x + c)^5 + 32340*a^3*cos(d*x + c)^3 - 6930*a^3*cos(d*x + c) - 3465*(a ^3*cos(d*x + c)^10 - 5*a^3*cos(d*x + c)^8 + 10*a^3*cos(d*x + c)^6 - 10*a^3 *cos(d*x + c)^4 + 5*a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2) + 3465*(a^3*cos(d*x + c)^10 - 5*a^3*cos(d*x + c)^8 + 10*a^3*cos(d*x + c)^6 - 10*a^3*cos(d*x + c)^4 + 5*a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c) + 1/2) + 2560*(5*a^3*cos(d*x + c)^9 - 12*a^3*cos(d*x + c)^7)*sin(d*x + c))/(d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*c os(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)
Timed out. \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.25 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {21 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 210 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {7680 \, a^{3}}{\tan \left (d x + c\right )^{7}} + \frac {2560 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{53760 \, d} \]
-1/53760*(21*a^3*(2*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 - 128*cos(d*x + c)^5 + 70*cos(d*x + c)^3 - 15*cos(d*x + c))/(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1) - 1 5*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 210*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*log(cos(d*x + c) - 1)) + 7680*a^3/tan(d*x + c)^7 + 2560*(9*tan(d*x + c)^2 + 7)*a^3/tan(d*x + c)^9)/d
Time = 0.52 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.56 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {42 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 525 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3570 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3360 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10500 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 55440 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 31920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {162382 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 31920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 10500 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 16800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3360 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3570 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 525 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 42 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{430080 \, d} \]
1/430080*(42*a^3*tan(1/2*d*x + 1/2*c)^10 + 280*a^3*tan(1/2*d*x + 1/2*c)^9 + 525*a^3*tan(1/2*d*x + 1/2*c)^8 - 600*a^3*tan(1/2*d*x + 1/2*c)^7 - 3570*a ^3*tan(1/2*d*x + 1/2*c)^6 - 3360*a^3*tan(1/2*d*x + 1/2*c)^5 + 5880*a^3*tan (1/2*d*x + 1/2*c)^4 + 16800*a^3*tan(1/2*d*x + 1/2*c)^3 + 10500*a^3*tan(1/2 *d*x + 1/2*c)^2 - 55440*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 31920*a^3*tan (1/2*d*x + 1/2*c) + (162382*a^3*tan(1/2*d*x + 1/2*c)^10 + 31920*a^3*tan(1/ 2*d*x + 1/2*c)^9 - 10500*a^3*tan(1/2*d*x + 1/2*c)^8 - 16800*a^3*tan(1/2*d* x + 1/2*c)^7 - 5880*a^3*tan(1/2*d*x + 1/2*c)^6 + 3360*a^3*tan(1/2*d*x + 1/ 2*c)^5 + 3570*a^3*tan(1/2*d*x + 1/2*c)^4 + 600*a^3*tan(1/2*d*x + 1/2*c)^3 - 525*a^3*tan(1/2*d*x + 1/2*c)^2 - 280*a^3*tan(1/2*d*x + 1/2*c) - 42*a^3)/ tan(1/2*d*x + 1/2*c)^10)/d
Time = 11.20 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.73 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}-\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {25\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {17\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}+\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {25\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {33\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}+\frac {19\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {19\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \]
(a^3*cot(c/2 + (d*x)/2)^5)/(128*d) - (5*a^3*cot(c/2 + (d*x)/2)^3)/(128*d) - (7*a^3*cot(c/2 + (d*x)/2)^4)/(512*d) - (25*a^3*cot(c/2 + (d*x)/2)^2)/(10 24*d) + (17*a^3*cot(c/2 + (d*x)/2)^6)/(2048*d) + (5*a^3*cot(c/2 + (d*x)/2) ^7)/(3584*d) - (5*a^3*cot(c/2 + (d*x)/2)^8)/(4096*d) - (a^3*cot(c/2 + (d*x )/2)^9)/(1536*d) - (a^3*cot(c/2 + (d*x)/2)^10)/(10240*d) + (25*a^3*tan(c/2 + (d*x)/2)^2)/(1024*d) + (5*a^3*tan(c/2 + (d*x)/2)^3)/(128*d) + (7*a^3*ta n(c/2 + (d*x)/2)^4)/(512*d) - (a^3*tan(c/2 + (d*x)/2)^5)/(128*d) - (17*a^3 *tan(c/2 + (d*x)/2)^6)/(2048*d) - (5*a^3*tan(c/2 + (d*x)/2)^7)/(3584*d) + (5*a^3*tan(c/2 + (d*x)/2)^8)/(4096*d) + (a^3*tan(c/2 + (d*x)/2)^9)/(1536*d ) + (a^3*tan(c/2 + (d*x)/2)^10)/(10240*d) - (33*a^3*log(tan(c/2 + (d*x)/2) ))/(256*d) + (19*a^3*cot(c/2 + (d*x)/2))/(256*d) - (19*a^3*tan(c/2 + (d*x) /2))/(256*d)