Integrand size = 29, antiderivative size = 286 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d} \]
27/1024*a^3*arctanh(cos(d*x+c))/d-4/7*a^3*cot(d*x+c)^7/d-a^3*cot(d*x+c)^9/ d-6/11*a^3*cot(d*x+c)^11/d-1/13*a^3*cot(d*x+c)^13/d+27/1024*a^3*cot(d*x+c) *csc(d*x+c)/d+9/512*a^3*cot(d*x+c)*csc(d*x+c)^3/d-3/128*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-3/64*a^3*cot(d*x+c)*csc(d*x+c)^7/d+1/8*a^3*cot(d*x+c)^3*csc(d* x+c)^7/d-1/4*a^3*cot(d*x+c)^5*csc(d*x+c)^7/d
Time = 10.46 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.72 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (138378240 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\cot (c+d x) \csc ^{12}(c+d x) (-200294400-243712000 \cos (2 (c+d x))-11079680 \cos (4 (c+d x))+43294720 \cos (6 (c+d x))+9420800 \cos (8 (c+d x))-1433600 \cos (10 (c+d x))+102400 \cos (12 (c+d x))-194159966 \sin (c+d x)-182107926 \sin (3 (c+d x))-123736613 \sin (5 (c+d x))+4571567 \sin (7 (c+d x))+1846845 \sin (9 (c+d x))-135135 \sin (11 (c+d x)))\right )}{5248122880 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
(a^3*(1 + Sin[c + d*x])^3*(138378240*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + Cot[c + d*x]*Csc[c + d*x]^12*(-200294400 - 243712000*Cos[2*(c + d*x)] - 11079680*Cos[4*(c + d*x)] + 43294720*Cos[6*(c + d*x)] + 9420800 *Cos[8*(c + d*x)] - 1433600*Cos[10*(c + d*x)] + 102400*Cos[12*(c + d*x)] - 194159966*Sin[c + d*x] - 182107926*Sin[3*(c + d*x)] - 123736613*Sin[5*(c + d*x)] + 4571567*Sin[7*(c + d*x)] + 1846845*Sin[9*(c + d*x)] - 135135*Sin [11*(c + d*x)])))/(5248122880*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.69 (sec) , antiderivative size = 286, 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.
| \(\displaystyle \int \cot ^6(c+d x) \csc ^8(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^3}{\sin (c+d x)^{14}}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \cot ^6(c+d x) \csc ^8(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^7(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^6(c+d x)+a^3 \cot ^6(c+d x) \csc ^5(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {27 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}\) |
(27*a^3*ArcTanh[Cos[c + d*x]])/(1024*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - ( a^3*Cot[c + d*x]^9)/d - (6*a^3*Cot[c + d*x]^11)/(11*d) - (a^3*Cot[c + d*x] ^13)/(13*d) + (27*a^3*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (9*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(512*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(128*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*a^3*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^3*Cot[ c + d*x]^3*Csc[c + d*x]^7)/(8*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^7)/(4* d)
3.7.22.3.1 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 = 0.90 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.70
| method | result | size |
| parallelrisch | \(-\frac {5 \left (\frac {8515584 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\left (\csc ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (11 d x +11 c \right )-\frac {\cos \left (13 d x +13 c \right )}{13}+484 \cos \left (d x +c \right )+\frac {957 \cos \left (3 d x +3 c \right )}{5}-\frac {121 \cos \left (5 d x +5 c \right )}{5}-\frac {198 \cos \left (7 d x +7 c \right )}{5}-6 \cos \left (9 d x +9 c \right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2079 \cos \left (11 d x +11 c \right )}{5120}+\frac {6102173 \cos \left (d x +c \right )}{2560}+\frac {16039023 \cos \left (3 d x +3 c \right )}{12800}+\frac {8551543 \cos \left (5 d x +5 c \right )}{25600}-\frac {614999 \cos \left (7 d x +7 c \right )}{25600}-\frac {4851 \cos \left (9 d x +9 c \right )}{1024}\right )\right ) a^{3}}{322961408 d}\) | \(201\) |
| risch | \(-\frac {a^{3} \left (135135 \,{\mathrm e}^{25 i \left (d x +c \right )}-1711710 \,{\mathrm e}^{23 i \left (d x +c \right )}+820019200 i {\mathrm e}^{14 i \left (d x +c \right )}-6418412 \,{\mathrm e}^{21 i \left (d x +c \right )}-82001920 i {\mathrm e}^{20 i \left (d x +c \right )}+119165046 \,{\mathrm e}^{19 i \left (d x +c \right )}+123002880 i {\mathrm e}^{16 i \left (d x +c \right )}+305844539 \,{\mathrm e}^{17 i \left (d x +c \right )}-105431040 i {\mathrm e}^{8 i \left (d x +c \right )}+376267892 \,{\mathrm e}^{15 i \left (d x +c \right )}+41000960 i {\mathrm e}^{18 i \left (d x +c \right )}-23429120 i {\mathrm e}^{6 i \left (d x +c \right )}-376267892 \,{\mathrm e}^{11 i \left (d x +c \right )}+468582400 i {\mathrm e}^{12 i \left (d x +c \right )}-305844539 \,{\mathrm e}^{9 i \left (d x +c \right )}+386580480 i {\mathrm e}^{10 i \left (d x +c \right )}-119165046 \,{\mathrm e}^{7 i \left (d x +c \right )}-15974400 i {\mathrm e}^{4 i \left (d x +c \right )}+6418412 \,{\mathrm e}^{5 i \left (d x +c \right )}+2662400 i {\mathrm e}^{2 i \left (d x +c \right )}+1711710 \,{\mathrm e}^{3 i \left (d x +c \right )}-204800 i-135135 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2562560 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{13}}+\frac {27 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{1024 d}-\frac {27 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{1024 d}\) | \(318\) |
| derivativedivides | \(\frac {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 )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{7}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{1024 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{1024}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{13 \sin \left (d x +c \right )^{13}}-\frac {6 \left (\cos ^{7}\left (d x +c \right )\right )}{143 \sin \left (d x +c \right )^{11}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{429 \sin \left (d x +c \right )^{9}}-\frac {16 \left (\cos ^{7}\left (d x +c \right )\right )}{3003 \sin \left (d x +c \right )^{7}}\right )}{d}\) | \(444\) |
| default | \(\frac {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 )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{7}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{1024 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{1024}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{13 \sin \left (d x +c \right )^{13}}-\frac {6 \left (\cos ^{7}\left (d x +c \right )\right )}{143 \sin \left (d x +c \right )^{11}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{429 \sin \left (d x +c \right )^{9}}-\frac {16 \left (\cos ^{7}\left (d x +c \right )\right )}{3003 \sin \left (d x +c \right )^{7}}\right )}{d}\) | \(444\) |
-5/322961408*(8515584/5*ln(tan(1/2*d*x+1/2*c))+csc(1/2*d*x+1/2*c)^12*sec(1 /2*d*x+1/2*c)^12*(sec(1/2*d*x+1/2*c)*(cos(11*d*x+11*c)-1/13*cos(13*d*x+13* c)+484*cos(d*x+c)+957/5*cos(3*d*x+3*c)-121/5*cos(5*d*x+5*c)-198/5*cos(7*d* x+7*c)-6*cos(9*d*x+9*c))*csc(1/2*d*x+1/2*c)+2079/5120*cos(11*d*x+11*c)+610 2173/2560*cos(d*x+c)+16039023/12800*cos(3*d*x+3*c)+8551543/25600*cos(5*d*x +5*c)-614999/25600*cos(7*d*x+7*c)-4851/1024*cos(9*d*x+9*c)))*a^3/d
Time = 0.31 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.46 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {409600 \, a^{3} \cos \left (d x + c\right )^{13} - 2662400 \, a^{3} \cos \left (d x + c\right )^{11} + 7321600 \, a^{3} \cos \left (d x + c\right )^{9} - 5857280 \, a^{3} \cos \left (d x + c\right )^{7} + 135135 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 135135 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2002 \, {\left (135 \, a^{3} \cos \left (d x + c\right )^{11} - 765 \, a^{3} \cos \left (d x + c\right )^{9} + 758 \, a^{3} \cos \left (d x + c\right )^{7} + 1782 \, a^{3} \cos \left (d x + c\right )^{5} - 765 \, a^{3} \cos \left (d x + c\right )^{3} + 135 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{10250240 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
1/10250240*(409600*a^3*cos(d*x + c)^13 - 2662400*a^3*cos(d*x + c)^11 + 732 1600*a^3*cos(d*x + c)^9 - 5857280*a^3*cos(d*x + c)^7 + 135135*(a^3*cos(d*x + c)^12 - 6*a^3*cos(d*x + c)^10 + 15*a^3*cos(d*x + c)^8 - 20*a^3*cos(d*x + c)^6 + 15*a^3*cos(d*x + c)^4 - 6*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos(d *x + c) + 1/2)*sin(d*x + c) - 135135*(a^3*cos(d*x + c)^12 - 6*a^3*cos(d*x + c)^10 + 15*a^3*cos(d*x + c)^8 - 20*a^3*cos(d*x + c)^6 + 15*a^3*cos(d*x + c)^4 - 6*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 2002*(135*a^3*cos(d*x + c)^11 - 765*a^3*cos(d*x + c)^9 + 758*a^3*cos (d*x + c)^7 + 1782*a^3*cos(d*x + c)^5 - 765*a^3*cos(d*x + c)^3 + 135*a^3*c os(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^12 - 6*d*cos(d*x + c)^10 + 15* d*cos(d*x + c)^8 - 20*d*cos(d*x + c)^6 + 15*d*cos(d*x + c)^4 - 6*d*cos(d*x + c)^2 + d)*sin(d*x + c))
Timed out. \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.29 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {15015 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{11} - 85 \, \cos \left (d x + c\right )^{9} + 198 \, \cos \left (d x + c\right )^{7} + 198 \, \cos \left (d x + c\right )^{5} - 85 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{12} - 6 \, \cos \left (d x + c\right )^{10} + 15 \, \cos \left (d x + c\right )^{8} - 20 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12012 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {133120 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}} + \frac {10240 \, {\left (429 \, \tan \left (d x + c\right )^{6} + 1001 \, \tan \left (d x + c\right )^{4} + 819 \, \tan \left (d x + c\right )^{2} + 231\right )} a^{3}}{\tan \left (d x + c\right )^{13}}}{30750720 \, d} \]
-1/30750720*(15015*a^3*(2*(15*cos(d*x + c)^11 - 85*cos(d*x + c)^9 + 198*co s(d*x + c)^7 + 198*cos(d*x + c)^5 - 85*cos(d*x + c)^3 + 15*cos(d*x + c))/( cos(d*x + c)^12 - 6*cos(d*x + c)^10 + 15*cos(d*x + c)^8 - 20*cos(d*x + c)^ 6 + 15*cos(d*x + c)^4 - 6*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 12012*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*c os(d*x + c)^2 - 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 133120*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^3/tan(d*x + c)^11 + 10240*(429*tan(d*x + c)^6 + 1001*tan(d*x + c)^4 + 819*tan(d*x + c)^2 + 231)*a^3/tan(d*x + c)^13)/d
Time = 0.56 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.58 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {770 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5005 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 11830 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 8008 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 20020 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 65065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 94380 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 40040 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 150150 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385385 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 450450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2162160 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1401400 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {6875958 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1401400 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 80080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 450450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 385385 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 150150 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 40040 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 94380 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 65065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20020 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8008 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11830 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5005 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 770 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13}}}{82001920 \, d} \]
1/82001920*(770*a^3*tan(1/2*d*x + 1/2*c)^13 + 5005*a^3*tan(1/2*d*x + 1/2*c )^12 + 11830*a^3*tan(1/2*d*x + 1/2*c)^11 + 8008*a^3*tan(1/2*d*x + 1/2*c)^1 0 - 20020*a^3*tan(1/2*d*x + 1/2*c)^9 - 65065*a^3*tan(1/2*d*x + 1/2*c)^8 - 94380*a^3*tan(1/2*d*x + 1/2*c)^7 - 40040*a^3*tan(1/2*d*x + 1/2*c)^6 + 1501 50*a^3*tan(1/2*d*x + 1/2*c)^5 + 385385*a^3*tan(1/2*d*x + 1/2*c)^4 + 450450 *a^3*tan(1/2*d*x + 1/2*c)^3 + 80080*a^3*tan(1/2*d*x + 1/2*c)^2 - 2162160*a ^3*log(abs(tan(1/2*d*x + 1/2*c))) - 1401400*a^3*tan(1/2*d*x + 1/2*c) + (68 75958*a^3*tan(1/2*d*x + 1/2*c)^13 + 1401400*a^3*tan(1/2*d*x + 1/2*c)^12 - 80080*a^3*tan(1/2*d*x + 1/2*c)^11 - 450450*a^3*tan(1/2*d*x + 1/2*c)^10 - 3 85385*a^3*tan(1/2*d*x + 1/2*c)^9 - 150150*a^3*tan(1/2*d*x + 1/2*c)^8 + 400 40*a^3*tan(1/2*d*x + 1/2*c)^7 + 94380*a^3*tan(1/2*d*x + 1/2*c)^6 + 65065*a ^3*tan(1/2*d*x + 1/2*c)^5 + 20020*a^3*tan(1/2*d*x + 1/2*c)^4 - 8008*a^3*ta n(1/2*d*x + 1/2*c)^3 - 11830*a^3*tan(1/2*d*x + 1/2*c)^2 - 5005*a^3*tan(1/2 *d*x + 1/2*c) - 770*a^3)/tan(1/2*d*x + 1/2*c)^13)/d
Time = 12.68 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.78 \[ \int \cot ^6(c+d x) \csc ^8(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 )}^6}{2048\,d}-\frac {45\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8192\,d}-\frac {77\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}-\frac {15\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {33\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{28672\,d}+\frac {13\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4096\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {13\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{90112\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{16384\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{106496\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {45\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8192\,d}+\frac {77\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}+\frac {15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {33\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{28672\,d}-\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4096\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{90112\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{16384\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{106496\,d}-\frac {27\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{1024\,d}+\frac {35\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2048\,d}-\frac {35\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2048\,d} \]
(a^3*cot(c/2 + (d*x)/2)^6)/(2048*d) - (45*a^3*cot(c/2 + (d*x)/2)^3)/(8192* d) - (77*a^3*cot(c/2 + (d*x)/2)^4)/(16384*d) - (15*a^3*cot(c/2 + (d*x)/2)^ 5)/(8192*d) - (a^3*cot(c/2 + (d*x)/2)^2)/(1024*d) + (33*a^3*cot(c/2 + (d*x )/2)^7)/(28672*d) + (13*a^3*cot(c/2 + (d*x)/2)^8)/(16384*d) + (a^3*cot(c/2 + (d*x)/2)^9)/(4096*d) - (a^3*cot(c/2 + (d*x)/2)^10)/(10240*d) - (13*a^3* cot(c/2 + (d*x)/2)^11)/(90112*d) - (a^3*cot(c/2 + (d*x)/2)^12)/(16384*d) - (a^3*cot(c/2 + (d*x)/2)^13)/(106496*d) + (a^3*tan(c/2 + (d*x)/2)^2)/(1024 *d) + (45*a^3*tan(c/2 + (d*x)/2)^3)/(8192*d) + (77*a^3*tan(c/2 + (d*x)/2)^ 4)/(16384*d) + (15*a^3*tan(c/2 + (d*x)/2)^5)/(8192*d) - (a^3*tan(c/2 + (d* x)/2)^6)/(2048*d) - (33*a^3*tan(c/2 + (d*x)/2)^7)/(28672*d) - (13*a^3*tan( c/2 + (d*x)/2)^8)/(16384*d) - (a^3*tan(c/2 + (d*x)/2)^9)/(4096*d) + (a^3*t an(c/2 + (d*x)/2)^10)/(10240*d) + (13*a^3*tan(c/2 + (d*x)/2)^11)/(90112*d) + (a^3*tan(c/2 + (d*x)/2)^12)/(16384*d) + (a^3*tan(c/2 + (d*x)/2)^13)/(10 6496*d) - (27*a^3*log(tan(c/2 + (d*x)/2)))/(1024*d) + (35*a^3*cot(c/2 + (d *x)/2))/(2048*d) - (35*a^3*tan(c/2 + (d*x)/2))/(2048*d)