Integrand size = 29, antiderivative size = 159 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3 x}{64 a^2}-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {3 \cos ^7(c+d x)}{7 a^2 d}-\frac {\cos ^9(c+d x)}{9 a^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{64 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{32 a^2 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{4 a^2 d} \]
-3/64*x/a^2-2/5*cos(d*x+c)^5/a^2/d+3/7*cos(d*x+c)^7/a^2/d-1/9*cos(d*x+c)^9 /a^2/d-3/64*cos(d*x+c)*sin(d*x+c)/a^2/d-1/32*cos(d*x+c)^3*sin(d*x+c)/a^2/d +1/8*cos(d*x+c)^5*sin(d*x+c)/a^2/d+1/4*cos(d*x+c)^5*sin(d*x+c)^3/a^2/d
Leaf count is larger than twice the leaf count of optimal. \(430\) vs. \(2(159)=318\).
Time = 3.79 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.70 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {420 (7+330 c+36 d x) \cos \left (\frac {c}{2}\right )+11340 \cos \left (\frac {c}{2}+d x\right )+11340 \cos \left (\frac {3 c}{2}+d x\right )+3360 \cos \left (\frac {5 c}{2}+3 d x\right )+3360 \cos \left (\frac {7 c}{2}+3 d x\right )-2520 \cos \left (\frac {7 c}{2}+4 d x\right )+2520 \cos \left (\frac {9 c}{2}+4 d x\right )-1008 \cos \left (\frac {9 c}{2}+5 d x\right )-1008 \cos \left (\frac {11 c}{2}+5 d x\right )-450 \cos \left (\frac {13 c}{2}+7 d x\right )-450 \cos \left (\frac {15 c}{2}+7 d x\right )+315 \cos \left (\frac {15 c}{2}+8 d x\right )-315 \cos \left (\frac {17 c}{2}+8 d x\right )+70 \cos \left (\frac {17 c}{2}+9 d x\right )+70 \cos \left (\frac {19 c}{2}+9 d x\right )-78960 \sin \left (\frac {c}{2}\right )+138600 c \sin \left (\frac {c}{2}\right )+15120 d x \sin \left (\frac {c}{2}\right )-11340 \sin \left (\frac {c}{2}+d x\right )+11340 \sin \left (\frac {3 c}{2}+d x\right )-3360 \sin \left (\frac {5 c}{2}+3 d x\right )+3360 \sin \left (\frac {7 c}{2}+3 d x\right )-2520 \sin \left (\frac {7 c}{2}+4 d x\right )-2520 \sin \left (\frac {9 c}{2}+4 d x\right )+1008 \sin \left (\frac {9 c}{2}+5 d x\right )-1008 \sin \left (\frac {11 c}{2}+5 d x\right )+450 \sin \left (\frac {13 c}{2}+7 d x\right )-450 \sin \left (\frac {15 c}{2}+7 d x\right )+315 \sin \left (\frac {15 c}{2}+8 d x\right )+315 \sin \left (\frac {17 c}{2}+8 d x\right )-70 \sin \left (\frac {17 c}{2}+9 d x\right )+70 \sin \left (\frac {19 c}{2}+9 d x\right )}{322560 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
-1/322560*(420*(7 + 330*c + 36*d*x)*Cos[c/2] + 11340*Cos[c/2 + d*x] + 1134 0*Cos[(3*c)/2 + d*x] + 3360*Cos[(5*c)/2 + 3*d*x] + 3360*Cos[(7*c)/2 + 3*d* x] - 2520*Cos[(7*c)/2 + 4*d*x] + 2520*Cos[(9*c)/2 + 4*d*x] - 1008*Cos[(9*c )/2 + 5*d*x] - 1008*Cos[(11*c)/2 + 5*d*x] - 450*Cos[(13*c)/2 + 7*d*x] - 45 0*Cos[(15*c)/2 + 7*d*x] + 315*Cos[(15*c)/2 + 8*d*x] - 315*Cos[(17*c)/2 + 8 *d*x] + 70*Cos[(17*c)/2 + 9*d*x] + 70*Cos[(19*c)/2 + 9*d*x] - 78960*Sin[c/ 2] + 138600*c*Sin[c/2] + 15120*d*x*Sin[c/2] - 11340*Sin[c/2 + d*x] + 11340 *Sin[(3*c)/2 + d*x] - 3360*Sin[(5*c)/2 + 3*d*x] + 3360*Sin[(7*c)/2 + 3*d*x ] - 2520*Sin[(7*c)/2 + 4*d*x] - 2520*Sin[(9*c)/2 + 4*d*x] + 1008*Sin[(9*c) /2 + 5*d*x] - 1008*Sin[(11*c)/2 + 5*d*x] + 450*Sin[(13*c)/2 + 7*d*x] - 450 *Sin[(15*c)/2 + 7*d*x] + 315*Sin[(15*c)/2 + 8*d*x] + 315*Sin[(17*c)/2 + 8* d*x] - 70*Sin[(17*c)/2 + 9*d*x] + 70*Sin[(19*c)/2 + 9*d*x])/(a^2*d*(Cos[c/ 2] + Sin[c/2]))
Time = 0.58 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3354, 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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^8(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^8}{(a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cos ^4(c+d x) \sin ^3(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos (c+d x)^4 \sin (c+d x)^3 (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^2 \cos ^4(c+d x) \sin ^5(c+d x)-2 a^2 \cos ^4(c+d x) \sin ^4(c+d x)+a^2 \cos ^4(c+d x) \sin ^3(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {a^2 \cos ^9(c+d x)}{9 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{4 d}+\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}-\frac {3 a^2 \sin (c+d x) \cos (c+d x)}{64 d}-\frac {3 a^2 x}{64}}{a^4}\) |
((-3*a^2*x)/64 - (2*a^2*Cos[c + d*x]^5)/(5*d) + (3*a^2*Cos[c + d*x]^7)/(7* d) - (a^2*Cos[c + d*x]^9)/(9*d) - (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(64*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(32*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x]^3)/(4*d))/a^4
3.8.24.3.1 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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Time = 0.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.56
| method | result | size |
| parallelrisch | \(\frac {-7560 d x +450 \cos \left (7 d x +7 c \right )+1008 \cos \left (5 d x +5 c \right )-3360 \cos \left (3 d x +3 c \right )-11340 \cos \left (d x +c \right )-70 \cos \left (9 d x +9 c \right )-315 \sin \left (8 d x +8 c \right )+2520 \sin \left (4 d x +4 c \right )-13312}{161280 d \,a^{2}}\) | \(89\) |
| risch | \(-\frac {3 x}{64 a^{2}}-\frac {9 \cos \left (d x +c \right )}{128 a^{2} d}-\frac {\cos \left (9 d x +9 c \right )}{2304 d \,a^{2}}-\frac {\sin \left (8 d x +8 c \right )}{512 d \,a^{2}}+\frac {5 \cos \left (7 d x +7 c \right )}{1792 d \,a^{2}}+\frac {\cos \left (5 d x +5 c \right )}{160 d \,a^{2}}+\frac {\sin \left (4 d x +4 c \right )}{64 d \,a^{2}}-\frac {\cos \left (3 d x +3 c \right )}{48 d \,a^{2}}\) | \(124\) |
| derivativedivides | \(\frac {\frac {16 \left (-\frac {13}{1260}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512}-\frac {13 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {17 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140}-\frac {155 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20}+\frac {169 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {41 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20}+\frac {3 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {169 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {11 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}+\frac {155 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {13 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {3 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}}{d \,a^{2}}\) | \(233\) |
| default | \(\frac {\frac {16 \left (-\frac {13}{1260}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512}-\frac {13 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {17 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140}-\frac {155 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20}+\frac {169 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {41 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20}+\frac {3 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {169 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {11 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}+\frac {155 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {13 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {3 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}}{d \,a^{2}}\) | \(233\) |
1/161280*(-7560*d*x+450*cos(7*d*x+7*c)+1008*cos(5*d*x+5*c)-3360*cos(3*d*x+ 3*c)-11340*cos(d*x+c)-70*cos(9*d*x+9*c)-315*sin(8*d*x+8*c)+2520*sin(4*d*x+ 4*c)-13312)/d/a^2
Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.57 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {2240 \, \cos \left (d x + c\right )^{9} - 8640 \, \cos \left (d x + c\right )^{7} + 8064 \, \cos \left (d x + c\right )^{5} + 945 \, d x + 315 \, {\left (16 \, \cos \left (d x + c\right )^{7} - 24 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20160 \, a^{2} d} \]
-1/20160*(2240*cos(d*x + c)^9 - 8640*cos(d*x + c)^7 + 8064*cos(d*x + c)^5 + 945*d*x + 315*(16*cos(d*x + c)^7 - 24*cos(d*x + c)^5 + 2*cos(d*x + c)^3 + 3*cos(d*x + c))*sin(d*x + c))/(a^2*d)
Timed out. \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (143) = 286\).
Time = 0.39 (sec) , antiderivative size = 542, normalized size of antiderivative = 3.41 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {14976 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {8190 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {19584 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {97650 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8064 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {106470 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {330624 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {120960 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {106470 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {147840 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {97650 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {40320 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} - \frac {8190 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {945 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - 1664}{a^{2} + \frac {9 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {36 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {84 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {126 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {126 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {84 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {36 \, a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {9 \, a^{2} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {a^{2} \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}}} - \frac {945 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{10080 \, d} \]
1/10080*((945*sin(d*x + c)/(cos(d*x + c) + 1) - 14976*sin(d*x + c)^2/(cos( d*x + c) + 1)^2 + 8190*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 19584*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 97650*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 8064*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 106470*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 330624*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 120960*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 106470*sin(d*x + c)^11/(cos(d*x + c) + 1) ^11 - 147840*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 97650*sin(d*x + c)^13 /(cos(d*x + c) + 1)^13 - 40320*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 819 0*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 945*sin(d*x + c)^17/(cos(d*x + c ) + 1)^17 - 1664)/(a^2 + 9*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 36*a^ 2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 84*a^2*sin(d*x + c)^6/(cos(d*x + c ) + 1)^6 + 126*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 126*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 84*a^2*sin(d*x + c)^12/(cos(d*x + c) + 1)^1 2 + 36*a^2*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 9*a^2*sin(d*x + c)^16/( cos(d*x + c) + 1)^16 + a^2*sin(d*x + c)^18/(cos(d*x + c) + 1)^18) - 945*ar ctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d
Time = 0.35 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {945 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 8190 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 40320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 97650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 147840 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 106470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 120960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 330624 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 106470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 8064 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 97650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 19584 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8190 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 14976 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1664\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{9} a^{2}}}{20160 \, d} \]
-1/20160*(945*(d*x + c)/a^2 + 2*(945*tan(1/2*d*x + 1/2*c)^17 + 8190*tan(1/ 2*d*x + 1/2*c)^15 + 40320*tan(1/2*d*x + 1/2*c)^14 - 97650*tan(1/2*d*x + 1/ 2*c)^13 + 147840*tan(1/2*d*x + 1/2*c)^12 + 106470*tan(1/2*d*x + 1/2*c)^11 - 120960*tan(1/2*d*x + 1/2*c)^10 + 330624*tan(1/2*d*x + 1/2*c)^8 - 106470* tan(1/2*d*x + 1/2*c)^7 - 8064*tan(1/2*d*x + 1/2*c)^6 + 97650*tan(1/2*d*x + 1/2*c)^5 + 19584*tan(1/2*d*x + 1/2*c)^4 - 8190*tan(1/2*d*x + 1/2*c)^3 + 1 4976*tan(1/2*d*x + 1/2*c)^2 - 945*tan(1/2*d*x + 1/2*c) + 1664)/((tan(1/2*d *x + 1/2*c)^2 + 1)^9*a^2))/d
Time = 12.16 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.42 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3\,x}{64\,a^2}-\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{32}+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{16}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-\frac {155\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{16}+\frac {44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+\frac {169\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{16}-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {164\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{5}-\frac {169\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {155\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {68\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}-\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{16}+\frac {52\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{32}+\frac {52}{315}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]
- (3*x)/(64*a^2) - ((52*tan(c/2 + (d*x)/2)^2)/35 - (3*tan(c/2 + (d*x)/2))/ 32 - (13*tan(c/2 + (d*x)/2)^3)/16 + (68*tan(c/2 + (d*x)/2)^4)/35 + (155*ta n(c/2 + (d*x)/2)^5)/16 - (4*tan(c/2 + (d*x)/2)^6)/5 - (169*tan(c/2 + (d*x) /2)^7)/16 + (164*tan(c/2 + (d*x)/2)^8)/5 - 12*tan(c/2 + (d*x)/2)^10 + (169 *tan(c/2 + (d*x)/2)^11)/16 + (44*tan(c/2 + (d*x)/2)^12)/3 - (155*tan(c/2 + (d*x)/2)^13)/16 + 4*tan(c/2 + (d*x)/2)^14 + (13*tan(c/2 + (d*x)/2)^15)/16 + (3*tan(c/2 + (d*x)/2)^17)/32 + 52/315)/(a^2*d*(tan(c/2 + (d*x)/2)^2 + 1 )^9)