Integrand size = 29, antiderivative size = 165 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 x}{256 a}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^9(c+d x)}{9 a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{256 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d} \]
-3/256*x/a-1/7*cos(d*x+c)^7/a/d+1/9*cos(d*x+c)^9/a/d-3/256*cos(d*x+c)*sin( d*x+c)/a/d-1/128*cos(d*x+c)^3*sin(d*x+c)/a/d-1/160*cos(d*x+c)^5*sin(d*x+c) /a/d+3/80*cos(d*x+c)^7*sin(d*x+c)/a/d+1/10*cos(d*x+c)^7*sin(d*x+c)^3/a/d
Leaf count is larger than twice the leaf count of optimal. \(533\) vs. \(2(165)=330\).
Time = 10.03 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.23 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {-1260 (25 c-12 d x) \cos \left (\frac {c}{2}\right )+15120 \cos \left (\frac {c}{2}+d x\right )+15120 \cos \left (\frac {3 c}{2}+d x\right )+1260 \cos \left (\frac {3 c}{2}+2 d x\right )-1260 \cos \left (\frac {5 c}{2}+2 d x\right )+6720 \cos \left (\frac {5 c}{2}+3 d x\right )+6720 \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 )-630 \cos \left (\frac {11 c}{2}+6 d x\right )+630 \cos \left (\frac {13 c}{2}+6 d x\right )-1080 \cos \left (\frac {13 c}{2}+7 d x\right )-1080 \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 )-280 \cos \left (\frac {17 c}{2}+9 d x\right )-280 \cos \left (\frac {19 c}{2}+9 d x\right )+126 \cos \left (\frac {19 c}{2}+10 d x\right )-126 \cos \left (\frac {21 c}{2}+10 d x\right )+37800 \sin \left (\frac {c}{2}\right )-31500 c \sin \left (\frac {c}{2}\right )+15120 d x \sin \left (\frac {c}{2}\right )-15120 \sin \left (\frac {c}{2}+d x\right )+15120 \sin \left (\frac {3 c}{2}+d x\right )+1260 \sin \left (\frac {3 c}{2}+2 d x\right )+1260 \sin \left (\frac {5 c}{2}+2 d x\right )-6720 \sin \left (\frac {5 c}{2}+3 d x\right )+6720 \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 )-630 \sin \left (\frac {11 c}{2}+6 d x\right )-630 \sin \left (\frac {13 c}{2}+6 d x\right )+1080 \sin \left (\frac {13 c}{2}+7 d x\right )-1080 \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 )+280 \sin \left (\frac {17 c}{2}+9 d x\right )-280 \sin \left (\frac {19 c}{2}+9 d x\right )+126 \sin \left (\frac {19 c}{2}+10 d x\right )+126 \sin \left (\frac {21 c}{2}+10 d x\right )}{1290240 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
-1/1290240*(-1260*(25*c - 12*d*x)*Cos[c/2] + 15120*Cos[c/2 + d*x] + 15120* Cos[(3*c)/2 + d*x] + 1260*Cos[(3*c)/2 + 2*d*x] - 1260*Cos[(5*c)/2 + 2*d*x] + 6720*Cos[(5*c)/2 + 3*d*x] + 6720*Cos[(7*c)/2 + 3*d*x] - 2520*Cos[(7*c)/ 2 + 4*d*x] + 2520*Cos[(9*c)/2 + 4*d*x] - 630*Cos[(11*c)/2 + 6*d*x] + 630*C os[(13*c)/2 + 6*d*x] - 1080*Cos[(13*c)/2 + 7*d*x] - 1080*Cos[(15*c)/2 + 7* d*x] + 315*Cos[(15*c)/2 + 8*d*x] - 315*Cos[(17*c)/2 + 8*d*x] - 280*Cos[(17 *c)/2 + 9*d*x] - 280*Cos[(19*c)/2 + 9*d*x] + 126*Cos[(19*c)/2 + 10*d*x] - 126*Cos[(21*c)/2 + 10*d*x] + 37800*Sin[c/2] - 31500*c*Sin[c/2] + 15120*d*x *Sin[c/2] - 15120*Sin[c/2 + d*x] + 15120*Sin[(3*c)/2 + d*x] + 1260*Sin[(3* c)/2 + 2*d*x] + 1260*Sin[(5*c)/2 + 2*d*x] - 6720*Sin[(5*c)/2 + 3*d*x] + 67 20*Sin[(7*c)/2 + 3*d*x] - 2520*Sin[(7*c)/2 + 4*d*x] - 2520*Sin[(9*c)/2 + 4 *d*x] - 630*Sin[(11*c)/2 + 6*d*x] - 630*Sin[(13*c)/2 + 6*d*x] + 1080*Sin[( 13*c)/2 + 7*d*x] - 1080*Sin[(15*c)/2 + 7*d*x] + 315*Sin[(15*c)/2 + 8*d*x] + 315*Sin[(17*c)/2 + 8*d*x] + 280*Sin[(17*c)/2 + 9*d*x] - 280*Sin[(19*c)/2 + 9*d*x] + 126*Sin[(19*c)/2 + 10*d*x] + 126*Sin[(21*c)/2 + 10*d*x])/(a*d* (Cos[c/2] + Sin[c/2]))
Time = 0.79 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.552, Rules used = {3042, 3318, 3042, 3045, 244, 2009, 3048, 3042, 3048, 3042, 3115, 3042, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^8(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^8}{a \sin (c+d x)+a}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cos ^6(c+d x) \sin ^3(c+d x)dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^4(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos (c+d x)^6 \sin (c+d x)^3dx}{a}-\frac {\int \cos (c+d x)^6 \sin (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle -\frac {\int \cos ^6(c+d x) \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{a d}-\frac {\int \cos (c+d x)^6 \sin (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {\int \left (\cos ^6(c+d x)-\cos ^8(c+d x)\right )d\cos (c+d x)}{a d}-\frac {\int \cos (c+d x)^6 \sin (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\int \cos (c+d x)^6 \sin (c+d x)^4dx}{a}-\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle -\frac {\frac {3}{10} \int \cos ^6(c+d x) \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}-\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{10} \int \cos (c+d x)^6 \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}-\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle -\frac {\frac {3}{10} \left (\frac {1}{8} \int \cos ^6(c+d x)dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}-\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{10} \left (\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}-\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}-\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}-\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}-\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}-\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}-\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}-\frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}\) |
-((Cos[c + d*x]^7/7 - Cos[c + d*x]^9/9)/(a*d)) - (-1/10*(Cos[c + d*x]^7*Si n[c + d*x]^3)/d + (3*(-1/8*(Cos[c + d*x]^7*Sin[c + d*x])/d + ((Cos[c + d*x ]^5*Sin[c + d*x])/(6*d) + (5*((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/ 2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/6)/8))/10)/a
3.8.7.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Time = 0.39 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.67
| method | result | size |
| parallelrisch | \(\frac {-7560 d x +1080 \cos \left (7 d x +7 c \right )-6720 \cos \left (3 d x +3 c \right )-15120 \cos \left (d x +c \right )-126 \sin \left (10 d x +10 c \right )+280 \cos \left (9 d x +9 c \right )-315 \sin \left (8 d x +8 c \right )+630 \sin \left (6 d x +6 c \right )+2520 \sin \left (4 d x +4 c \right )-1260 \sin \left (2 d x +2 c \right )-20480}{645120 d a}\) | \(111\) |
| risch | \(-\frac {3 x}{256 a}-\frac {3 \cos \left (d x +c \right )}{128 a d}-\frac {\sin \left (10 d x +10 c \right )}{5120 d a}+\frac {\cos \left (9 d x +9 c \right )}{2304 a d}-\frac {\sin \left (8 d x +8 c \right )}{2048 d a}+\frac {3 \cos \left (7 d x +7 c \right )}{1792 a d}+\frac {\sin \left (6 d x +6 c \right )}{1024 d a}+\frac {\sin \left (4 d x +4 c \right )}{256 d a}-\frac {\cos \left (3 d x +3 c \right )}{96 a d}-\frac {\sin \left (2 d x +2 c \right )}{512 d a}\) | \(158\) |
| derivativedivides | \(\frac {\frac {16 \left (-\frac {1}{252}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2048}-\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{126}+\frac {29 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}-\frac {867 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2560}-\frac {9 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}+\frac {519 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}-\frac {1879 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {1879 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {5 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {519 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}+\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {867 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2560}-\frac {\left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {29 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}-\frac {3 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}}{d a}\) | \(259\) |
| default | \(\frac {\frac {16 \left (-\frac {1}{252}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2048}-\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{126}+\frac {29 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}-\frac {867 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2560}-\frac {9 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}+\frac {519 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}-\frac {1879 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {1879 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {5 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {519 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}+\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {867 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2560}-\frac {\left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {29 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}-\frac {3 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}}{d a}\) | \(259\) |
1/645120*(-7560*d*x+1080*cos(7*d*x+7*c)-6720*cos(3*d*x+3*c)-15120*cos(d*x+ c)-126*sin(10*d*x+10*c)+280*cos(9*d*x+9*c)-315*sin(8*d*x+8*c)+630*sin(6*d* x+6*c)+2520*sin(4*d*x+4*c)-1260*sin(2*d*x+2*c)-20480)/d/a
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.55 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {8960 \, \cos \left (d x + c\right )^{9} - 11520 \, \cos \left (d x + c\right )^{7} - 945 \, d x - 63 \, {\left (128 \, \cos \left (d x + c\right )^{9} - 176 \, \cos \left (d x + c\right )^{7} + 8 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, a d} \]
1/80640*(8960*cos(d*x + c)^9 - 11520*cos(d*x + c)^7 - 945*d*x - 63*(128*co s(d*x + c)^9 - 176*cos(d*x + c)^7 + 8*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c))/(a*d)
Leaf count of result is larger than twice the leaf count of optimal. 5501 vs. \(2 (138) = 276\).
Time = 118.03 (sec) , antiderivative size = 5501, normalized size of antiderivative = 33.34 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
Piecewise((-945*d*x*tan(c/2 + d*x/2)**20/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676 800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 2032128 0*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a* d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c /2 + d*x/2)**2 + 80640*a*d) - 9450*d*x*tan(c/2 + d*x/2)**18/(80640*a*d*tan (c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d *x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x /2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 42525*d*x*tan(c/2 + d*x/2 )**16/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 1693 4400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 169344 00*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d *tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 11340 0*d*x*tan(c/2 + d*x/2)**14/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*ta n(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/ 2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 +...
Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (149) = 298\).
Time = 0.43 (sec) , antiderivative size = 583, normalized size of antiderivative = 3.53 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {25600 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {9135 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {46080 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {218484 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {414720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {653940 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {1183770 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {322560 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {1183770 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {537600 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {653940 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} + \frac {107520 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {218484 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {161280 \, \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} - \frac {9135 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - \frac {945 \, \sin \left (d x + c\right )^{19}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{19}} - 2560}{a + \frac {10 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {120 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {210 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {252 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {210 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {120 \, a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {45 \, a \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {10 \, a \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}} + \frac {a \sin \left (d x + c\right )^{20}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{20}}} - \frac {945 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{40320 \, d} \]
1/40320*((945*sin(d*x + c)/(cos(d*x + c) + 1) - 25600*sin(d*x + c)^2/(cos( d*x + c) + 1)^2 + 9135*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 46080*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 218484*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 414720*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 653940*sin(d*x + c)^7/(cos( d*x + c) + 1)^7 - 1183770*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 322560*sin (d*x + c)^10/(cos(d*x + c) + 1)^10 + 1183770*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 537600*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 653940*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 + 107520*sin(d*x + c)^14/(cos(d*x + c) + 1)^1 4 + 218484*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 161280*sin(d*x + c)^16/ (cos(d*x + c) + 1)^16 - 9135*sin(d*x + c)^17/(cos(d*x + c) + 1)^17 - 945*s in(d*x + c)^19/(cos(d*x + c) + 1)^19 - 2560)/(a + 10*a*sin(d*x + c)^2/(cos (d*x + c) + 1)^2 + 45*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 120*a*sin(d* x + c)^6/(cos(d*x + c) + 1)^6 + 210*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 252*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 210*a*sin(d*x + c)^12/(cos (d*x + c) + 1)^12 + 120*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 45*a*sin (d*x + c)^16/(cos(d*x + c) + 1)^16 + 10*a*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 + a*sin(d*x + c)^20/(cos(d*x + c) + 1)^20) - 945*arctan(sin(d*x + c) /(cos(d*x + c) + 1))/a)/d
Time = 0.34 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.56 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {945 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} + 9135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 161280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} - 218484 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 107520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 653940 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 537600 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 1183770 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 322560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1183770 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 653940 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 414720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 218484 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 46080 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 9135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 25600 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2560\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{10} a}}{80640 \, d} \]
-1/80640*(945*(d*x + c)/a + 2*(945*tan(1/2*d*x + 1/2*c)^19 + 9135*tan(1/2* d*x + 1/2*c)^17 + 161280*tan(1/2*d*x + 1/2*c)^16 - 218484*tan(1/2*d*x + 1/ 2*c)^15 - 107520*tan(1/2*d*x + 1/2*c)^14 + 653940*tan(1/2*d*x + 1/2*c)^13 + 537600*tan(1/2*d*x + 1/2*c)^12 - 1183770*tan(1/2*d*x + 1/2*c)^11 + 32256 0*tan(1/2*d*x + 1/2*c)^10 + 1183770*tan(1/2*d*x + 1/2*c)^9 - 653940*tan(1/ 2*d*x + 1/2*c)^7 + 414720*tan(1/2*d*x + 1/2*c)^6 + 218484*tan(1/2*d*x + 1/ 2*c)^5 - 46080*tan(1/2*d*x + 1/2*c)^4 - 9135*tan(1/2*d*x + 1/2*c)^3 + 2560 0*tan(1/2*d*x + 1/2*c)^2 - 945*tan(1/2*d*x + 1/2*c) + 2560)/((tan(1/2*d*x + 1/2*c)^2 + 1)^10*a))/d
Time = 13.21 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.52 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3\,x}{256\,a}-\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-\frac {867\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}+\frac {519\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}-\frac {1879\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {1879\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {519\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\frac {72\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}+\frac {867\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}-\frac {29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}+\frac {40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {4}{63}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]
- (3*x)/(256*a) - ((40*tan(c/2 + (d*x)/2)^2)/63 - (3*tan(c/2 + (d*x)/2))/1 28 - (29*tan(c/2 + (d*x)/2)^3)/128 - (8*tan(c/2 + (d*x)/2)^4)/7 + (867*tan (c/2 + (d*x)/2)^5)/160 + (72*tan(c/2 + (d*x)/2)^6)/7 - (519*tan(c/2 + (d*x )/2)^7)/32 + (1879*tan(c/2 + (d*x)/2)^9)/64 + 8*tan(c/2 + (d*x)/2)^10 - (1 879*tan(c/2 + (d*x)/2)^11)/64 + (40*tan(c/2 + (d*x)/2)^12)/3 + (519*tan(c/ 2 + (d*x)/2)^13)/32 - (8*tan(c/2 + (d*x)/2)^14)/3 - (867*tan(c/2 + (d*x)/2 )^15)/160 + 4*tan(c/2 + (d*x)/2)^16 + (29*tan(c/2 + (d*x)/2)^17)/128 + (3* tan(c/2 + (d*x)/2)^19)/128 + 4/63)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^10)