Integrand size = 29, antiderivative size = 183 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 x}{256 a}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {2 \cos ^9(c+d x)}{9 a d}+\frac {\cos ^{11}(c+d x)}{11 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{256 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}-\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}-\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d} \] Output:
3/256*x/a+1/7*cos(d*x+c)^7/a/d-2/9*cos(d*x+c)^9/a/d+1/11*cos(d*x+c)^11/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*c os(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. \(573\) vs. \(2(183)=366\).
Time = 13.69 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.13 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {97020 c}{a d}+\frac {83160 x}{a}-\frac {103950 \cos (c) \cos (d x)}{a d}+\frac {66990 \cos (3 c) \cos (3 d x)}{a d}-\frac {24948 \cos (5 c) \cos (5 d x)}{a d}+\frac {1980 \cos (7 c) \cos (7 d x)}{a d}+\frac {173250 \cos (c+d x)}{a d}-\frac {43890 \cos (3 (c+d x))}{a d}+\frac {18018 \cos (5 (c+d x))}{a d}-\frac {6930 \cos (7 (c+d x))}{a d}+\frac {770 \cos (9 (c+d x))}{a d}+\frac {630 \cos (11 (c+d x))}{a d}+\frac {90090 \cos (2 d x) \sin (2 c)}{a d}-\frac {55440 \cos (4 d x) \sin (4 c)}{a d}+\frac {4620 \cos (6 d x) \sin (6 c)}{a d}+\frac {103950 \sin (c) \sin (d x)}{a d}+\frac {90090 \cos (2 c) \sin (2 d x)}{a d}-\frac {66990 \sin (3 c) \sin (3 d x)}{a d}-\frac {55440 \cos (4 c) \sin (4 d x)}{a d}+\frac {24948 \sin (5 c) \sin (5 d x)}{a d}+\frac {4620 \cos (6 c) \sin (6 d x)}{a d}-\frac {1980 \sin (7 c) \sin (7 d x)}{a d}-\frac {76230 \sin \left (\frac {d x}{2}\right )}{a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {20790 \sin \left (\frac {1}{2} (c+d x)\right )}{a d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {48510 \sin (c+d x)}{a d (1+\sin (c+d x))}+\frac {97020 \sin ^2\left (\frac {1}{2} (c+d x)\right )}{d (a+a \sin (c+d x))}-\frac {76230 \sin (2 (c+d x))}{a d}+\frac {27720 \sin (4 (c+d x))}{a d}-\frac {11550 \sin (6 (c+d x))}{a d}+\frac {3465 \sin (8 (c+d x))}{a d}+\frac {1386 \sin (10 (c+d x))}{a d}}{7096320} \] Input:
Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
Output:
((97020*c)/(a*d) + (83160*x)/a - (103950*Cos[c]*Cos[d*x])/(a*d) + (66990*C os[3*c]*Cos[3*d*x])/(a*d) - (24948*Cos[5*c]*Cos[5*d*x])/(a*d) + (1980*Cos[ 7*c]*Cos[7*d*x])/(a*d) + (173250*Cos[c + d*x])/(a*d) - (43890*Cos[3*(c + d *x)])/(a*d) + (18018*Cos[5*(c + d*x)])/(a*d) - (6930*Cos[7*(c + d*x)])/(a* d) + (770*Cos[9*(c + d*x)])/(a*d) + (630*Cos[11*(c + d*x)])/(a*d) + (90090 *Cos[2*d*x]*Sin[2*c])/(a*d) - (55440*Cos[4*d*x]*Sin[4*c])/(a*d) + (4620*Co s[6*d*x]*Sin[6*c])/(a*d) + (103950*Sin[c]*Sin[d*x])/(a*d) + (90090*Cos[2*c ]*Sin[2*d*x])/(a*d) - (66990*Sin[3*c]*Sin[3*d*x])/(a*d) - (55440*Cos[4*c]* Sin[4*d*x])/(a*d) + (24948*Sin[5*c]*Sin[5*d*x])/(a*d) + (4620*Cos[6*c]*Sin [6*d*x])/(a*d) - (1980*Sin[7*c]*Sin[7*d*x])/(a*d) - (76230*Sin[(d*x)/2])/( a*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) - (20790* Sin[(c + d*x)/2])/(a*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (48510*Sin [c + d*x])/(a*d*(1 + Sin[c + d*x])) + (97020*Sin[(c + d*x)/2]^2)/(d*(a + a *Sin[c + d*x])) - (76230*Sin[2*(c + d*x)])/(a*d) + (27720*Sin[4*(c + d*x)] )/(a*d) - (11550*Sin[6*(c + d*x)])/(a*d) + (3465*Sin[8*(c + d*x)])/(a*d) + (1386*Sin[10*(c + d*x)])/(a*d))/7096320
Time = 0.86 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {\sin ^4(c+d x) \cos ^8(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^4 \cos (c+d x)^8}{a \sin (c+d x)+a}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cos ^6(c+d x) \sin ^4(c+d x)dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^5(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos (c+d x)^6 \sin (c+d x)^4dx}{a}-\frac {\int \cos (c+d x)^6 \sin (c+d x)^5dx}{a}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle \frac {\int \cos ^6(c+d x) \left (1-\cos ^2(c+d x)\right )^2d\cos (c+d x)}{a d}+\frac {\int \cos (c+d x)^6 \sin (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \left (\cos ^{10}(c+d x)-2 \cos ^8(c+d x)+\cos ^6(c+d x)\right )d\cos (c+d x)}{a d}+\frac {\int \cos (c+d x)^6 \sin (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \cos (c+d x)^6 \sin (c+d x)^4dx}{a}+\frac {\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\frac {3}{10} \int \cos ^6(c+d x) \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}+\frac {\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{10} \int \cos (c+d x)^6 \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}+\frac {\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\frac {3}{10} \left (\frac {1}{8} \int \cos ^6(c+d x)dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}+\frac {\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{10} \left (\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}+\frac {\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}+\frac {\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}+\frac {\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}+\frac {\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}+\frac {\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}+\frac {\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{11} \cos ^{11}(c+d x)-\frac {2}{9} \cos ^9(c+d x)+\frac {1}{7} \cos ^7(c+d x)}{a d}+\frac {\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}}{a}\) |
Input:
Int[(Cos[c + d*x]^8*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
Output:
(Cos[c + d*x]^7/7 - (2*Cos[c + d*x]^9)/9 + Cos[c + d*x]^11/11)/(a*d) + (-1 /10*(Cos[c + d*x]^7*Sin[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))/1 0)/a
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 = 2.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.73
| method | result | size |
| parallelrisch | \(\frac {81920-6930 \sin \left (6 d x +6 c \right )+13860 \sin \left (2 d x +2 c \right )+23100 \cos \left (3 d x +3 c \right )-27720 \sin \left (4 d x +4 c \right )+3465 \sin \left (8 d x +8 c \right )+83160 d x +69300 \cos \left (d x +c \right )-6930 \cos \left (5 d x +5 c \right )-4950 \cos \left (7 d x +7 c \right )+630 \cos \left (11 d x +11 c \right )+770 \cos \left (9 d x +9 c \right )+1386 \sin \left (10 d x +10 c \right )}{7096320 d a}\) | \(133\) |
| risch | \(\frac {3 x}{256 a}+\frac {5 \cos \left (d x +c \right )}{512 a d}+\frac {\cos \left (11 d x +11 c \right )}{11264 a d}+\frac {\sin \left (10 d x +10 c \right )}{5120 d a}+\frac {\cos \left (9 d x +9 c \right )}{9216 a d}+\frac {\sin \left (8 d x +8 c \right )}{2048 d a}-\frac {5 \cos \left (7 d x +7 c \right )}{7168 a d}-\frac {\sin \left (6 d x +6 c \right )}{1024 d a}-\frac {\cos \left (5 d x +5 c \right )}{1024 a d}-\frac {\sin \left (4 d x +4 c \right )}{256 d a}+\frac {5 \cos \left (3 d x +3 c \right )}{1536 a d}+\frac {\sin \left (2 d x +2 c \right )}{512 d a}\) | \(192\) |
| derivativedivides | \(\frac {\frac {32 \left (\frac {1}{1386}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4096}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{126}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{128}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{126}+\frac {3323 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20480}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{14}-\frac {27 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{80}+\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{14}+\frac {841 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{2048}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{6}-\frac {841 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{2048}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{6}+\frac {27 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{80}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{3}-\frac {3323 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{20480}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{19}}{128}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{21}}{4096}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{11}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}}{d a}\) | \(272\) |
| default | \(\frac {\frac {32 \left (\frac {1}{1386}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4096}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{126}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{128}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{126}+\frac {3323 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20480}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{14}-\frac {27 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{80}+\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{14}+\frac {841 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{2048}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{6}-\frac {841 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{2048}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{6}+\frac {27 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{80}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{3}-\frac {3323 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{20480}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{19}}{128}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{21}}{4096}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{11}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}}{d a}\) | \(272\) |
Input:
int(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/7096320*(81920-6930*sin(6*d*x+6*c)+13860*sin(2*d*x+2*c)+23100*cos(3*d*x+ 3*c)-27720*sin(4*d*x+4*c)+3465*sin(8*d*x+8*c)+83160*d*x+69300*cos(d*x+c)-6 930*cos(5*d*x+5*c)-4950*cos(7*d*x+7*c)+630*cos(11*d*x+11*c)+770*cos(9*d*x+ 9*c)+1386*sin(10*d*x+10*c))/d/a
Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.55 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {80640 \, \cos \left (d x + c\right )^{11} - 197120 \, \cos \left (d x + c\right )^{9} + 126720 \, \cos \left (d x + c\right )^{7} + 10395 \, d x + 693 \, {\left (128 \, \cos \left (d x + c\right )^{9} - 176 \, \cos \left (d x + c\right )^{7} + 8 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, a d} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/887040*(80640*cos(d*x + c)^11 - 197120*cos(d*x + c)^9 + 126720*cos(d*x + c)^7 + 10395*d*x + 693*(128*cos(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)
Timed out. \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**8*sin(d*x+c)**4/(a+a*sin(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (165) = 330\).
Time = 0.14 (sec) , antiderivative size = 624, normalized size of antiderivative = 3.41 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/443520*((10395*sin(d*x + c)/(cos(d*x + c) + 1) - 112640*sin(d*x + c)^2/ (cos(d*x + c) + 1)^2 + 110880*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 563200 *sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 2302839*sin(d*x + c)^5/(cos(d*x + c ) + 1)^5 + 3041280*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 4790016*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 15206400*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 5828130*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 21288960*sin(d*x + c)^10/( cos(d*x + c) + 1)^10 - 26019840*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 58 28130*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 + 11827200*sin(d*x + c)^14/(co s(d*x + c) + 1)^14 - 4790016*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 47308 80*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 + 2302839*sin(d*x + c)^17/(cos(d* x + c) + 1)^17 - 110880*sin(d*x + c)^19/(cos(d*x + c) + 1)^19 - 10395*sin( d*x + c)^21/(cos(d*x + c) + 1)^21 - 10240)/(a + 11*a*sin(d*x + c)^2/(cos(d *x + c) + 1)^2 + 55*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 165*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 330*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 462*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 462*a*sin(d*x + c)^12/(cos(d *x + c) + 1)^12 + 330*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 165*a*sin( d*x + c)^16/(cos(d*x + c) + 1)^16 + 55*a*sin(d*x + c)^18/(cos(d*x + c) + 1 )^18 + 11*a*sin(d*x + c)^20/(cos(d*x + c) + 1)^20 + a*sin(d*x + c)^22/(cos (d*x + c) + 1)^22) - 10395*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d
Time = 0.15 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.48 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {10395 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (10395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{21} + 110880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} - 2302839 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 4730880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} + 4790016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 11827200 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 5828130 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 26019840 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 21288960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 5828130 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 15206400 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 4790016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3041280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2302839 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 563200 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 110880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10240\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{11} a}}{887040 \, d} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/887040*(10395*(d*x + c)/a + 2*(10395*tan(1/2*d*x + 1/2*c)^21 + 110880*ta n(1/2*d*x + 1/2*c)^19 - 2302839*tan(1/2*d*x + 1/2*c)^17 + 4730880*tan(1/2* d*x + 1/2*c)^16 + 4790016*tan(1/2*d*x + 1/2*c)^15 - 11827200*tan(1/2*d*x + 1/2*c)^14 - 5828130*tan(1/2*d*x + 1/2*c)^13 + 26019840*tan(1/2*d*x + 1/2* c)^12 - 21288960*tan(1/2*d*x + 1/2*c)^10 + 5828130*tan(1/2*d*x + 1/2*c)^9 + 15206400*tan(1/2*d*x + 1/2*c)^8 - 4790016*tan(1/2*d*x + 1/2*c)^7 - 30412 80*tan(1/2*d*x + 1/2*c)^6 + 2302839*tan(1/2*d*x + 1/2*c)^5 + 563200*tan(1/ 2*d*x + 1/2*c)^4 - 110880*tan(1/2*d*x + 1/2*c)^3 + 112640*tan(1/2*d*x + 1/ 2*c)^2 - 10395*tan(1/2*d*x + 1/2*c) + 10240)/((tan(1/2*d*x + 1/2*c)^2 + 1) ^11*a))/d
Time = 34.38 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3\,x}{256\,a}+\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{128}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{4}-\frac {3323\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{640}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{3}+\frac {54\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}-\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}-\frac {841\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {176\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}-48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {841\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {240\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}-\frac {54\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}-\frac {48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}+\frac {3323\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640}+\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{63}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {16}{693}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \] Input:
int((cos(c + d*x)^8*sin(c + d*x)^4)/(a + a*sin(c + d*x)),x)
Output:
(3*x)/(256*a) + ((16*tan(c/2 + (d*x)/2)^2)/63 - (3*tan(c/2 + (d*x)/2))/128 - tan(c/2 + (d*x)/2)^3/4 + (80*tan(c/2 + (d*x)/2)^4)/63 + (3323*tan(c/2 + (d*x)/2)^5)/640 - (48*tan(c/2 + (d*x)/2)^6)/7 - (54*tan(c/2 + (d*x)/2)^7) /5 + (240*tan(c/2 + (d*x)/2)^8)/7 + (841*tan(c/2 + (d*x)/2)^9)/64 - 48*tan (c/2 + (d*x)/2)^10 + (176*tan(c/2 + (d*x)/2)^12)/3 - (841*tan(c/2 + (d*x)/ 2)^13)/64 - (80*tan(c/2 + (d*x)/2)^14)/3 + (54*tan(c/2 + (d*x)/2)^15)/5 + (32*tan(c/2 + (d*x)/2)^16)/3 - (3323*tan(c/2 + (d*x)/2)^17)/640 + tan(c/2 + (d*x)/2)^19/4 + (3*tan(c/2 + (d*x)/2)^21)/128 + 16/693)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^11)
Time = 0.41 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-80640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10}+88704 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}+206080 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-232848 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-144640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+171864 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+3840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-6930 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+5120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-10395 \cos \left (d x +c \right ) \sin \left (d x +c \right )+10240 \cos \left (d x +c \right )+10395 d x -10240}{887040 a d} \] Input:
int(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c)),x)
Output:
( - 80640*cos(c + d*x)*sin(c + d*x)**10 + 88704*cos(c + d*x)*sin(c + d*x)* *9 + 206080*cos(c + d*x)*sin(c + d*x)**8 - 232848*cos(c + d*x)*sin(c + d*x )**7 - 144640*cos(c + d*x)*sin(c + d*x)**6 + 171864*cos(c + d*x)*sin(c + d *x)**5 + 3840*cos(c + d*x)*sin(c + d*x)**4 - 6930*cos(c + d*x)*sin(c + d*x )**3 + 5120*cos(c + d*x)*sin(c + d*x)**2 - 10395*cos(c + d*x)*sin(c + d*x) + 10240*cos(c + d*x) + 10395*d*x - 10240)/(887040*a*d)