Integrand size = 29, antiderivative size = 203 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3 x}{128 a^2}-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {5 \cos ^7(c+d x)}{7 a^2 d}-\frac {4 \cos ^9(c+d x)}{9 a^2 d}+\frac {\cos ^{11}(c+d x)}{11 a^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{128 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{5 a^2 d} \] Output:
-3/128*x/a^2-2/5*cos(d*x+c)^5/a^2/d+5/7*cos(d*x+c)^7/a^2/d-4/9*cos(d*x+c)^ 9/a^2/d+1/11*cos(d*x+c)^11/a^2/d-3/128*cos(d*x+c)*sin(d*x+c)/a^2/d-1/64*co s(d*x+c)^3*sin(d*x+c)/a^2/d+1/16*cos(d*x+c)^5*sin(d*x+c)/a^2/d+1/8*cos(d*x +c)^5*sin(d*x+c)^3/a^2/d+1/5*cos(d*x+c)^5*sin(d*x+c)^5/a^2/d
Leaf count is larger than twice the leaf count of optimal. \(1453\) vs. \(2(203)=406\).
Time = 11.59 (sec) , antiderivative size = 1453, normalized size of antiderivative = 7.16 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]
Output:
(-5*Cos[c + d*x]*(1 + 2*Sin[c + d*x]))/(3072*a^2*d*(1 + Sin[c + d*x])^2) + (27720*(c + d*x) + 41580*Cos[c + d*x] - 7056*Cos[3*(c + d*x)] + 1764*Cos[ 5*(c + d*x)] - 360*Cos[7*(c + d*x)] + 28*Cos[9*(c + d*x)] + (42*Sin[(c + d *x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 21/(Cos[(c + d*x)/2] + S in[(c + d*x)/2])^2 - (15204*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 15120*Sin[2*(c + d*x)] + 3528*Sin[4*(c + d*x)] - 840*Sin[6*(c + d*x)] + 126*Sin[8*(c + d*x)])/(86016*a^2*d) + (-360360*(c + d*x) - 56628 0*Cos[c + d*x] + 108900*Cos[3*(c + d*x)] - 33264*Cos[5*(c + d*x)] + 9900*C os[7*(c + d*x)] - 2200*Cos[9*(c + d*x)] + 180*Cos[11*(c + d*x)] - (330*Sin [(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 165/(Cos[(c + d*x )/2] + Sin[(c + d*x)/2])^2 + (166980*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 217800*Sin[2*(c + d*x)] - 59400*Sin[4*(c + d*x)] + 18 480*Sin[6*(c + d*x)] - 4950*Sin[8*(c + d*x)] + 792*Sin[10*(c + d*x)])/(202 7520*a^2*d) + (25*(36*d*x*Cos[(d*x)/2] - 21*Cos[c + (d*x)/2] + 35*Cos[c + (3*d*x)/2] - 12*d*x*Cos[2*c + (3*d*x)/2] - 3*Cos[3*c + (5*d*x)/2] - 57*Sin [(d*x)/2] + 36*d*x*Sin[c + (d*x)/2] + 12*d*x*Sin[c + (3*d*x)/2] + 9*Sin[2* c + (3*d*x)/2] + 3*Sin[2*c + (5*d*x)/2]))/(12288*a^2*d*(Cos[c/2] + Sin[c/2 ])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (5*(180*d*x*Cos[(d*x)/2] - 2 1*Cos[c + (d*x)/2] + 147*Cos[c + (3*d*x)/2] - 60*d*x*Cos[2*c + (3*d*x)/2] - 15*Cos[3*c + (5*d*x)/2] + 3*Cos[3*c + (7*d*x)/2] + Cos[5*c + (9*d*x)/...
Time = 0.69 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int \frac {\sin ^5(c+d x) \cos ^8(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^5 \cos (c+d x)^8}{(a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cos ^4(c+d x) \sin ^5(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos (c+d x)^4 \sin (c+d x)^5 (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^2 \cos ^4(c+d x) \sin ^7(c+d x)-2 a^2 \cos ^4(c+d x) \sin ^6(c+d x)+a^2 \cos ^4(c+d x) \sin ^5(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^2 \cos ^{11}(c+d x)}{11 d}-\frac {4 a^2 \cos ^9(c+d x)}{9 d}+\frac {5 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {a^2 \sin ^5(c+d x) \cos ^5(c+d x)}{5 d}+\frac {a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}+\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}-\frac {3 a^2 \sin (c+d x) \cos (c+d x)}{128 d}-\frac {3 a^2 x}{128}}{a^4}\) |
Input:
Int[(Cos[c + d*x]^8*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]
Output:
((-3*a^2*x)/128 - (2*a^2*Cos[c + d*x]^5)/(5*d) + (5*a^2*Cos[c + d*x]^7)/(7 *d) - (4*a^2*Cos[c + d*x]^9)/(9*d) + (a^2*Cos[c + d*x]^11)/(11*d) - (3*a^2 *Cos[c + d*x]*Sin[c + d*x])/(128*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(6 4*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) + (a^2*Cos[c + d*x]^5*Sin[ c + d*x]^3)/(8*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x]^5)/(5*d))/a^4
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 = 2.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.66
| method | result | size |
| parallelrisch | \(\frac {-139264-6930 \sin \left (6 d x +6 c \right )+13860 \sin \left (2 d x +2 c \right )-25410 \cos \left (3 d x +3 c \right )+27720 \sin \left (4 d x +4 c \right )-3465 \sin \left (8 d x +8 c \right )-83160 d x -131670 \cos \left (d x +c \right )+18711 \cos \left (5 d x +5 c \right )+1485 \cos \left (7 d x +7 c \right )+315 \cos \left (11 d x +11 c \right )-2695 \cos \left (9 d x +9 c \right )+1386 \sin \left (10 d x +10 c \right )}{3548160 d \,a^{2}}\) | \(133\) |
| risch | \(-\frac {19 \cos \left (d x +c \right )}{512 a^{2} d}-\frac {3 x}{128 a^{2}}+\frac {\cos \left (11 d x +11 c \right )}{11264 d \,a^{2}}+\frac {\sin \left (10 d x +10 c \right )}{2560 d \,a^{2}}-\frac {7 \cos \left (9 d x +9 c \right )}{9216 d \,a^{2}}-\frac {\sin \left (8 d x +8 c \right )}{1024 d \,a^{2}}+\frac {3 \cos \left (7 d x +7 c \right )}{7168 d \,a^{2}}-\frac {\sin \left (6 d x +6 c \right )}{512 d \,a^{2}}+\frac {27 \cos \left (5 d x +5 c \right )}{5120 a^{2} d}+\frac {\sin \left (4 d x +4 c \right )}{128 d \,a^{2}}-\frac {11 \cos \left (3 d x +3 c \right )}{1536 d \,a^{2}}+\frac {\sin \left (2 d x +2 c \right )}{256 d \,a^{2}}\) | \(192\) |
| derivativedivides | \(\frac {\frac {64 \left (-\frac {17}{13860}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4096}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{1260}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{128}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{252}+\frac {773 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20480}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{28}-\frac {37 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{80}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{28}+\frac {1207 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{2048}-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{20}+\frac {53 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{60}-\frac {1207 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{2048}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{12}+\frac {37 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{80}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{6}-\frac {773 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{20480}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{19}}{128}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{21}}{4096}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{11}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d \,a^{2}}\) | \(272\) |
| default | \(\frac {\frac {64 \left (-\frac {17}{13860}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4096}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{1260}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{128}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{252}+\frac {773 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20480}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{28}-\frac {37 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{80}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{28}+\frac {1207 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{2048}-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{20}+\frac {53 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{60}-\frac {1207 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{2048}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{12}+\frac {37 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{80}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{6}-\frac {773 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{20480}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{19}}{128}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{21}}{4096}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{11}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d \,a^{2}}\) | \(272\) |
Input:
int(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/3548160*(-139264-6930*sin(6*d*x+6*c)+13860*sin(2*d*x+2*c)-25410*cos(3*d* x+3*c)+27720*sin(4*d*x+4*c)-3465*sin(8*d*x+8*c)-83160*d*x-131670*cos(d*x+c )+18711*cos(5*d*x+5*c)+1485*cos(7*d*x+7*c)+315*cos(11*d*x+11*c)-2695*cos(9 *d*x+9*c)+1386*sin(10*d*x+10*c))/d/a^2
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.54 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {40320 \, \cos \left (d x + c\right )^{11} - 197120 \, \cos \left (d x + c\right )^{9} + 316800 \, \cos \left (d x + c\right )^{7} - 177408 \, \cos \left (d x + c\right )^{5} - 10395 \, d x + 693 \, {\left (128 \, \cos \left (d x + c\right )^{9} - 336 \, \cos \left (d x + c\right )^{7} + 248 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{443520 \, a^{2} d} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
1/443520*(40320*cos(d*x + c)^11 - 197120*cos(d*x + c)^9 + 316800*cos(d*x + c)^7 - 177408*cos(d*x + c)^5 - 10395*d*x + 693*(128*cos(d*x + c)^9 - 336* cos(d*x + c)^7 + 248*cos(d*x + c)^5 - 10*cos(d*x + c)^3 - 15*cos(d*x + c)) *sin(d*x + c))/(a^2*d)
Timed out. \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**8*sin(d*x+c)**5/(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 648 vs. \(2 (183) = 366\).
Time = 0.15 (sec) , antiderivative size = 648, normalized size of antiderivative = 3.19 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
1/221760*((10395*sin(d*x + c)/(cos(d*x + c) + 1) - 191488*sin(d*x + c)^2/( cos(d*x + c) + 1)^2 + 110880*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 957440* sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 535689*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 506880*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 6564096*sin(d*x + c) ^7/(cos(d*x + c) + 1)^7 + 2534400*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 83 64510*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 20579328*sin(d*x + c)^10/(cos( d*x + c) + 1)^10 + 12536832*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 836451 0*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 8279040*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 6564096*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 2365440*si n(d*x + c)^16/(cos(d*x + c) + 1)^16 - 535689*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 - 17408)/(a^2 + 11*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 55*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 165*a^2*sin(d* x + c)^6/(cos(d*x + c) + 1)^6 + 330*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^ 8 + 462*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 462*a^2*sin(d*x + c)^1 2/(cos(d*x + c) + 1)^12 + 330*a^2*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 165*a^2*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 + 55*a^2*sin(d*x + c)^18/(co s(d*x + c) + 1)^18 + 11*a^2*sin(d*x + c)^20/(cos(d*x + c) + 1)^20 + a^2*si n(d*x + c)^22/(cos(d*x + c) + 1)^22) - 10395*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d
Time = 0.17 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {10395 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (10395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{21} + 110880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} + 535689 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 2365440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} - 6564096 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 8279040 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 8364510 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 12536832 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 20579328 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 8364510 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 2534400 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6564096 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 506880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 535689 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 957440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 110880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 191488 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17408\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{11} a^{2}}}{443520 \, d} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
-1/443520*(10395*(d*x + c)/a^2 + 2*(10395*tan(1/2*d*x + 1/2*c)^21 + 110880 *tan(1/2*d*x + 1/2*c)^19 + 535689*tan(1/2*d*x + 1/2*c)^17 + 2365440*tan(1/ 2*d*x + 1/2*c)^16 - 6564096*tan(1/2*d*x + 1/2*c)^15 + 8279040*tan(1/2*d*x + 1/2*c)^14 + 8364510*tan(1/2*d*x + 1/2*c)^13 - 12536832*tan(1/2*d*x + 1/2 *c)^12 + 20579328*tan(1/2*d*x + 1/2*c)^10 - 8364510*tan(1/2*d*x + 1/2*c)^9 - 2534400*tan(1/2*d*x + 1/2*c)^8 + 6564096*tan(1/2*d*x + 1/2*c)^7 + 50688 0*tan(1/2*d*x + 1/2*c)^6 - 535689*tan(1/2*d*x + 1/2*c)^5 + 957440*tan(1/2* d*x + 1/2*c)^4 - 110880*tan(1/2*d*x + 1/2*c)^3 + 191488*tan(1/2*d*x + 1/2* c)^2 - 10395*tan(1/2*d*x + 1/2*c) + 17408)/((tan(1/2*d*x + 1/2*c)^2 + 1)^1 1*a^2))/d
Time = 34.44 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.30 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3\,x}{128\,a^2}-\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{64}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{2}+\frac {773\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{320}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{3}-\frac {148\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\frac {112\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}+\frac {1207\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {848\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{15}+\frac {464\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5}-\frac {1207\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{32}-\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}+\frac {148\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}-\frac {773\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}+\frac {272\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{63}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {272\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{315}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {272}{3465}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \] Input:
int((cos(c + d*x)^8*sin(c + d*x)^5)/(a + a*sin(c + d*x))^2,x)
Output:
- (3*x)/(128*a^2) - ((272*tan(c/2 + (d*x)/2)^2)/315 - (3*tan(c/2 + (d*x)/2 ))/64 - tan(c/2 + (d*x)/2)^3/2 + (272*tan(c/2 + (d*x)/2)^4)/63 - (773*tan( c/2 + (d*x)/2)^5)/320 + (16*tan(c/2 + (d*x)/2)^6)/7 + (148*tan(c/2 + (d*x) /2)^7)/5 - (80*tan(c/2 + (d*x)/2)^8)/7 - (1207*tan(c/2 + (d*x)/2)^9)/32 + (464*tan(c/2 + (d*x)/2)^10)/5 - (848*tan(c/2 + (d*x)/2)^12)/15 + (1207*tan (c/2 + (d*x)/2)^13)/32 + (112*tan(c/2 + (d*x)/2)^14)/3 - (148*tan(c/2 + (d *x)/2)^15)/5 + (32*tan(c/2 + (d*x)/2)^16)/3 + (773*tan(c/2 + (d*x)/2)^17)/ 320 + tan(c/2 + (d*x)/2)^19/2 + (3*tan(c/2 + (d*x)/2)^21)/64 + 272/3465)/( a^2*d*(tan(c/2 + (d*x)/2)^2 + 1)^11)
Time = 1.91 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {-40320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10}+88704 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}+4480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-121968 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+68480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+5544 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-6528 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+6930 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-8704 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+10395 \cos \left (d x +c \right ) \sin \left (d x +c \right )-17408 \cos \left (d x +c \right )-10395 d x +17408}{443520 a^{2} d} \] Input:
int(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x)
Output:
( - 40320*cos(c + d*x)*sin(c + d*x)**10 + 88704*cos(c + d*x)*sin(c + d*x)* *9 + 4480*cos(c + d*x)*sin(c + d*x)**8 - 121968*cos(c + d*x)*sin(c + d*x)* *7 + 68480*cos(c + d*x)*sin(c + d*x)**6 + 5544*cos(c + d*x)*sin(c + d*x)** 5 - 6528*cos(c + d*x)*sin(c + d*x)**4 + 6930*cos(c + d*x)*sin(c + d*x)**3 - 8704*cos(c + d*x)*sin(c + d*x)**2 + 10395*cos(c + d*x)*sin(c + d*x) - 17 408*cos(c + d*x) - 10395*d*x + 17408)/(443520*a**2*d)