Integrand size = 27, antiderivative size = 131 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {7 x}{16 a^3}-\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )} \] Output:
-7/16*x/a^3-7/30*cos(d*x+c)^5/a^3/d-7/16*cos(d*x+c)*sin(d*x+c)/a^3/d-7/24* cos(d*x+c)^3*sin(d*x+c)/a^3/d-1/3*cos(d*x+c)^9/d/(a+a*sin(d*x+c))^3-1/6*co s(d*x+c)^7/d/(a^3+a^3*sin(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(366\) vs. \(2(131)=262\).
Time = 2.81 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.79 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-21 (1+40 d x) \cos \left (\frac {c}{2}\right )-600 \cos \left (\frac {c}{2}+d x\right )-600 \cos \left (\frac {3 c}{2}+d x\right )+15 \cos \left (\frac {3 c}{2}+2 d x\right )-15 \cos \left (\frac {5 c}{2}+2 d x\right )-140 \cos \left (\frac {5 c}{2}+3 d x\right )-140 \cos \left (\frac {7 c}{2}+3 d x\right )+105 \cos \left (\frac {7 c}{2}+4 d x\right )-105 \cos \left (\frac {9 c}{2}+4 d x\right )+36 \cos \left (\frac {9 c}{2}+5 d x\right )+36 \cos \left (\frac {11 c}{2}+5 d x\right )-5 \cos \left (\frac {11 c}{2}+6 d x\right )+5 \cos \left (\frac {13 c}{2}+6 d x\right )+21 \sin \left (\frac {c}{2}\right )-840 d x \sin \left (\frac {c}{2}\right )+600 \sin \left (\frac {c}{2}+d x\right )-600 \sin \left (\frac {3 c}{2}+d x\right )+15 \sin \left (\frac {3 c}{2}+2 d x\right )+15 \sin \left (\frac {5 c}{2}+2 d x\right )+140 \sin \left (\frac {5 c}{2}+3 d x\right )-140 \sin \left (\frac {7 c}{2}+3 d x\right )+105 \sin \left (\frac {7 c}{2}+4 d x\right )+105 \sin \left (\frac {9 c}{2}+4 d x\right )-36 \sin \left (\frac {9 c}{2}+5 d x\right )+36 \sin \left (\frac {11 c}{2}+5 d x\right )-5 \sin \left (\frac {11 c}{2}+6 d x\right )-5 \sin \left (\frac {13 c}{2}+6 d x\right )}{1920 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \] Input:
Integrate[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]
Output:
(-21*(1 + 40*d*x)*Cos[c/2] - 600*Cos[c/2 + d*x] - 600*Cos[(3*c)/2 + d*x] + 15*Cos[(3*c)/2 + 2*d*x] - 15*Cos[(5*c)/2 + 2*d*x] - 140*Cos[(5*c)/2 + 3*d *x] - 140*Cos[(7*c)/2 + 3*d*x] + 105*Cos[(7*c)/2 + 4*d*x] - 105*Cos[(9*c)/ 2 + 4*d*x] + 36*Cos[(9*c)/2 + 5*d*x] + 36*Cos[(11*c)/2 + 5*d*x] - 5*Cos[(1 1*c)/2 + 6*d*x] + 5*Cos[(13*c)/2 + 6*d*x] + 21*Sin[c/2] - 840*d*x*Sin[c/2] + 600*Sin[c/2 + d*x] - 600*Sin[(3*c)/2 + d*x] + 15*Sin[(3*c)/2 + 2*d*x] + 15*Sin[(5*c)/2 + 2*d*x] + 140*Sin[(5*c)/2 + 3*d*x] - 140*Sin[(7*c)/2 + 3* d*x] + 105*Sin[(7*c)/2 + 4*d*x] + 105*Sin[(9*c)/2 + 4*d*x] - 36*Sin[(9*c)/ 2 + 5*d*x] + 36*Sin[(11*c)/2 + 5*d*x] - 5*Sin[(11*c)/2 + 6*d*x] - 5*Sin[(1 3*c)/2 + 6*d*x])/(1920*a^3*d*(Cos[c/2] + Sin[c/2]))
Time = 0.70 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {3042, 3338, 3042, 3158, 3042, 3161, 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.
| \(\displaystyle \int \frac {\sin (c+d x) \cos ^8(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x) \cos (c+d x)^8}{(a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle -\frac {\int \frac {\cos ^8(c+d x)}{(\sin (c+d x) a+a)^2}dx}{a}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\cos (c+d x)^8}{(\sin (c+d x) a+a)^2}dx}{a}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle -\frac {\frac {7 \int \frac {\cos ^6(c+d x)}{\sin (c+d x) a+a}dx}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}}{a}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {7 \int \frac {\cos (c+d x)^6}{\sin (c+d x) a+a}dx}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}}{a}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle -\frac {\frac {7 \left (\frac {\int \cos ^4(c+d x)dx}{a}+\frac {\cos ^5(c+d x)}{5 a d}\right )}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}}{a}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {7 \left (\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx}{a}+\frac {\cos ^5(c+d x)}{5 a d}\right )}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}}{a}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {7 \left (\frac {\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}}{a}+\frac {\cos ^5(c+d x)}{5 a d}\right )}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}}{a}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {7 \left (\frac {\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}}{a}+\frac {\cos ^5(c+d x)}{5 a d}\right )}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}}{a}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {7 \left (\frac {\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}}{a}+\frac {\cos ^5(c+d x)}{5 a d}\right )}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}}{a}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {7 \left (\frac {\cos ^5(c+d x)}{5 a d}+\frac {\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a}\right )}{6 a}}{a}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}\) |
Input:
Int[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]
Output:
-1/3*Cos[c + d*x]^9/(d*(a + a*Sin[c + d*x])^3) - (Cos[c + d*x]^7/(6*d*(a^2 + a^2*Sin[c + d*x])) + (7*(Cos[c + d*x]^5/(5*a*d) + ((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)/a))/(6* a))/a
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_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + p))), x] + Simp[g^2*((p - 1)/(a*(m + p))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p, 0] && In tegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[g*((g*Cos[e + f*x])^(p - 1)/(b*f*(p - 1))), x] + Si mp[g^2/a Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g}, x ] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1) )), x] + Simp[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0 ]) && NeQ[2*m + p + 1, 0]
Time = 1.58 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60
| method | result | size |
| parallelrisch | \(\frac {-420 d x +36 \cos \left (5 d x +5 c \right )-140 \cos \left (3 d x +3 c \right )-600 \cos \left (d x +c \right )-5 \sin \left (6 d x +6 c \right )+105 \sin \left (4 d x +4 c \right )+15 \sin \left (2 d x +2 c \right )-704}{960 d \,a^{3}}\) | \(78\) |
| risch | \(-\frac {7 x}{16 a^{3}}-\frac {5 \cos \left (d x +c \right )}{8 a^{3} d}-\frac {\sin \left (6 d x +6 c \right )}{192 d \,a^{3}}+\frac {3 \cos \left (5 d x +5 c \right )}{80 a^{3} d}+\frac {7 \sin \left (4 d x +4 c \right )}{64 d \,a^{3}}-\frac {7 \cos \left (3 d x +3 c \right )}{48 a^{3} d}+\frac {\sin \left (2 d x +2 c \right )}{64 d \,a^{3}}\) | \(107\) |
| derivativedivides | \(\frac {\frac {4 \left (-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{32}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}+\frac {73 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{96}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {37 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{16}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}-\frac {37 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{16}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {73 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{96}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{10}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}-\frac {11}{30}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{3}}\) | \(181\) |
| default | \(\frac {\frac {4 \left (-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{32}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}+\frac {73 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{96}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {37 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{16}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}-\frac {37 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{16}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {73 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{96}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{10}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}-\frac {11}{30}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{3}}\) | \(181\) |
Input:
int(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/960*(-420*d*x+36*cos(5*d*x+5*c)-140*cos(3*d*x+3*c)-600*cos(d*x+c)-5*sin( 6*d*x+6*c)+105*sin(4*d*x+4*c)+15*sin(2*d*x+2*c)-704)/d/a^3
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.53 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {144 \, \cos \left (d x + c\right )^{5} - 320 \, \cos \left (d x + c\right )^{3} - 105 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 50 \, \cos \left (d x + c\right )^{3} + 21 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{3} d} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
1/240*(144*cos(d*x + c)^5 - 320*cos(d*x + c)^3 - 105*d*x - 5*(8*cos(d*x + c)^5 - 50*cos(d*x + c)^3 + 21*cos(d*x + c))*sin(d*x + c))/(a^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 2535 vs. \(2 (121) = 242\).
Time = 134.71 (sec) , antiderivative size = 2535, normalized size of antiderivative = 19.35 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)**8*sin(d*x+c)/(a+a*sin(d*x+c))**3,x)
Output:
Piecewise((-105*d*x*tan(c/2 + d*x/2)**12/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 480 0*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3 *d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 630*d*x*tan(c/2 + d*x/2)**10/(240*a **3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3* d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan( c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 1575*d*x *tan(c/2 + d*x/2)**8/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/ 2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d *x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)* *2 + 240*a**3*d) - 2100*d*x*tan(c/2 + d*x/2)**6/(240*a**3*d*tan(c/2 + d*x/ 2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)** 8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 14 40*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 1575*d*x*tan(c/2 + d*x/2)**4 /(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 360 0*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3 *d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 6 30*d*x*tan(c/2 + d*x/2)**2/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d* tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c /2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 ...
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (119) = 238\).
Time = 0.12 (sec) , antiderivative size = 393, normalized size of antiderivative = 3.00 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {816 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {365 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {480 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {1110 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1760 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1110 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {2160 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {365 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {240 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {105 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 176}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{120 \, d} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
1/120*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 816*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 365*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 480*sin(d*x + c)^4 /(cos(d*x + c) + 1)^4 - 1110*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1760*si n(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1110*sin(d*x + c)^7/(cos(d*x + c) + 1) ^7 - 2160*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 365*sin(d*x + c)^9/(cos(d* x + c) + 1)^9 - 240*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 105*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 176)/(a^3 + 6*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 20*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 6 *a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 105*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d
Time = 0.13 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.37 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {105 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 365 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1760 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 365 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 816 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 176\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{3}}}{240 \, d} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
-1/240*(105*(d*x + c)/a^3 + 2*(105*tan(1/2*d*x + 1/2*c)^11 + 240*tan(1/2*d *x + 1/2*c)^10 - 365*tan(1/2*d*x + 1/2*c)^9 + 2160*tan(1/2*d*x + 1/2*c)^8 - 1110*tan(1/2*d*x + 1/2*c)^7 + 1760*tan(1/2*d*x + 1/2*c)^6 + 1110*tan(1/2 *d*x + 1/2*c)^5 + 480*tan(1/2*d*x + 1/2*c)^4 + 365*tan(1/2*d*x + 1/2*c)^3 + 816*tan(1/2*d*x + 1/2*c)^2 - 105*tan(1/2*d*x + 1/2*c) + 176)/((tan(1/2*d *x + 1/2*c)^2 + 1)^6*a^3))/d
Time = 36.06 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.32 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {7\,x}{16\,a^3}-\frac {\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {73\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {37\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {37\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {73\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {34\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {22}{15}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \] Input:
int((cos(c + d*x)^8*sin(c + d*x))/(a + a*sin(c + d*x))^3,x)
Output:
- (7*x)/(16*a^3) - ((34*tan(c/2 + (d*x)/2)^2)/5 - (7*tan(c/2 + (d*x)/2))/8 + (73*tan(c/2 + (d*x)/2)^3)/24 + 4*tan(c/2 + (d*x)/2)^4 + (37*tan(c/2 + ( d*x)/2)^5)/4 + (44*tan(c/2 + (d*x)/2)^6)/3 - (37*tan(c/2 + (d*x)/2)^7)/4 + 18*tan(c/2 + (d*x)/2)^8 - (73*tan(c/2 + (d*x)/2)^9)/24 + 2*tan(c/2 + (d*x )/2)^10 + (7*tan(c/2 + (d*x)/2)^11)/8 + 22/15)/(a^3*d*(tan(c/2 + (d*x)/2)^ 2 + 1)^6)
Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-170 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+105 \cos \left (d x +c \right ) \sin \left (d x +c \right )-176 \cos \left (d x +c \right )-105 d x +176}{240 a^{3} d} \] Input:
int(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c))^3,x)
Output:
( - 40*cos(c + d*x)*sin(c + d*x)**5 + 144*cos(c + d*x)*sin(c + d*x)**4 - 1 70*cos(c + d*x)*sin(c + d*x)**3 + 32*cos(c + d*x)*sin(c + d*x)**2 + 105*co s(c + d*x)*sin(c + d*x) - 176*cos(c + d*x) - 105*d*x + 176)/(240*a**3*d)