Integrand size = 29, antiderivative size = 139 \[ \int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 x}{128 a}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {\cos ^9(c+d x)}{9 a d}+\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}-\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d} \] Output:
5/128*x/a+1/7*cos(d*x+c)^7/a/d-1/9*cos(d*x+c)^9/a/d+5/128*cos(d*x+c)*sin(d *x+c)/a/d+5/192*cos(d*x+c)^3*sin(d*x+c)/a/d+1/48*cos(d*x+c)^5*sin(d*x+c)/a /d-1/8*cos(d*x+c)^7*sin(d*x+c)/a/d
Leaf count is larger than twice the leaf count of optimal. \(479\) vs. \(2(139)=278\).
Time = 11.75 (sec) , antiderivative size = 479, normalized size of antiderivative = 3.45 \[ \int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2520 (c-2 d x) \cos \left (\frac {c}{2}\right )-1512 \cos \left (\frac {c}{2}+d x\right )-1512 \cos \left (\frac {3 c}{2}+d x\right )-1008 \cos \left (\frac {3 c}{2}+2 d x\right )+1008 \cos \left (\frac {5 c}{2}+2 d x\right )-672 \cos \left (\frac {5 c}{2}+3 d x\right )-672 \cos \left (\frac {7 c}{2}+3 d x\right )+504 \cos \left (\frac {7 c}{2}+4 d x\right )-504 \cos \left (\frac {9 c}{2}+4 d x\right )+336 \cos \left (\frac {11 c}{2}+6 d x\right )-336 \cos \left (\frac {13 c}{2}+6 d x\right )+108 \cos \left (\frac {13 c}{2}+7 d x\right )+108 \cos \left (\frac {15 c}{2}+7 d x\right )+63 \cos \left (\frac {15 c}{2}+8 d x\right )-63 \cos \left (\frac {17 c}{2}+8 d x\right )+28 \cos \left (\frac {17 c}{2}+9 d x\right )+28 \cos \left (\frac {19 c}{2}+9 d x\right )-7560 \sin \left (\frac {c}{2}\right )+2520 c \sin \left (\frac {c}{2}\right )-5040 d x \sin \left (\frac {c}{2}\right )+1512 \sin \left (\frac {c}{2}+d x\right )-1512 \sin \left (\frac {3 c}{2}+d x\right )-1008 \sin \left (\frac {3 c}{2}+2 d x\right )-1008 \sin \left (\frac {5 c}{2}+2 d x\right )+672 \sin \left (\frac {5 c}{2}+3 d x\right )-672 \sin \left (\frac {7 c}{2}+3 d x\right )+504 \sin \left (\frac {7 c}{2}+4 d x\right )+504 \sin \left (\frac {9 c}{2}+4 d x\right )+336 \sin \left (\frac {11 c}{2}+6 d x\right )+336 \sin \left (\frac {13 c}{2}+6 d x\right )-108 \sin \left (\frac {13 c}{2}+7 d x\right )+108 \sin \left (\frac {15 c}{2}+7 d x\right )+63 \sin \left (\frac {15 c}{2}+8 d x\right )+63 \sin \left (\frac {17 c}{2}+8 d x\right )-28 \sin \left (\frac {17 c}{2}+9 d x\right )+28 \sin \left (\frac {19 c}{2}+9 d x\right )}{129024 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \] Input:
Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
Output:
-1/129024*(2520*(c - 2*d*x)*Cos[c/2] - 1512*Cos[c/2 + d*x] - 1512*Cos[(3*c )/2 + d*x] - 1008*Cos[(3*c)/2 + 2*d*x] + 1008*Cos[(5*c)/2 + 2*d*x] - 672*C os[(5*c)/2 + 3*d*x] - 672*Cos[(7*c)/2 + 3*d*x] + 504*Cos[(7*c)/2 + 4*d*x] - 504*Cos[(9*c)/2 + 4*d*x] + 336*Cos[(11*c)/2 + 6*d*x] - 336*Cos[(13*c)/2 + 6*d*x] + 108*Cos[(13*c)/2 + 7*d*x] + 108*Cos[(15*c)/2 + 7*d*x] + 63*Cos[ (15*c)/2 + 8*d*x] - 63*Cos[(17*c)/2 + 8*d*x] + 28*Cos[(17*c)/2 + 9*d*x] + 28*Cos[(19*c)/2 + 9*d*x] - 7560*Sin[c/2] + 2520*c*Sin[c/2] - 5040*d*x*Sin[ c/2] + 1512*Sin[c/2 + d*x] - 1512*Sin[(3*c)/2 + d*x] - 1008*Sin[(3*c)/2 + 2*d*x] - 1008*Sin[(5*c)/2 + 2*d*x] + 672*Sin[(5*c)/2 + 3*d*x] - 672*Sin[(7 *c)/2 + 3*d*x] + 504*Sin[(7*c)/2 + 4*d*x] + 504*Sin[(9*c)/2 + 4*d*x] + 336 *Sin[(11*c)/2 + 6*d*x] + 336*Sin[(13*c)/2 + 6*d*x] - 108*Sin[(13*c)/2 + 7* d*x] + 108*Sin[(15*c)/2 + 7*d*x] + 63*Sin[(15*c)/2 + 8*d*x] + 63*Sin[(17*c )/2 + 8*d*x] - 28*Sin[(17*c)/2 + 9*d*x] + 28*Sin[(19*c)/2 + 9*d*x])/(a*d*( Cos[c/2] + Sin[c/2]))
Time = 0.72 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {3042, 3318, 3042, 3045, 244, 2009, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^8(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^8}{a \sin (c+d x)+a}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cos ^6(c+d x) \sin ^2(c+d x)dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^3(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos (c+d x)^6 \sin (c+d x)^2dx}{a}-\frac {\int \cos (c+d x)^6 \sin (c+d x)^3dx}{a}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle \frac {\int \cos ^6(c+d x) \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{a d}+\frac {\int \cos (c+d x)^6 \sin (c+d x)^2dx}{a}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \left (\cos ^6(c+d x)-\cos ^8(c+d x)\right )d\cos (c+d x)}{a d}+\frac {\int \cos (c+d x)^6 \sin (c+d x)^2dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \cos (c+d x)^6 \sin (c+d x)^2dx}{a}+\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\frac {1}{8} \int \cos ^6(c+d x)dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}+\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}+\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}+\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}+\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}+\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}+\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}+\frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)}{a d}+\frac {\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}\) |
Input:
Int[(Cos[c + d*x]^8*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
Output:
(Cos[c + d*x]^7/7 - Cos[c + d*x]^9/9)/(a*d) + (-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) /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 = 1.72 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.72
| method | result | size |
| parallelrisch | \(\frac {2520 d x -28 \cos \left (9 d x +9 c \right )-63 \sin \left (8 d x +8 c \right )-336 \sin \left (6 d x +6 c \right )-504 \sin \left (4 d x +4 c \right )+1008 \sin \left (2 d x +2 c \right )-108 \cos \left (7 d x +7 c \right )+672 \cos \left (3 d x +3 c \right )+1512 \cos \left (d x +c \right )+2048}{64512 d a}\) | \(100\) |
| risch | \(\frac {5 x}{128 a}+\frac {3 \cos \left (d x +c \right )}{128 a d}-\frac {\cos \left (9 d x +9 c \right )}{2304 a d}-\frac {\sin \left (8 d x +8 c \right )}{1024 d a}-\frac {3 \cos \left (7 d x +7 c \right )}{1792 a d}-\frac {\sin \left (6 d x +6 c \right )}{192 d a}-\frac {\sin \left (4 d x +4 c \right )}{128 d a}+\frac {\cos \left (3 d x +3 c \right )}{96 a d}+\frac {\sin \left (2 d x +2 c \right )}{64 d a}\) | \(141\) |
| derivativedivides | \(\frac {\frac {8 \left (\frac {1}{126}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{14}+\frac {191 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{768}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{14}-\frac {83 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{256}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}+\frac {145 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{256}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}-\frac {145 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{256}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{6}+\frac {83 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{256}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{2}-\frac {191 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{768}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{512}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}+\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(233\) |
| default | \(\frac {\frac {8 \left (\frac {1}{126}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{14}+\frac {191 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{768}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{14}-\frac {83 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{256}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}+\frac {145 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{256}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}-\frac {145 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{256}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{6}+\frac {83 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{256}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{2}-\frac {191 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{768}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{512}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}+\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(233\) |
Input:
int(cos(d*x+c)^8*sin(d*x+c)^2/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/64512*(2520*d*x-28*cos(9*d*x+9*c)-63*sin(8*d*x+8*c)-336*sin(6*d*x+6*c)-5 04*sin(4*d*x+4*c)+1008*sin(2*d*x+2*c)-108*cos(7*d*x+7*c)+672*cos(3*d*x+3*c )+1512*cos(d*x+c)+2048)/d/a
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.58 \[ \int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {896 \, \cos \left (d x + c\right )^{9} - 1152 \, \cos \left (d x + c\right )^{7} - 315 \, d x + 21 \, {\left (48 \, \cos \left (d x + c\right )^{7} - 8 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, a d} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/8064*(896*cos(d*x + c)^9 - 1152*cos(d*x + c)^7 - 315*d*x + 21*(48*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. 4490 vs. \(2 (116) = 232\).
Time = 80.86 (sec) , antiderivative size = 4490, normalized size of antiderivative = 32.30 \[ \int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)**8*sin(d*x+c)**2/(a+a*sin(d*x+c)),x)
Output:
Piecewise((315*d*x*tan(c/2 + d*x/2)**18/(8064*a*d*tan(c/2 + d*x/2)**18 + 7 2576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a *d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*t an(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 2835*d*x*tan(c/2 + d*x/2)**16/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)** 16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1 016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376 *a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan( c/2 + d*x/2)**2 + 8064*a*d) + 11340*d*x*tan(c/2 + d*x/2)**14/(8064*a*d*tan (c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2 )**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d ) + 26460*d*x*tan(c/2 + d*x/2)**12/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576* a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*ta n(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/ 2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/ 2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 39690*d*x*tan(c/2 + d* x/2)**10/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**1...
Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (125) = 250\).
Time = 0.14 (sec) , antiderivative size = 522, normalized size of antiderivative = 3.76 \[ \int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/4032*((315*sin(d*x + c)/(cos(d*x + c) + 1) - 2304*sin(d*x + c)^2/(cos(d *x + c) + 1)^2 - 8022*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 6912*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10458*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 4 8384*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 18270*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 48384*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 80640*sin(d*x + c )^10/(cos(d*x + c) + 1)^10 + 18270*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 26880*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 10458*sin(d*x + c)^13/(cos( d*x + c) + 1)^13 - 16128*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 8022*sin( d*x + c)^15/(cos(d*x + c) + 1)^15 - 315*sin(d*x + c)^17/(cos(d*x + c) + 1) ^17 - 256)/(a + 9*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 36*a*sin(d*x + c )^4/(cos(d*x + c) + 1)^4 + 84*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 126* a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 126*a*sin(d*x + c)^10/(cos(d*x + c ) + 1)^10 + 84*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 36*a*sin(d*x + c) ^14/(cos(d*x + c) + 1)^14 + 9*a*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 + a* sin(d*x + c)^18/(cos(d*x + c) + 1)^18) - 315*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d
Time = 0.13 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.66 \[ \int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {315 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} - 8022 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 16128 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 10458 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 26880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 18270 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 80640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 48384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 18270 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 48384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 10458 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6912 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8022 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2304 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 256\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{9} a}}{8064 \, d} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/8064*(315*(d*x + c)/a + 2*(315*tan(1/2*d*x + 1/2*c)^17 - 8022*tan(1/2*d* x + 1/2*c)^15 + 16128*tan(1/2*d*x + 1/2*c)^14 + 10458*tan(1/2*d*x + 1/2*c) ^13 - 26880*tan(1/2*d*x + 1/2*c)^12 - 18270*tan(1/2*d*x + 1/2*c)^11 + 8064 0*tan(1/2*d*x + 1/2*c)^10 - 48384*tan(1/2*d*x + 1/2*c)^8 + 18270*tan(1/2*d *x + 1/2*c)^7 + 48384*tan(1/2*d*x + 1/2*c)^6 - 10458*tan(1/2*d*x + 1/2*c)^ 5 - 6912*tan(1/2*d*x + 1/2*c)^4 + 8022*tan(1/2*d*x + 1/2*c)^3 + 2304*tan(1 /2*d*x + 1/2*c)^2 - 315*tan(1/2*d*x + 1/2*c) + 256)/((tan(1/2*d*x + 1/2*c) ^2 + 1)^9*a))/d
Time = 34.84 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.61 \[ \int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5\,x}{128\,a}+\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}-\frac {191\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{96}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {83\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}-\frac {145\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {145\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {83\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}-\frac {12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}+\frac {191\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {4}{63}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \] Input:
int((cos(c + d*x)^8*sin(c + d*x)^2)/(a + a*sin(c + d*x)),x)
Output:
(5*x)/(128*a) + ((4*tan(c/2 + (d*x)/2)^2)/7 - (5*tan(c/2 + (d*x)/2))/64 + (191*tan(c/2 + (d*x)/2)^3)/96 - (12*tan(c/2 + (d*x)/2)^4)/7 - (83*tan(c/2 + (d*x)/2)^5)/32 + 12*tan(c/2 + (d*x)/2)^6 + (145*tan(c/2 + (d*x)/2)^7)/32 - 12*tan(c/2 + (d*x)/2)^8 + 20*tan(c/2 + (d*x)/2)^10 - (145*tan(c/2 + (d* x)/2)^11)/32 - (20*tan(c/2 + (d*x)/2)^12)/3 + (83*tan(c/2 + (d*x)/2)^13)/3 2 + 4*tan(c/2 + (d*x)/2)^14 - (191*tan(c/2 + (d*x)/2)^15)/96 + (5*tan(c/2 + (d*x)/2)^17)/64 + 4/63)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^9)
Time = 0.18 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-896 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}+1008 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+2432 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-2856 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-1920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+2478 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+128 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-315 \cos \left (d x +c \right ) \sin \left (d x +c \right )+256 \cos \left (d x +c \right )+315 d x -256}{8064 a d} \] Input:
int(cos(d*x+c)^8*sin(d*x+c)^2/(a+a*sin(d*x+c)),x)
Output:
( - 896*cos(c + d*x)*sin(c + d*x)**8 + 1008*cos(c + d*x)*sin(c + d*x)**7 + 2432*cos(c + d*x)*sin(c + d*x)**6 - 2856*cos(c + d*x)*sin(c + d*x)**5 - 1 920*cos(c + d*x)*sin(c + d*x)**4 + 2478*cos(c + d*x)*sin(c + d*x)**3 + 128 *cos(c + d*x)*sin(c + d*x)**2 - 315*cos(c + d*x)*sin(c + d*x) + 256*cos(c + d*x) + 315*d*x - 256)/(8064*a*d)