Integrand size = 21, antiderivative size = 183 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}-\frac {5 \cos ^3(c+d x)}{143 a d (a+a \sin (c+d x))^7}-\frac {20 \cos ^3(c+d x)}{1287 a^2 d (a+a \sin (c+d x))^6}-\frac {20 \cos ^3(c+d x)}{3003 a^3 d (a+a \sin (c+d x))^5}-\frac {8 \cos ^3(c+d x)}{3003 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {8 \cos ^3(c+d x)}{9009 a^2 d \left (a^2+a^2 \sin (c+d x)\right )^3} \]
-1/13*cos(d*x+c)^3/d/(a+a*sin(d*x+c))^8-5/143*cos(d*x+c)^3/a/d/(a+a*sin(d* x+c))^7-20/1287*cos(d*x+c)^3/a^2/d/(a+a*sin(d*x+c))^6-20/3003*cos(d*x+c)^3 /a^3/d/(a+a*sin(d*x+c))^5-8/3003*cos(d*x+c)^3/d/(a^2+a^2*sin(d*x+c))^4-8/9 009*cos(d*x+c)^3/a^2/d/(a^2+a^2*sin(d*x+c))^3
Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.43 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {\cos ^3(c+d x) \left (1240+911 \sin (c+d x)+544 \sin ^2(c+d x)+236 \sin ^3(c+d x)+64 \sin ^4(c+d x)+8 \sin ^5(c+d x)\right )}{9009 a^8 d (1+\sin (c+d x))^8} \]
-1/9009*(Cos[c + d*x]^3*(1240 + 911*Sin[c + d*x] + 544*Sin[c + d*x]^2 + 23 6*Sin[c + d*x]^3 + 64*Sin[c + d*x]^4 + 8*Sin[c + d*x]^5))/(a^8*d*(1 + Sin[ c + d*x])^8)
Time = 0.89 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3151, 3042, 3151, 3042, 3151, 3042, 3151, 3042, 3151, 3042, 3150}
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.
| \(\displaystyle \int \frac {\cos ^2(c+d x)}{(a \sin (c+d x)+a)^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2}{(a \sin (c+d x)+a)^8}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {5 \int \frac {\cos ^2(c+d x)}{(\sin (c+d x) a+a)^7}dx}{13 a}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \int \frac {\cos (c+d x)^2}{(\sin (c+d x) a+a)^7}dx}{13 a}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {5 \left (\frac {4 \int \frac {\cos ^2(c+d x)}{(\sin (c+d x) a+a)^6}dx}{11 a}-\frac {\cos ^3(c+d x)}{11 d (a \sin (c+d x)+a)^7}\right )}{13 a}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {4 \int \frac {\cos (c+d x)^2}{(\sin (c+d x) a+a)^6}dx}{11 a}-\frac {\cos ^3(c+d x)}{11 d (a \sin (c+d x)+a)^7}\right )}{13 a}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\int \frac {\cos ^2(c+d x)}{(\sin (c+d x) a+a)^5}dx}{3 a}-\frac {\cos ^3(c+d x)}{9 d (a \sin (c+d x)+a)^6}\right )}{11 a}-\frac {\cos ^3(c+d x)}{11 d (a \sin (c+d x)+a)^7}\right )}{13 a}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\int \frac {\cos (c+d x)^2}{(\sin (c+d x) a+a)^5}dx}{3 a}-\frac {\cos ^3(c+d x)}{9 d (a \sin (c+d x)+a)^6}\right )}{11 a}-\frac {\cos ^3(c+d x)}{11 d (a \sin (c+d x)+a)^7}\right )}{13 a}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\frac {2 \int \frac {\cos ^2(c+d x)}{(\sin (c+d x) a+a)^4}dx}{7 a}-\frac {\cos ^3(c+d x)}{7 d (a \sin (c+d x)+a)^5}}{3 a}-\frac {\cos ^3(c+d x)}{9 d (a \sin (c+d x)+a)^6}\right )}{11 a}-\frac {\cos ^3(c+d x)}{11 d (a \sin (c+d x)+a)^7}\right )}{13 a}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\frac {2 \int \frac {\cos (c+d x)^2}{(\sin (c+d x) a+a)^4}dx}{7 a}-\frac {\cos ^3(c+d x)}{7 d (a \sin (c+d x)+a)^5}}{3 a}-\frac {\cos ^3(c+d x)}{9 d (a \sin (c+d x)+a)^6}\right )}{11 a}-\frac {\cos ^3(c+d x)}{11 d (a \sin (c+d x)+a)^7}\right )}{13 a}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\frac {2 \left (\frac {\int \frac {\cos ^2(c+d x)}{(\sin (c+d x) a+a)^3}dx}{5 a}-\frac {\cos ^3(c+d x)}{5 d (a \sin (c+d x)+a)^4}\right )}{7 a}-\frac {\cos ^3(c+d x)}{7 d (a \sin (c+d x)+a)^5}}{3 a}-\frac {\cos ^3(c+d x)}{9 d (a \sin (c+d x)+a)^6}\right )}{11 a}-\frac {\cos ^3(c+d x)}{11 d (a \sin (c+d x)+a)^7}\right )}{13 a}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\frac {2 \left (\frac {\int \frac {\cos (c+d x)^2}{(\sin (c+d x) a+a)^3}dx}{5 a}-\frac {\cos ^3(c+d x)}{5 d (a \sin (c+d x)+a)^4}\right )}{7 a}-\frac {\cos ^3(c+d x)}{7 d (a \sin (c+d x)+a)^5}}{3 a}-\frac {\cos ^3(c+d x)}{9 d (a \sin (c+d x)+a)^6}\right )}{11 a}-\frac {\cos ^3(c+d x)}{11 d (a \sin (c+d x)+a)^7}\right )}{13 a}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\frac {2 \left (-\frac {\cos ^3(c+d x)}{15 a d (a \sin (c+d x)+a)^3}-\frac {\cos ^3(c+d x)}{5 d (a \sin (c+d x)+a)^4}\right )}{7 a}-\frac {\cos ^3(c+d x)}{7 d (a \sin (c+d x)+a)^5}}{3 a}-\frac {\cos ^3(c+d x)}{9 d (a \sin (c+d x)+a)^6}\right )}{11 a}-\frac {\cos ^3(c+d x)}{11 d (a \sin (c+d x)+a)^7}\right )}{13 a}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}\) |
-1/13*Cos[c + d*x]^3/(d*(a + a*Sin[c + d*x])^8) + (5*(-1/11*Cos[c + d*x]^3 /(d*(a + a*Sin[c + d*x])^7) + (4*(-1/9*Cos[c + d*x]^3/(d*(a + a*Sin[c + d* x])^6) + (-1/7*Cos[c + d*x]^3/(d*(a + a*Sin[c + d*x])^5) + (2*(-1/5*Cos[c + d*x]^3/(d*(a + a*Sin[c + d*x])^4) - Cos[c + d*x]^3/(15*a*d*(a + a*Sin[c + d*x])^3)))/(7*a))/(3*a)))/(11*a)))/(13*a)
3.1.94.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[2*m + p + 1]) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] , x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Result contains complex when optimal does not.
Time = 1.30 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(\frac {16 i \left (-4290 i {\mathrm e}^{6 i \left (d x +c \right )}+6006 \,{\mathrm e}^{7 i \left (d x +c \right )}-715 i {\mathrm e}^{4 i \left (d x +c \right )}-1287 \,{\mathrm e}^{5 i \left (d x +c \right )}+78 i {\mathrm e}^{2 i \left (d x +c \right )}+286 \,{\mathrm e}^{3 i \left (d x +c \right )}-i-13 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{9009 d \,a^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{13}}\) | \(107\) |
| parallelrisch | \(\frac {-\frac {2480}{9009}-180 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {290 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {122 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-10 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1220 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {6310 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63}-\frac {5020 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21}-\frac {700 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {2650 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{231}-\frac {2398 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63}-\frac {1094 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{693}}{d \,a^{8} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{13}}\) | \(178\) |
| derivativedivides | \(\frac {\frac {864}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {4544}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}-\frac {480}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{12}}+\frac {200}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2672}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {256}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{13}}-\frac {9056}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {188}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {1472}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {11680}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}}{a^{8} d}\) | \(205\) |
| default | \(\frac {\frac {864}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {4544}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}-\frac {480}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{12}}+\frac {200}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2672}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {256}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{13}}-\frac {9056}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {188}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {1472}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {11680}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}}{a^{8} d}\) | \(205\) |
16/9009*I*(-4290*I*exp(6*I*(d*x+c))+6006*exp(7*I*(d*x+c))-715*I*exp(4*I*(d *x+c))-1287*exp(5*I*(d*x+c))+78*I*exp(2*I*(d*x+c))+286*exp(3*I*(d*x+c))-I- 13*exp(I*(d*x+c)))/d/a^8/(exp(I*(d*x+c))+I)^13
Time = 0.30 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.85 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {8 \, \cos \left (d x + c\right )^{7} - 48 \, \cos \left (d x + c\right )^{6} - 196 \, \cos \left (d x + c\right )^{5} + 280 \, \cos \left (d x + c\right )^{4} + 735 \, \cos \left (d x + c\right )^{3} - 378 \, \cos \left (d x + c\right )^{2} - {\left (8 \, \cos \left (d x + c\right )^{6} + 56 \, \cos \left (d x + c\right )^{5} - 140 \, \cos \left (d x + c\right )^{4} - 420 \, \cos \left (d x + c\right )^{3} + 315 \, \cos \left (d x + c\right )^{2} + 693 \, \cos \left (d x + c\right ) + 1386\right )} \sin \left (d x + c\right ) + 693 \, \cos \left (d x + c\right ) + 1386}{9009 \, {\left (a^{8} d \cos \left (d x + c\right )^{7} + 7 \, a^{8} d \cos \left (d x + c\right )^{6} - 18 \, a^{8} d \cos \left (d x + c\right )^{5} - 56 \, a^{8} d \cos \left (d x + c\right )^{4} + 48 \, a^{8} d \cos \left (d x + c\right )^{3} + 112 \, a^{8} d \cos \left (d x + c\right )^{2} - 32 \, a^{8} d \cos \left (d x + c\right ) - 64 \, a^{8} d + {\left (a^{8} d \cos \left (d x + c\right )^{6} - 6 \, a^{8} d \cos \left (d x + c\right )^{5} - 24 \, a^{8} d \cos \left (d x + c\right )^{4} + 32 \, a^{8} d \cos \left (d x + c\right )^{3} + 80 \, a^{8} d \cos \left (d x + c\right )^{2} - 32 \, a^{8} d \cos \left (d x + c\right ) - 64 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \]
1/9009*(8*cos(d*x + c)^7 - 48*cos(d*x + c)^6 - 196*cos(d*x + c)^5 + 280*co s(d*x + c)^4 + 735*cos(d*x + c)^3 - 378*cos(d*x + c)^2 - (8*cos(d*x + c)^6 + 56*cos(d*x + c)^5 - 140*cos(d*x + c)^4 - 420*cos(d*x + c)^3 + 315*cos(d *x + c)^2 + 693*cos(d*x + c) + 1386)*sin(d*x + c) + 693*cos(d*x + c) + 138 6)/(a^8*d*cos(d*x + c)^7 + 7*a^8*d*cos(d*x + c)^6 - 18*a^8*d*cos(d*x + c)^ 5 - 56*a^8*d*cos(d*x + c)^4 + 48*a^8*d*cos(d*x + c)^3 + 112*a^8*d*cos(d*x + c)^2 - 32*a^8*d*cos(d*x + c) - 64*a^8*d + (a^8*d*cos(d*x + c)^6 - 6*a^8* d*cos(d*x + c)^5 - 24*a^8*d*cos(d*x + c)^4 + 32*a^8*d*cos(d*x + c)^3 + 80* a^8*d*cos(d*x + c)^2 - 32*a^8*d*cos(d*x + c) - 64*a^8*d)*sin(d*x + c))
Leaf count of result is larger than twice the leaf count of optimal. 3405 vs. \(2 (170) = 340\).
Time = 84.94 (sec) , antiderivative size = 3405, normalized size of antiderivative = 18.61 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\text {Too large to display} \]
Piecewise((-18018*tan(c/2 + d*x/2)**12/(9009*a**8*d*tan(c/2 + d*x/2)**13 + 117117*a**8*d*tan(c/2 + d*x/2)**12 + 702702*a**8*d*tan(c/2 + d*x/2)**11 + 2576574*a**8*d*tan(c/2 + d*x/2)**10 + 6441435*a**8*d*tan(c/2 + d*x/2)**9 + 11594583*a**8*d*tan(c/2 + d*x/2)**8 + 15459444*a**8*d*tan(c/2 + d*x/2)** 7 + 15459444*a**8*d*tan(c/2 + d*x/2)**6 + 11594583*a**8*d*tan(c/2 + d*x/2) **5 + 6441435*a**8*d*tan(c/2 + d*x/2)**4 + 2576574*a**8*d*tan(c/2 + d*x/2) **3 + 702702*a**8*d*tan(c/2 + d*x/2)**2 + 117117*a**8*d*tan(c/2 + d*x/2) + 9009*a**8*d) - 90090*tan(c/2 + d*x/2)**11/(9009*a**8*d*tan(c/2 + d*x/2)** 13 + 117117*a**8*d*tan(c/2 + d*x/2)**12 + 702702*a**8*d*tan(c/2 + d*x/2)** 11 + 2576574*a**8*d*tan(c/2 + d*x/2)**10 + 6441435*a**8*d*tan(c/2 + d*x/2) **9 + 11594583*a**8*d*tan(c/2 + d*x/2)**8 + 15459444*a**8*d*tan(c/2 + d*x/ 2)**7 + 15459444*a**8*d*tan(c/2 + d*x/2)**6 + 11594583*a**8*d*tan(c/2 + d* x/2)**5 + 6441435*a**8*d*tan(c/2 + d*x/2)**4 + 2576574*a**8*d*tan(c/2 + d* x/2)**3 + 702702*a**8*d*tan(c/2 + d*x/2)**2 + 117117*a**8*d*tan(c/2 + d*x/ 2) + 9009*a**8*d) - 366366*tan(c/2 + d*x/2)**10/(9009*a**8*d*tan(c/2 + d*x /2)**13 + 117117*a**8*d*tan(c/2 + d*x/2)**12 + 702702*a**8*d*tan(c/2 + d*x /2)**11 + 2576574*a**8*d*tan(c/2 + d*x/2)**10 + 6441435*a**8*d*tan(c/2 + d *x/2)**9 + 11594583*a**8*d*tan(c/2 + d*x/2)**8 + 15459444*a**8*d*tan(c/2 + d*x/2)**7 + 15459444*a**8*d*tan(c/2 + d*x/2)**6 + 11594583*a**8*d*tan(c/2 + d*x/2)**5 + 6441435*a**8*d*tan(c/2 + d*x/2)**4 + 2576574*a**8*d*tan(...
Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (171) = 342\).
Time = 0.21 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.99 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {2 \, {\left (\frac {7111 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {51675 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {171457 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {451165 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {785070 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1076790 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1051050 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {810810 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {435435 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {183183 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {45045 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {9009 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + 1240\right )}}{9009 \, {\left (a^{8} + \frac {13 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {78 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {286 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {715 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1287 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1716 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1716 \, a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {1287 \, a^{8} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {715 \, a^{8} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {286 \, a^{8} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {78 \, a^{8} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {13 \, a^{8} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a^{8} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}}\right )} d} \]
-2/9009*(7111*sin(d*x + c)/(cos(d*x + c) + 1) + 51675*sin(d*x + c)^2/(cos( d*x + c) + 1)^2 + 171457*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 451165*sin( d*x + c)^4/(cos(d*x + c) + 1)^4 + 785070*sin(d*x + c)^5/(cos(d*x + c) + 1) ^5 + 1076790*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1051050*sin(d*x + c)^7/ (cos(d*x + c) + 1)^7 + 810810*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 435435 *sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 183183*sin(d*x + c)^10/(cos(d*x + c ) + 1)^10 + 45045*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 9009*sin(d*x + c )^12/(cos(d*x + c) + 1)^12 + 1240)/((a^8 + 13*a^8*sin(d*x + c)/(cos(d*x + c) + 1) + 78*a^8*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 286*a^8*sin(d*x + c )^3/(cos(d*x + c) + 1)^3 + 715*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1 287*a^8*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 1716*a^8*sin(d*x + c)^6/(cos (d*x + c) + 1)^6 + 1716*a^8*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 1287*a^8 *sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 715*a^8*sin(d*x + c)^9/(cos(d*x + c ) + 1)^9 + 286*a^8*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 78*a^8*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 13*a^8*sin(d*x + c)^12/(cos(d*x + c) + 1)^ 12 + a^8*sin(d*x + c)^13/(cos(d*x + c) + 1)^13)*d)
Time = 0.39 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {2 \, {\left (9009 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 45045 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 183183 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 435435 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 810810 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1051050 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1076790 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 785070 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 451165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 171457 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 51675 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 7111 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1240\right )}}{9009 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{13}} \]
-2/9009*(9009*tan(1/2*d*x + 1/2*c)^12 + 45045*tan(1/2*d*x + 1/2*c)^11 + 18 3183*tan(1/2*d*x + 1/2*c)^10 + 435435*tan(1/2*d*x + 1/2*c)^9 + 810810*tan( 1/2*d*x + 1/2*c)^8 + 1051050*tan(1/2*d*x + 1/2*c)^7 + 1076790*tan(1/2*d*x + 1/2*c)^6 + 785070*tan(1/2*d*x + 1/2*c)^5 + 451165*tan(1/2*d*x + 1/2*c)^4 + 171457*tan(1/2*d*x + 1/2*c)^3 + 51675*tan(1/2*d*x + 1/2*c)^2 + 7111*tan (1/2*d*x + 1/2*c) + 1240)/(a^8*d*(tan(1/2*d*x + 1/2*c) + 1)^13)
Time = 9.98 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {\sqrt {2}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {14983\,\cos \left (c+d\,x\right )}{2}-\frac {63921\,\sin \left (c+d\,x\right )}{2}+17605\,\cos \left (2\,c+2\,d\,x\right )-\frac {15365\,\cos \left (3\,c+3\,d\,x\right )}{4}-\frac {6943\,\cos \left (4\,c+4\,d\,x\right )}{4}+\frac {937\,\cos \left (5\,c+5\,d\,x\right )}{4}+\frac {77\,\cos \left (6\,c+6\,d\,x\right )}{4}+\frac {28743\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {27027\,\sin \left (3\,c+3\,d\,x\right )}{4}-\frac {5005\,\sin \left (4\,c+4\,d\,x\right )}{4}-\frac {1079\,\sin \left (5\,c+5\,d\,x\right )}{4}+\frac {39\,\sin \left (6\,c+6\,d\,x\right )}{2}-21013\right )}{576576\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^{13}} \]
(2^(1/2)*cos(c/2 + (d*x)/2)*((14983*cos(c + d*x))/2 - (63921*sin(c + d*x)) /2 + 17605*cos(2*c + 2*d*x) - (15365*cos(3*c + 3*d*x))/4 - (6943*cos(4*c + 4*d*x))/4 + (937*cos(5*c + 5*d*x))/4 + (77*cos(6*c + 6*d*x))/4 + (28743*s in(2*c + 2*d*x))/4 + (27027*sin(3*c + 3*d*x))/4 - (5005*sin(4*c + 4*d*x))/ 4 - (1079*sin(5*c + 5*d*x))/4 + (39*sin(6*c + 6*d*x))/2 - 21013))/(576576* a^8*d*cos(c/2 - pi/4 + (d*x)/2)^13)