Integrand size = 21, antiderivative size = 155 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {67 \sin (c+d x)}{315 a^2 d (a+a \cos (c+d x))^3}-\frac {142 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac {83 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \]
-1/9*cos(d*x+c)^3*sin(d*x+c)/d/(a+a*cos(d*x+c))^5-11/63*cos(d*x+c)^2*sin(d *x+c)/a/d/(a+a*cos(d*x+c))^4+67/315*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^3-14 2/315*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^2+83/315*sin(d*x+c)/d/(a^5+a^5*cos (d*x+c))
Time = 2.74 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\left (8+40 \cos (c+d x)+84 \cos ^2(c+d x)+100 \cos ^3(c+d x)+83 \cos ^4(c+d x)\right ) \sin (c+d x)}{315 a^5 d (1+\cos (c+d x))^5} \]
((8 + 40*Cos[c + d*x] + 84*Cos[c + d*x]^2 + 100*Cos[c + d*x]^3 + 83*Cos[c + d*x]^4)*Sin[c + d*x])/(315*a^5*d*(1 + Cos[c + d*x])^5)
Time = 0.95 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3244, 3042, 3456, 3042, 3447, 3042, 3498, 27, 3042, 3229, 3042, 3127}
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.
| \(\displaystyle \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+a)^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^5}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) (3 a-8 a \cos (c+d x))}{(\cos (c+d x) a+a)^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (3 a-8 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\int \frac {\cos (c+d x) \left (22 a^2-45 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {11 a \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (22 a^2-45 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}+\frac {11 a \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle -\frac {\frac {\int \frac {22 a^2 \cos (c+d x)-45 a^2 \cos ^2(c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {11 a \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {22 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )-45 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}+\frac {11 a \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3498 |
| \(\displaystyle -\frac {\frac {-\frac {\int -\frac {3 \left (67 a^3-75 a^3 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {67 a^2 \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {11 a \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {3 \int \frac {67 a^3-75 a^3 \cos (c+d x)}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {67 a^2 \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {11 a \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {3 \int \frac {67 a^3-75 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {67 a^2 \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {11 a \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3229 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {142 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}-\frac {83}{3} a^2 \int \frac {1}{\cos (c+d x) a+a}dx\right )}{5 a^2}-\frac {67 a^2 \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {11 a \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {142 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}-\frac {83}{3} a^2 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx\right )}{5 a^2}-\frac {67 a^2 \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {11 a \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {142 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}-\frac {83 a^2 \sin (c+d x)}{3 d (a \cos (c+d x)+a)}\right )}{5 a^2}-\frac {67 a^2 \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {11 a \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
-1/9*(Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^5) - ((11*a*Cos [c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((-67*a^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (3*((142*a^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (83*a^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x]))))/( 5*a^2))/(7*a^2))/(9*a^2)
3.1.85.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f* x])^m/(a*f*(2*m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Simp[1/(a*b* (2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] & & NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[c, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b - a* B + b*C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(a*f*(2*m + 1))), x] + Simp[1 /(a^2*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*A*(m + 1) + m*(b *B - a*C) + b*C*(2*m + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
Time = 0.74 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.46
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(71\) |
| default | \(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(71\) |
| parallelrisch | \(\frac {35 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-180 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+378 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-420 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5040 a^{5} d}\) | \(73\) |
| risch | \(\frac {2 i \left (315 \,{\mathrm e}^{8 i \left (d x +c \right )}+1260 \,{\mathrm e}^{7 i \left (d x +c \right )}+3360 \,{\mathrm e}^{6 i \left (d x +c \right )}+5040 \,{\mathrm e}^{5 i \left (d x +c \right )}+5418 \,{\mathrm e}^{4 i \left (d x +c \right )}+3612 \,{\mathrm e}^{3 i \left (d x +c \right )}+1728 \,{\mathrm e}^{2 i \left (d x +c \right )}+432 \,{\mathrm e}^{i \left (d x +c \right )}+83\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(113\) |
| norman | \(\frac {\frac {\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{6 d a}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d a}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{70 d a}+\frac {109 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2520 d a}+\frac {19 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{630 d a}-\frac {11 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{420 d a}-\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{126 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} a^{4}}\) | \(190\) |
1/16/d/a^5*(1/9*tan(1/2*d*x+1/2*c)^9-4/7*tan(1/2*d*x+1/2*c)^7+6/5*tan(1/2* d*x+1/2*c)^5-4/3*tan(1/2*d*x+1/2*c)^3+tan(1/2*d*x+1/2*c))
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {{\left (83 \, \cos \left (d x + c\right )^{4} + 100 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} + 40 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{315 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
1/315*(83*cos(d*x + c)^4 + 100*cos(d*x + c)^3 + 84*cos(d*x + c)^2 + 40*cos (d*x + c) + 8)*sin(d*x + c)/(a^5*d*cos(d*x + c)^5 + 5*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos(d*x + c)^3 + 10*a^5*d*cos(d*x + c)^2 + 5*a^5*d*cos(d*x + c ) + a^5*d)
Time = 5.44 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} - \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{28 a^{5} d} + \frac {3 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{5} d} - \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{12 a^{5} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \]
Piecewise((tan(c/2 + d*x/2)**9/(144*a**5*d) - tan(c/2 + d*x/2)**7/(28*a**5 *d) + 3*tan(c/2 + d*x/2)**5/(40*a**5*d) - tan(c/2 + d*x/2)**3/(12*a**5*d) + tan(c/2 + d*x/2)/(16*a**5*d), Ne(d, 0)), (x*cos(c)**4/(a*cos(c) + a)**5, True))
Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {420 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {180 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{5040 \, a^{5} d} \]
1/5040*(315*sin(d*x + c)/(cos(d*x + c) + 1) - 420*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 378*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 180*sin(d*x + c)^7 /(cos(d*x + c) + 1)^7 + 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/(a^5*d)
Time = 0.42 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.46 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 180 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 378 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 420 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{5040 \, a^{5} d} \]
1/5040*(35*tan(1/2*d*x + 1/2*c)^9 - 180*tan(1/2*d*x + 1/2*c)^7 + 378*tan(1 /2*d*x + 1/2*c)^5 - 420*tan(1/2*d*x + 1/2*c)^3 + 315*tan(1/2*d*x + 1/2*c)) /(a^5*d)
Time = 14.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.82 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+378\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-180\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\right )}{5040\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]