Integrand size = 21, antiderivative size = 184 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {130 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^3}-\frac {268 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^2}+\frac {146 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))}-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4} \]
130/693*sin(d*x+c)/a^6/d/(1+cos(d*x+c))^3-268/693*sin(d*x+c)/a^6/d/(1+cos( d*x+c))^2+146/693*sin(d*x+c)/a^6/d/(1+cos(d*x+c))-1/11*cos(d*x+c)^4*sin(d* x+c)/d/(a+a*cos(d*x+c))^6-14/99*cos(d*x+c)^3*sin(d*x+c)/a/d/(a+a*cos(d*x+c ))^5-118/693*cos(d*x+c)^2*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^4
Time = 5.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.41 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\left (8+48 \cos (c+d x)+124 \cos ^2(c+d x)+184 \cos ^3(c+d x)+183 \cos ^4(c+d x)+146 \cos ^5(c+d x)\right ) \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^6} \]
((8 + 48*Cos[c + d*x] + 124*Cos[c + d*x]^2 + 184*Cos[c + d*x]^3 + 183*Cos[ c + d*x]^4 + 146*Cos[c + d*x]^5)*Sin[c + d*x])/(693*a^6*d*(1 + Cos[c + d*x ])^6)
Time = 1.24 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.15, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 3244, 27, 3042, 3456, 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.
| \(\displaystyle \int \frac {\cos ^5(c+d x)}{(a \cos (c+d x)+a)^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^5}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^6}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int \frac {2 \cos ^3(c+d x) (2 a-5 a \cos (c+d x))}{(\cos (c+d x) a+a)^5}dx}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int \frac {\cos ^3(c+d x) (2 a-5 a \cos (c+d x))}{(\cos (c+d x) a+a)^5}dx}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (2 a-5 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^5}dx}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {2 \left (\frac {\int \frac {\cos ^2(c+d x) \left (21 a^2-38 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^4}dx}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (21 a^2-38 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\int \frac {\cos (c+d x) \left (118 a^3-207 a^3 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {59 a^2 \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (118 a^3-207 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}+\frac {59 a^2 \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\int \frac {118 a^3 \cos (c+d x)-207 a^3 \cos ^2(c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {59 a^2 \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\int \frac {118 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )-207 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}+\frac {59 a^2 \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3498 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {-\frac {\int -\frac {15 \left (65 a^4-69 a^4 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {65 \sin (c+d x)}{d (\cos (c+d x)+1)^3}}{7 a^2}+\frac {59 a^2 \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\frac {3 \int \frac {65 a^4-69 a^4 \cos (c+d x)}{(\cos (c+d x) a+a)^2}dx}{a^2}-\frac {65 \sin (c+d x)}{d (\cos (c+d x)+1)^3}}{7 a^2}+\frac {59 a^2 \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\frac {3 \int \frac {65 a^4-69 a^4 \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{a^2}-\frac {65 \sin (c+d x)}{d (\cos (c+d x)+1)^3}}{7 a^2}+\frac {59 a^2 \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3229 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\frac {3 \left (\frac {134 a^4 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}-\frac {73}{3} a^3 \int \frac {1}{\cos (c+d x) a+a}dx\right )}{a^2}-\frac {65 \sin (c+d x)}{d (\cos (c+d x)+1)^3}}{7 a^2}+\frac {59 a^2 \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\frac {3 \left (\frac {134 a^4 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}-\frac {73}{3} a^3 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx\right )}{a^2}-\frac {65 \sin (c+d x)}{d (\cos (c+d x)+1)^3}}{7 a^2}+\frac {59 a^2 \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {59 a^2 \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {\frac {3 \left (\frac {134 a^4 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}-\frac {73 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)}\right )}{a^2}-\frac {65 \sin (c+d x)}{d (\cos (c+d x)+1)^3}}{7 a^2}}{9 a^2}+\frac {7 a \sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
-1/11*(Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^6) - (2*((7*a* Cos[c + d*x]^3*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) + ((59*a^2*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((-65*Sin[c + d*x]) /(d*(1 + Cos[c + d*x])^3) + (3*((134*a^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (73*a^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x]))))/a^2)/(7*a^2 ))/(9*a^2)))/(11*a^2)
3.1.93.3.1 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 = 83, normalized size of antiderivative = 0.45
| method | result | size |
| parallelrisch | \(-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {55 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {110 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-22 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {55 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-11\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{352 a^{6} d}\) | \(83\) |
| derivativedivides | \(\frac {-\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}+\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d \,a^{6}}\) | \(84\) |
| default | \(\frac {-\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}+\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d \,a^{6}}\) | \(84\) |
| risch | \(\frac {2 i \left (693 \,{\mathrm e}^{10 i \left (d x +c \right )}+3465 \,{\mathrm e}^{9 i \left (d x +c \right )}+11550 \,{\mathrm e}^{8 i \left (d x +c \right )}+23100 \,{\mathrm e}^{7 i \left (d x +c \right )}+33726 \,{\mathrm e}^{6 i \left (d x +c \right )}+33726 \,{\mathrm e}^{5 i \left (d x +c \right )}+25080 \,{\mathrm e}^{4 i \left (d x +c \right )}+12540 \,{\mathrm e}^{3 i \left (d x +c \right )}+4565 \,{\mathrm e}^{2 i \left (d x +c \right )}+913 \,{\mathrm e}^{i \left (d x +c \right )}+146\right )}{693 d \,a^{6} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{11}}\) | \(135\) |
-1/352*(tan(1/2*d*x+1/2*c)^10-55/9*tan(1/2*d*x+1/2*c)^8+110/7*tan(1/2*d*x+ 1/2*c)^6-22*tan(1/2*d*x+1/2*c)^4+55/3*tan(1/2*d*x+1/2*c)^2-11)*tan(1/2*d*x +1/2*c)/a^6/d
Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {{\left (146 \, \cos \left (d x + c\right )^{5} + 183 \, \cos \left (d x + c\right )^{4} + 184 \, \cos \left (d x + c\right )^{3} + 124 \, \cos \left (d x + c\right )^{2} + 48 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{693 \, {\left (a^{6} d \cos \left (d x + c\right )^{6} + 6 \, a^{6} d \cos \left (d x + c\right )^{5} + 15 \, a^{6} d \cos \left (d x + c\right )^{4} + 20 \, a^{6} d \cos \left (d x + c\right )^{3} + 15 \, a^{6} d \cos \left (d x + c\right )^{2} + 6 \, a^{6} d \cos \left (d x + c\right ) + a^{6} d\right )}} \]
1/693*(146*cos(d*x + c)^5 + 183*cos(d*x + c)^4 + 184*cos(d*x + c)^3 + 124* cos(d*x + c)^2 + 48*cos(d*x + c) + 8)*sin(d*x + c)/(a^6*d*cos(d*x + c)^6 + 6*a^6*d*cos(d*x + c)^5 + 15*a^6*d*cos(d*x + c)^4 + 20*a^6*d*cos(d*x + c)^ 3 + 15*a^6*d*cos(d*x + c)^2 + 6*a^6*d*cos(d*x + c) + a^6*d)
Time = 11.97 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\begin {cases} - \frac {\tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{352 a^{6} d} + \frac {5 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{288 a^{6} d} - \frac {5 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{112 a^{6} d} + \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{6} d} - \frac {5 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{96 a^{6} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32 a^{6} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{6}} & \text {otherwise} \end {cases} \]
Piecewise((-tan(c/2 + d*x/2)**11/(352*a**6*d) + 5*tan(c/2 + d*x/2)**9/(288 *a**6*d) - 5*tan(c/2 + d*x/2)**7/(112*a**6*d) + tan(c/2 + d*x/2)**5/(16*a* *6*d) - 5*tan(c/2 + d*x/2)**3/(96*a**6*d) + tan(c/2 + d*x/2)/(32*a**6*d), Ne(d, 0)), (x*cos(c)**5/(a*cos(c) + a)**6, True))
Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\frac {693 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1155 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1386 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {990 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {385 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {63 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{22176 \, a^{6} d} \]
1/22176*(693*sin(d*x + c)/(cos(d*x + c) + 1) - 1155*sin(d*x + c)^3/(cos(d* x + c) + 1)^3 + 1386*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 990*sin(d*x + c )^7/(cos(d*x + c) + 1)^7 + 385*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 63*si n(d*x + c)^11/(cos(d*x + c) + 1)^11)/(a^6*d)
Time = 0.42 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.46 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=-\frac {63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 385 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 990 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1386 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 693 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{22176 \, a^{6} d} \]
-1/22176*(63*tan(1/2*d*x + 1/2*c)^11 - 385*tan(1/2*d*x + 1/2*c)^9 + 990*ta n(1/2*d*x + 1/2*c)^7 - 1386*tan(1/2*d*x + 1/2*c)^5 + 1155*tan(1/2*d*x + 1/ 2*c)^3 - 693*tan(1/2*d*x + 1/2*c))/(a^6*d)
Time = 15.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.41 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\frac {495\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}+\frac {495\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{16}+\frac {275\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}+\frac {55\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{8}+\frac {73\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{16}}{22176\,a^6\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}} \]