Integrand size = 21, antiderivative size = 176 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=-\frac {241 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^3}+\frac {61 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^2}+\frac {61 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))}-\frac {\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}+\frac {9 \sin (c+d x)}{77 a^2 d (a+a \cos (c+d x))^4} \]
-241/1155*sin(d*x+c)/a^6/d/(1+cos(d*x+c))^3+61/1155*sin(d*x+c)/a^6/d/(1+co s(d*x+c))^2+61/1155*sin(d*x+c)/a^6/d/(1+cos(d*x+c))-1/11*cos(d*x+c)^3*sin( d*x+c)/d/(a+a*cos(d*x+c))^6-4/33*cos(d*x+c)^2*sin(d*x+c)/a/d/(a+a*cos(d*x+ c))^5+9/77*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 = 0.43 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\left (16+96 \cos (c+d x)+248 \cos ^2(c+d x)+368 \cos ^3(c+d x)+366 \cos ^4(c+d x)+61 \cos ^5(c+d x)\right ) \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^6} \]
((16 + 96*Cos[c + d*x] + 248*Cos[c + d*x]^2 + 368*Cos[c + d*x]^3 + 366*Cos [c + d*x]^4 + 61*Cos[c + d*x]^5)*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d* x])^6)
Time = 1.09 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.16, 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, 3447, 3042, 3498, 25, 3042, 3229, 3042, 3129, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+a)^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^6}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int \frac {3 \cos ^2(c+d x) (a-3 a \cos (c+d x))}{(\cos (c+d x) a+a)^5}dx}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \int \frac {\cos ^2(c+d x) (a-3 a \cos (c+d x))}{(\cos (c+d x) a+a)^5}dx}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a-3 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^5}dx}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {\cos (c+d x) \left (8 a^2-19 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^4}dx}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (8 a^2-19 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {8 a^2 \cos (c+d x)-19 a^2 \cos ^2(c+d x)}{(\cos (c+d x) a+a)^4}dx}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {8 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )-19 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3498 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {\int -\frac {108 a^3-133 a^3 \cos (c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {27 a^2 \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \left (\frac {\frac {\int \frac {108 a^3-133 a^3 \cos (c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {27 a^2 \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (\frac {\frac {\int \frac {108 a^3-133 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {27 a^2 \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3229 |
| \(\displaystyle -\frac {3 \left (\frac {\frac {\frac {241 \sin (c+d x)}{5 d (\cos (c+d x)+1)^3}-\frac {183}{5} a^2 \int \frac {1}{(\cos (c+d x) a+a)^2}dx}{7 a^2}-\frac {27 a^2 \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (\frac {\frac {\frac {241 \sin (c+d x)}{5 d (\cos (c+d x)+1)^3}-\frac {183}{5} a^2 \int \frac {1}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{7 a^2}-\frac {27 a^2 \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle -\frac {3 \left (\frac {\frac {\frac {241 \sin (c+d x)}{5 d (\cos (c+d x)+1)^3}-\frac {183}{5} a^2 \left (\frac {\int \frac {1}{\cos (c+d x) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{7 a^2}-\frac {27 a^2 \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (\frac {\frac {\frac {241 \sin (c+d x)}{5 d (\cos (c+d x)+1)^3}-\frac {183}{5} a^2 \left (\frac {\int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{7 a^2}-\frac {27 a^2 \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle -\frac {3 \left (\frac {\frac {\frac {241 \sin (c+d x)}{5 d (\cos (c+d x)+1)^3}-\frac {183}{5} a^2 \left (\frac {\sin (c+d x)}{3 a d (a \cos (c+d x)+a)}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{7 a^2}-\frac {27 a^2 \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}+\frac {4 a \sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\right )}{11 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}\) |
-1/11*(Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^6) - (3*((4*a* Cos[c + d*x]^2*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) + ((-27*a^2*Sin[ c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((241*Sin[c + d*x])/(5*d*(1 + Cos [c + d*x])^3) - (183*a^2*(Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2) + Sin[ c + d*x]/(3*a*d*(a + a*Cos[c + d*x]))))/5)/(7*a^2))/(9*a^2)))/(11*a^2)
3.1.94.3.1 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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
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 = 0.87 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.48
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d \,a^{6}}\) | \(84\) |
| default | \(\frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d \,a^{6}}\) | \(84\) |
| parallelrisch | \(\frac {105 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-385 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+330 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+462 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1155 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1155 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{36960 a^{6} d}\) | \(86\) |
| risch | \(\frac {2 i \left (1155 \,{\mathrm e}^{9 i \left (d x +c \right )}+3465 \,{\mathrm e}^{8 i \left (d x +c \right )}+9240 \,{\mathrm e}^{7 i \left (d x +c \right )}+12936 \,{\mathrm e}^{6 i \left (d x +c \right )}+15246 \,{\mathrm e}^{5 i \left (d x +c \right )}+10890 \,{\mathrm e}^{4 i \left (d x +c \right )}+6600 \,{\mathrm e}^{3 i \left (d x +c \right )}+2200 \,{\mathrm e}^{2 i \left (d x +c \right )}+671 \,{\mathrm e}^{i \left (d x +c \right )}+61\right )}{1155 d \,a^{6} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{11}}\) | \(124\) |
1/32/d/a^6*(1/11*tan(1/2*d*x+1/2*c)^11-1/3*tan(1/2*d*x+1/2*c)^9+2/7*tan(1/ 2*d*x+1/2*c)^7+2/5*tan(1/2*d*x+1/2*c)^5-tan(1/2*d*x+1/2*c)^3+tan(1/2*d*x+1 /2*c))
Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {{\left (61 \, \cos \left (d x + c\right )^{5} + 366 \, \cos \left (d x + c\right )^{4} + 368 \, \cos \left (d x + c\right )^{3} + 248 \, \cos \left (d x + c\right )^{2} + 96 \, \cos \left (d x + c\right ) + 16\right )} \sin \left (d x + c\right )}{1155 \, {\left (a^{6} d \cos \left (d x + c\right )^{6} + 6 \, a^{6} d \cos \left (d x + c\right )^{5} + 15 \, a^{6} d \cos \left (d x + c\right )^{4} + 20 \, a^{6} d \cos \left (d x + c\right )^{3} + 15 \, a^{6} d \cos \left (d x + c\right )^{2} + 6 \, a^{6} d \cos \left (d x + c\right ) + a^{6} d\right )}} \]
1/1155*(61*cos(d*x + c)^5 + 366*cos(d*x + c)^4 + 368*cos(d*x + c)^3 + 248* cos(d*x + c)^2 + 96*cos(d*x + c) + 16)*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 = 9.67 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\begin {cases} \frac {\tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{352 a^{6} d} - \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{96 a^{6} d} + \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{112 a^{6} d} + \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{80 a^{6} d} - \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32 a^{6} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32 a^{6} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{6}} & \text {otherwise} \end {cases} \]
Piecewise((tan(c/2 + d*x/2)**11/(352*a**6*d) - tan(c/2 + d*x/2)**9/(96*a** 6*d) + tan(c/2 + d*x/2)**7/(112*a**6*d) + tan(c/2 + d*x/2)**5/(80*a**6*d) - tan(c/2 + d*x/2)**3/(32*a**6*d) + tan(c/2 + d*x/2)/(32*a**6*d), Ne(d, 0) ), (x*cos(c)**4/(a*cos(c) + a)**6, True))
Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\frac {1155 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1155 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {462 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {330 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {385 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {105 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{36960 \, a^{6} d} \]
1/36960*(1155*sin(d*x + c)/(cos(d*x + c) + 1) - 1155*sin(d*x + c)^3/(cos(d *x + c) + 1)^3 + 462*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 330*sin(d*x + c )^7/(cos(d*x + c) + 1)^7 - 385*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 105*s in(d*x + c)^11/(cos(d*x + c) + 1)^11)/(a^6*d)
Time = 0.41 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.48 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 385 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 462 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{36960 \, a^{6} d} \]
1/36960*(105*tan(1/2*d*x + 1/2*c)^11 - 385*tan(1/2*d*x + 1/2*c)^9 + 330*ta n(1/2*d*x + 1/2*c)^7 + 462*tan(1/2*d*x + 1/2*c)^5 - 1155*tan(1/2*d*x + 1/2 *c)^3 + 1155*tan(1/2*d*x + 1/2*c))/(a^6*d)
Time = 14.79 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+462\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+330\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-385\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+105\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\right )}{36960\,a^6\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}} \]
(sin(c/2 + (d*x)/2)*(1155*cos(c/2 + (d*x)/2)^10 + 105*sin(c/2 + (d*x)/2)^1 0 - 385*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^8 + 330*cos(c/2 + (d*x)/2) ^4*sin(c/2 + (d*x)/2)^6 + 462*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^4 - 1155*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^2))/(36960*a^6*d*cos(c/2 + (d *x)/2)^11)