Integrand size = 21, antiderivative size = 127 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {x}{a^4}+\frac {11 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))^2}-\frac {43 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))}-\frac {\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3} \]
x/a^4+11/21*sin(d*x+c)/a^4/d/(1+cos(d*x+c))^2-43/21*sin(d*x+c)/a^4/d/(1+co s(d*x+c))-1/7*cos(d*x+c)^3*sin(d*x+c)/d/(a+a*cos(d*x+c))^4-2/7*cos(d*x+c)^ 2*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^3
Time = 3.19 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {32 \cos \left (\frac {1}{2} (c+d x)\right ) \csc ^8(c+d x) \sin ^9\left (\frac {1}{2} (c+d x)\right ) \left (336 \arcsin (\cos (c+d x)) \cos ^8\left (\frac {1}{2} (c+d x)\right )+(94+146 \cos (c+d x)+62 \cos (2 (c+d x))+13 \cos (3 (c+d x))) \sqrt {\sin ^2(c+d x)}\right )}{21 a^4 d \sqrt {\sin ^2(c+d x)}} \]
(-32*Cos[(c + d*x)/2]*Csc[c + d*x]^8*Sin[(c + d*x)/2]^9*(336*ArcSin[Cos[c + d*x]]*Cos[(c + d*x)/2]^8 + (94 + 146*Cos[c + d*x] + 62*Cos[2*(c + d*x)] + 13*Cos[3*(c + d*x)])*Sqrt[Sin[c + d*x]^2]))/(21*a^4*d*Sqrt[Sin[c + d*x]^ 2])
Time = 0.96 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3244, 3042, 3456, 27, 3042, 3447, 3042, 3498, 25, 3042, 3214, 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.
\(\displaystyle \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+a)^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
\(\Big \downarrow \) 3244 |
\(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) (3 a-7 a \cos (c+d x))}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (3 a-7 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3456 |
\(\displaystyle -\frac {\frac {\int \frac {5 \cos (c+d x) \left (4 a^2-7 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {2 a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\int \frac {\cos (c+d x) \left (4 a^2-7 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{a^2}+\frac {2 a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (4 a^2-7 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{a^2}+\frac {2 a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3447 |
\(\displaystyle -\frac {\frac {\int \frac {4 a^2 \cos (c+d x)-7 a^2 \cos ^2(c+d x)}{(\cos (c+d x) a+a)^2}dx}{a^2}+\frac {2 a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {\int \frac {4 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )-7 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{a^2}+\frac {2 a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3498 |
\(\displaystyle -\frac {\frac {-\frac {\int -\frac {22 a^3-21 a^3 \cos (c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {11 \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{a^2}+\frac {2 a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {\frac {\int \frac {22 a^3-21 a^3 \cos (c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {11 \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{a^2}+\frac {2 a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {\frac {\int \frac {22 a^3-21 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}-\frac {11 \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{a^2}+\frac {2 a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle -\frac {\frac {\frac {43 a^3 \int \frac {1}{\cos (c+d x) a+a}dx-21 a^2 x}{3 a^2}-\frac {11 \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{a^2}+\frac {2 a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {\frac {43 a^3 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-21 a^2 x}{3 a^2}-\frac {11 \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{a^2}+\frac {2 a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle -\frac {\frac {\frac {\frac {43 a^3 \sin (c+d x)}{d (a \cos (c+d x)+a)}-21 a^2 x}{3 a^2}-\frac {11 \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{a^2}+\frac {2 a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
-1/7*(Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^4) - ((2*a*Cos[ c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^3) + ((-11*Sin[c + d*x])/ (3*d*(1 + Cos[c + d*x])^2) + (-21*a^2*x + (43*a^3*Sin[c + d*x])/(d*(a + a* Cos[c + d*x])))/(3*a^2))/a^2)/(7*a^2)
3.1.74.3.1 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_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
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 = 0.65 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.50
method | result | size |
parallelrisch | \(\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-21 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+77 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+168 d x -315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{168 a^{4} d}\) | \(64\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+16 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(72\) |
default | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+16 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(72\) |
risch | \(\frac {x}{a^{4}}-\frac {4 i \left (42 \,{\mathrm e}^{6 i \left (d x +c \right )}+189 \,{\mathrm e}^{5 i \left (d x +c \right )}+413 \,{\mathrm e}^{4 i \left (d x +c \right )}+497 \,{\mathrm e}^{3 i \left (d x +c \right )}+357 \,{\mathrm e}^{2 i \left (d x +c \right )}+140 \,{\mathrm e}^{i \left (d x +c \right )}+26\right )}{21 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(97\) |
norman | \(\frac {\frac {x}{a}+\frac {x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {169 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {229 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {293 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d a}-\frac {121 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{168 d a}+\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{168 d a}-\frac {3 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 d a}+\frac {4 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {6 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {4 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} a^{3}}\) | \(243\) |
1/168*(3*tan(1/2*d*x+1/2*c)^7-21*tan(1/2*d*x+1/2*c)^5+77*tan(1/2*d*x+1/2*c )^3+168*d*x-315*tan(1/2*d*x+1/2*c))/a^4/d
Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.20 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {21 \, d x \cos \left (d x + c\right )^{4} + 84 \, d x \cos \left (d x + c\right )^{3} + 126 \, d x \cos \left (d x + c\right )^{2} + 84 \, d x \cos \left (d x + c\right ) + 21 \, d x - {\left (52 \, \cos \left (d x + c\right )^{3} + 124 \, \cos \left (d x + c\right )^{2} + 107 \, \cos \left (d x + c\right ) + 32\right )} \sin \left (d x + c\right )}{21 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
1/21*(21*d*x*cos(d*x + c)^4 + 84*d*x*cos(d*x + c)^3 + 126*d*x*cos(d*x + c) ^2 + 84*d*x*cos(d*x + c) + 21*d*x - (52*cos(d*x + c)^3 + 124*cos(d*x + c)^ 2 + 107*cos(d*x + c) + 32)*sin(d*x + c))/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*c os(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c) + a^4*d)
Time = 3.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.75 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} \frac {x}{a^{4}} + \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {11 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} - \frac {15 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Piecewise((x/a**4 + tan(c/2 + d*x/2)**7/(56*a**4*d) - tan(c/2 + d*x/2)**5/ (8*a**4*d) + 11*tan(c/2 + d*x/2)**3/(24*a**4*d) - 15*tan(c/2 + d*x/2)/(8*a **4*d), Ne(d, 0)), (x*cos(c)**4/(a*cos(c) + a)**4, True))
Time = 0.33 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{168 \, d} \]
-1/168*((315*sin(d*x + c)/(cos(d*x + c) + 1) - 77*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 3*sin(d*x + c)^7/(c os(d*x + c) + 1)^7)/a^4 - 336*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4) /d
Time = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {168 \, {\left (d x + c\right )}}{a^{4}} + \frac {3 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 21 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 77 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{168 \, d} \]
1/168*(168*(d*x + c)/a^4 + (3*a^24*tan(1/2*d*x + 1/2*c)^7 - 21*a^24*tan(1/ 2*d*x + 1/2*c)^5 + 77*a^24*tan(1/2*d*x + 1/2*c)^3 - 315*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 14.84 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {x}{a^4}+\frac {-\frac {52\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{21}+\frac {16\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{21}-\frac {5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{28}+\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]