Integrand size = 23, antiderivative size = 150 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {7}{8} a^4 (5 A+4 B) x+\frac {8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac {27 a^4 (5 A+4 B) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {a^4 (5 A+4 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}-\frac {4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d} \]
7/8*a^4*(5*A+4*B)*x+8/5*a^4*(5*A+4*B)*sin(d*x+c)/d+27/40*a^4*(5*A+4*B)*cos (d*x+c)*sin(d*x+c)/d+1/20*a^4*(5*A+4*B)*cos(d*x+c)^3*sin(d*x+c)/d+1/5*B*(a +a*cos(d*x+c))^4*sin(d*x+c)/d-4/15*a^4*(5*A+4*B)*sin(d*x+c)^3/d
Time = 0.58 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.89 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {a^4 \sin (c+d x) \left (210 (5 A+4 B) \arcsin \left (\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )+\left (800 A+664 B+15 (27 A+28 B) \cos (c+d x)+16 (10 A+17 B) \cos ^2(c+d x)+30 (A+4 B) \cos ^3(c+d x)+24 B \cos ^4(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{120 d \sqrt {\sin ^2(c+d x)}} \]
(a^4*Sin[c + d*x]*(210*(5*A + 4*B)*ArcSin[Sqrt[Sin[(c + d*x)/2]^2]] + (800 *A + 664*B + 15*(27*A + 28*B)*Cos[c + d*x] + 16*(10*A + 17*B)*Cos[c + d*x] ^2 + 30*(A + 4*B)*Cos[c + d*x]^3 + 24*B*Cos[c + d*x]^4)*Sqrt[Sin[c + d*x]^ 2]))/(120*d*Sqrt[Sin[c + d*x]^2])
Time = 0.39 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 3230, 3042, 3124, 2009}
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.
| \(\displaystyle \int (a \cos (c+d x)+a)^4 (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {1}{5} (5 A+4 B) \int (\cos (c+d x) a+a)^4dx+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} (5 A+4 B) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4dx+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3124 |
| \(\displaystyle \frac {1}{5} (5 A+4 B) \int \left (\cos ^4(c+d x) a^4+4 \cos ^3(c+d x) a^4+6 \cos ^2(c+d x) a^4+4 \cos (c+d x) a^4+a^4\right )dx+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} (5 A+4 B) \left (-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8}\right )+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
(B*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + ((5*A + 4*B)*((35*a^4*x)/8 + (8*a^4*Sin[c + d*x])/d + (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^ 4*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (4*a^4*Sin[c + d*x]^3)/(3*d)))/5
3.1.30.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTri g[(a + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 3.67 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.63
| method | result | size |
| parallelrisch | \(\frac {\left (\left (56 A +64 B \right ) \sin \left (2 d x +2 c \right )+\left (\frac {32 A}{3}+\frac {58 B}{3}\right ) \sin \left (3 d x +3 c \right )+\left (A +4 B \right ) \sin \left (4 d x +4 c \right )+\frac {2 B \sin \left (5 d x +5 c \right )}{5}+\left (224 A +196 B \right ) \sin \left (d x +c \right )+140 \left (A +\frac {4 B}{5}\right ) x d \right ) a^{4}}{32 d}\) | \(94\) |
| risch | \(\frac {35 a^{4} x A}{8}+\frac {7 a^{4} B x}{2}+\frac {7 \sin \left (d x +c \right ) a^{4} A}{d}+\frac {49 \sin \left (d x +c \right ) B \,a^{4}}{8 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{80 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} A}{32 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{4}}{8 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} A}{3 d}+\frac {29 \sin \left (3 d x +3 c \right ) B \,a^{4}}{48 d}+\frac {7 \sin \left (2 d x +2 c \right ) a^{4} A}{4 d}+\frac {2 \sin \left (2 d x +2 c \right ) B \,a^{4}}{d}\) | \(172\) |
| parts | \(a^{4} x A +\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(187\) |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} A \left (d x +c \right )+B \,a^{4} \sin \left (d x +c \right )}{d}\) | \(248\) |
| default | \(\frac {a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} A \left (d x +c \right )+B \,a^{4} \sin \left (d x +c \right )}{d}\) | \(248\) |
| norman | \(\frac {\frac {7 a^{4} \left (5 A +4 B \right ) x}{8}+\frac {79 a^{4} \left (5 A +4 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {224 a^{4} \left (5 A +4 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {49 a^{4} \left (5 A +4 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {7 a^{4} \left (5 A +4 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {35 a^{4} \left (5 A +4 B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a^{4} \left (5 A +4 B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {35 a^{4} \left (5 A +4 B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {35 a^{4} \left (5 A +4 B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a^{4} \left (5 A +4 B \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{4} \left (93 A +100 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(279\) |
1/32*((56*A+64*B)*sin(2*d*x+2*c)+(32/3*A+58/3*B)*sin(3*d*x+3*c)+(A+4*B)*si n(4*d*x+4*c)+2/5*B*sin(5*d*x+5*c)+(224*A+196*B)*sin(d*x+c)+140*(A+4/5*B)*x *d)*a^4/d
Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.73 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {105 \, {\left (5 \, A + 4 \, B\right )} a^{4} d x + {\left (24 \, B a^{4} \cos \left (d x + c\right )^{4} + 30 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \, {\left (10 \, A + 17 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (27 \, A + 28 \, B\right )} a^{4} \cos \left (d x + c\right ) + 8 \, {\left (100 \, A + 83 \, B\right )} a^{4}\right )} \sin \left (d x + c\right )}{120 \, d} \]
1/120*(105*(5*A + 4*B)*a^4*d*x + (24*B*a^4*cos(d*x + c)^4 + 30*(A + 4*B)*a ^4*cos(d*x + c)^3 + 16*(10*A + 17*B)*a^4*cos(d*x + c)^2 + 15*(27*A + 28*B) *a^4*cos(d*x + c) + 8*(100*A + 83*B)*a^4)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (141) = 282\).
Time = 0.31 (sec) , antiderivative size = 544, normalized size of antiderivative = 3.63 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\begin {cases} \frac {3 A a^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + 3 A a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 A a^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + 3 A a^{4} x \cos ^{2}{\left (c + d x \right )} + A a^{4} x + \frac {3 A a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {8 A a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 A a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {4 A a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 A a^{4} \sin {\left (c + d x \right )}}{d} + \frac {3 B a^{4} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 B a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + 2 B a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 B a^{4} x \cos ^{4}{\left (c + d x \right )}}{2} + 2 B a^{4} x \cos ^{2}{\left (c + d x \right )} + \frac {8 B a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 B a^{4} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {B a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 B a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {6 B a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 B a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {B a^{4} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{4} & \text {otherwise} \end {cases} \]
Piecewise((3*A*a**4*x*sin(c + d*x)**4/8 + 3*A*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*A*a**4*x*sin(c + d*x)**2 + 3*A*a**4*x*cos(c + d*x)**4/8 + 3*A*a**4*x*cos(c + d*x)**2 + A*a**4*x + 3*A*a**4*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 8*A*a**4*sin(c + d*x)**3/(3*d) + 5*A*a**4*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 4*A*a**4*sin(c + d*x)*cos(c + d*x)**2/d + 3*A*a**4*sin(c + d*x)*cos(c + d*x)/d + 4*A*a**4*sin(c + d*x)/d + 3*B*a**4*x*sin(c + d*x) **4/2 + 3*B*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2 + 2*B*a**4*x*sin(c + d* x)**2 + 3*B*a**4*x*cos(c + d*x)**4/2 + 2*B*a**4*x*cos(c + d*x)**2 + 8*B*a* *4*sin(c + d*x)**5/(15*d) + 4*B*a**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 3*B*a**4*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 4*B*a**4*sin(c + d*x)**3/ d + B*a**4*sin(c + d*x)*cos(c + d*x)**4/d + 5*B*a**4*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 6*B*a**4*sin(c + d*x)*cos(c + d*x)**2/d + 2*B*a**4*sin(c + d*x)*cos(c + d*x)/d + B*a**4*sin(c + d*x)/d, Ne(d, 0)), (x*(A + B*cos(c)) *(a*cos(c) + a)**4, True))
Time = 0.22 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.57 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=-\frac {640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 480 \, {\left (d x + c\right )} A a^{4} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 1920 \, A a^{4} \sin \left (d x + c\right ) - 480 \, B a^{4} \sin \left (d x + c\right )}{480 \, d} \]
-1/480*(640*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 - 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 - 720*(2*d*x + 2*c + sin(2*d* x + 2*c))*A*a^4 - 480*(d*x + c)*A*a^4 - 32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^4 + 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B* a^4 - 60*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 - 4 80*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4 - 1920*A*a^4*sin(d*x + c) - 480* B*a^4*sin(d*x + c))/d
Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.93 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {B a^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {7}{8} \, {\left (5 \, A a^{4} + 4 \, B a^{4}\right )} x + \frac {{\left (A a^{4} + 4 \, B a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (16 \, A a^{4} + 29 \, B a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (7 \, A a^{4} + 8 \, B a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {7 \, {\left (8 \, A a^{4} + 7 \, B a^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \]
1/80*B*a^4*sin(5*d*x + 5*c)/d + 7/8*(5*A*a^4 + 4*B*a^4)*x + 1/32*(A*a^4 + 4*B*a^4)*sin(4*d*x + 4*c)/d + 1/48*(16*A*a^4 + 29*B*a^4)*sin(3*d*x + 3*c)/ d + 1/4*(7*A*a^4 + 8*B*a^4)*sin(2*d*x + 2*c)/d + 7/8*(8*A*a^4 + 7*B*a^4)*s in(d*x + c)/d
Time = 1.64 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.85 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {245\,A\,a^4}{6}+\frac {98\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {224\,A\,a^4}{3}+\frac {896\,B\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {395\,A\,a^4}{6}+\frac {158\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {93\,A\,a^4}{4}+25\,B\,a^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {7\,a^4\,\left (5\,A+4\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,A+4\,B\right )}{4\,\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4\right )}\right )\,\left (5\,A+4\,B\right )}{4\,d} \]
(tan(c/2 + (d*x)/2)*((93*A*a^4)/4 + 25*B*a^4) + tan(c/2 + (d*x)/2)^9*((35* A*a^4)/4 + 7*B*a^4) + tan(c/2 + (d*x)/2)^7*((245*A*a^4)/6 + (98*B*a^4)/3) + tan(c/2 + (d*x)/2)^3*((395*A*a^4)/6 + (158*B*a^4)/3) + tan(c/2 + (d*x)/2 )^5*((224*A*a^4)/3 + (896*B*a^4)/15))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan( c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan( c/2 + (d*x)/2)^10 + 1)) - (7*a^4*(5*A + 4*B)*(atan(tan(c/2 + (d*x)/2)) - ( d*x)/2))/(4*d) + (7*a^4*atan((7*a^4*tan(c/2 + (d*x)/2)*(5*A + 4*B))/(4*((3 5*A*a^4)/4 + 7*B*a^4)))*(5*A + 4*B))/(4*d)