\(\int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx\) [241]

Optimal result

Integrand size = 29, antiderivative size = 325 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {1}{16} \left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right ) x+\frac {\left (24 a^4 A b+224 a^2 A b^3+32 A b^5-4 a^5 B+121 a^3 b^2 B+128 a b^4 B\right ) \sin (c+d x)}{60 b d}+\frac {\left (48 a^3 A b+232 a A b^3-8 a^4 B+178 a^2 b^2 B+75 b^4 B\right ) \cos (c+d x) \sin (c+d x)}{240 d}+\frac {\left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{120 b d}+\frac {\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \cos (c+d x))^3 \sin (c+d x)}{120 b d}+\frac {(6 A b-a B) (a+b \cos (c+d x))^4 \sin (c+d x)}{30 b d}+\frac {B (a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d} \] Output:

1/16*(32*A*a^3*b+24*A*a*b^3+8*B*a^4+36*B*a^2*b^2+5*B*b^4)*x+1/60*(24*A*a^4 
*b+224*A*a^2*b^3+32*A*b^5-4*B*a^5+121*B*a^3*b^2+128*B*a*b^4)*sin(d*x+c)/b/ 
d+1/240*(48*A*a^3*b+232*A*a*b^3-8*B*a^4+178*B*a^2*b^2+75*B*b^4)*cos(d*x+c) 
*sin(d*x+c)/d+1/120*(24*A*a^2*b+32*A*b^3-4*B*a^3+53*B*a*b^2)*(a+b*cos(d*x+ 
c))^2*sin(d*x+c)/b/d+1/120*(24*A*a*b-4*B*a^2+25*B*b^2)*(a+b*cos(d*x+c))^3* 
sin(d*x+c)/b/d+1/30*(6*A*b-B*a)*(a+b*cos(d*x+c))^4*sin(d*x+c)/b/d+1/6*B*(a 
+b*cos(d*x+c))^5*sin(d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 4.21 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.02 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {1920 a^3 A b c+1440 a A b^3 c+480 a^4 B c+2160 a^2 b^2 B c+300 b^4 B c+1920 a^3 A b d x+1440 a A b^3 d x+480 a^4 B d x+2160 a^2 b^2 B d x+300 b^4 B d x+120 \left (8 a^4 A+36 a^2 A b^2+5 A b^4+24 a^3 b B+20 a b^3 B\right ) \sin (c+d x)+15 \left (64 a^3 A b+64 a A b^3+16 a^4 B+96 a^2 b^2 B+15 b^4 B\right ) \sin (2 (c+d x))+480 a^2 A b^2 \sin (3 (c+d x))+100 A b^4 \sin (3 (c+d x))+320 a^3 b B \sin (3 (c+d x))+400 a b^3 B \sin (3 (c+d x))+120 a A b^3 \sin (4 (c+d x))+180 a^2 b^2 B \sin (4 (c+d x))+45 b^4 B \sin (4 (c+d x))+12 A b^4 \sin (5 (c+d x))+48 a b^3 B \sin (5 (c+d x))+5 b^4 B \sin (6 (c+d x))}{960 d} \] Input:

Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]
 

Output:

(1920*a^3*A*b*c + 1440*a*A*b^3*c + 480*a^4*B*c + 2160*a^2*b^2*B*c + 300*b^ 
4*B*c + 1920*a^3*A*b*d*x + 1440*a*A*b^3*d*x + 480*a^4*B*d*x + 2160*a^2*b^2 
*B*d*x + 300*b^4*B*d*x + 120*(8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b* 
B + 20*a*b^3*B)*Sin[c + d*x] + 15*(64*a^3*A*b + 64*a*A*b^3 + 16*a^4*B + 96 
*a^2*b^2*B + 15*b^4*B)*Sin[2*(c + d*x)] + 480*a^2*A*b^2*Sin[3*(c + d*x)] + 
 100*A*b^4*Sin[3*(c + d*x)] + 320*a^3*b*B*Sin[3*(c + d*x)] + 400*a*b^3*B*S 
in[3*(c + d*x)] + 120*a*A*b^3*Sin[4*(c + d*x)] + 180*a^2*b^2*B*Sin[4*(c + 
d*x)] + 45*b^4*B*Sin[4*(c + d*x)] + 12*A*b^4*Sin[5*(c + d*x)] + 48*a*b^3*B 
*Sin[5*(c + d*x)] + 5*b^4*B*Sin[6*(c + d*x)])/(960*d)
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {3042, 3447, 3042, 3502, 3042, 3232, 3042, 3232, 27, 3042, 3232, 3042, 3213}

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.

\(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\)

\(\Big \downarrow \) 3447

\(\displaystyle \int (a+b \cos (c+d x))^4 \left (A \cos (c+d x)+B \cos ^2(c+d x)\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )+B \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\int (a+b \cos (c+d x))^4 (5 b B+(6 A b-a B) \cos (c+d x))dx}{6 b}+\frac {B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (5 b B+(6 A b-a B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{6 b}+\frac {B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {1}{5} \int (a+b \cos (c+d x))^3 \left (3 b (8 A b+7 a B)+\left (-4 B a^2+24 A b a+25 b^2 B\right ) \cos (c+d x)\right )dx+\frac {(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (3 b (8 A b+7 a B)+\left (-4 B a^2+24 A b a+25 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int 3 (a+b \cos (c+d x))^2 \left (b \left (24 B a^2+56 A b a+25 b^2 B\right )+\left (-4 B a^3+24 A b a^2+53 b^2 B a+32 A b^3\right ) \cos (c+d x)\right )dx+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int (a+b \cos (c+d x))^2 \left (b \left (24 B a^2+56 A b a+25 b^2 B\right )+\left (-4 B a^3+24 A b a^2+53 b^2 B a+32 A b^3\right ) \cos (c+d x)\right )dx+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b \left (24 B a^2+56 A b a+25 b^2 B\right )+\left (-4 B a^3+24 A b a^2+53 b^2 B a+32 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x)) \left (b \left (64 B a^3+216 A b a^2+181 b^2 B a+64 A b^3\right )+\left (-8 B a^4+48 A b a^3+178 b^2 B a^2+232 A b^3 a+75 b^4 B\right ) \cos (c+d x)\right )dx+\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left (64 B a^3+216 A b a^2+181 b^2 B a+64 A b^3\right )+\left (-8 B a^4+48 A b a^3+178 b^2 B a^2+232 A b^3 a+75 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3213

\(\displaystyle \frac {\frac {1}{5} \left (\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {3}{4} \left (\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {b \left (-8 a^4 B+48 a^3 A b+178 a^2 b^2 B+232 a A b^3+75 b^4 B\right ) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {15}{2} b x \left (8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right )+\frac {2 \left (-4 a^5 B+24 a^4 A b+121 a^3 b^2 B+224 a^2 A b^3+128 a b^4 B+32 A b^5\right ) \sin (c+d x)}{d}\right )\right )\right )+\frac {(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

Input:

Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]
 

Output:

(B*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d) + (((6*A*b - a*B)*(a + b*C 
os[c + d*x])^4*Sin[c + d*x])/(5*d) + (((24*a*A*b - 4*a^2*B + 25*b^2*B)*(a 
+ b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (3*(((24*a^2*A*b + 32*A*b^3 - 4* 
a^3*B + 53*a*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + ((15*b*(3 
2*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*x)/2 + (2*(24*a 
^4*A*b + 224*a^2*A*b^3 + 32*A*b^5 - 4*a^5*B + 121*a^3*b^2*B + 128*a*b^4*B) 
*Sin[c + d*x])/d + (b*(48*a^3*A*b + 232*a*A*b^3 - 8*a^4*B + 178*a^2*b^2*B 
+ 75*b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d))/3))/4)/5)/(6*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3213

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) 
*(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co 
s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free 
Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
 

rule 3232

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[1/(m + 1)   Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* 
d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ 
[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 
 0] && IntegerQ[2*m]
 

rule 3447

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]
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

Maple [A] (verified)

Time = 212.17 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.74

method	result	size
parts	\(\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {B \,b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {\sin \left (d x +c \right ) a^{4} A}{d}\)	\(242\)
parallelrisch	\(\frac {\left (960 A \,a^{3} b +960 A a \,b^{3}+240 B \,a^{4}+1440 B \,a^{2} b^{2}+225 B \,b^{4}\right ) \sin \left (2 d x +2 c \right )+\left (480 A \,a^{2} b^{2}+100 A \,b^{4}+320 B \,a^{3} b +400 B a \,b^{3}\right ) \sin \left (3 d x +3 c \right )+\left (120 A a \,b^{3}+180 B \,a^{2} b^{2}+45 B \,b^{4}\right ) \sin \left (4 d x +4 c \right )+\left (12 A \,b^{4}+48 B a \,b^{3}\right ) \sin \left (5 d x +5 c \right )+5 B \,b^{4} \sin \left (6 d x +6 c \right )+\left (960 a^{4} A +4320 A \,a^{2} b^{2}+600 A \,b^{4}+2880 B \,a^{3} b +2400 B a \,b^{3}\right ) \sin \left (d x +c \right )+1920 \left (A \,a^{3} b +\frac {3}{4} A a \,b^{3}+\frac {1}{4} B \,a^{4}+\frac {9}{8} B \,a^{2} b^{2}+\frac {5}{32} B \,b^{4}\right ) x d}{960 d}\)	\(247\)
derivativedivides	\(\frac {a^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \,a^{3} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{3} b \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+2 A \,a^{2} b^{2} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )+6 B \,a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 A a \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B a \,b^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {A \,b^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(316\)
default	\(\frac {a^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \,a^{3} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{3} b \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+2 A \,a^{2} b^{2} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )+6 B \,a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 A a \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B a \,b^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {A \,b^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(316\)
risch	\(\frac {5 b^{4} B x}{16}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{4}}{4 d}+2 x A \,a^{3} b +\frac {3 x A a \,b^{3}}{2}+\frac {9 x B \,a^{2} b^{2}}{4}+\frac {\sin \left (d x +c \right ) a^{4} A}{d}+\frac {5 \sin \left (d x +c \right ) A \,b^{4}}{8 d}+\frac {\sin \left (5 d x +5 c \right ) A \,b^{4}}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) B \,b^{4}}{64 d}+\frac {5 \sin \left (3 d x +3 c \right ) A \,b^{4}}{48 d}+\frac {a^{4} B x}{2}+\frac {B \,b^{4} \sin \left (6 d x +6 c \right )}{192 d}+\frac {9 \sin \left (d x +c \right ) A \,a^{2} b^{2}}{2 d}+\frac {3 \sin \left (d x +c \right ) B \,a^{3} b}{d}+\frac {5 \sin \left (d x +c \right ) B a \,b^{3}}{2 d}+\frac {\sin \left (5 d x +5 c \right ) B a \,b^{3}}{20 d}+\frac {\sin \left (4 d x +4 c \right ) A a \,b^{3}}{8 d}+\frac {3 \sin \left (4 d x +4 c \right ) B \,a^{2} b^{2}}{16 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{2} b^{2}}{2 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3} b}{3 d}+\frac {5 \sin \left (3 d x +3 c \right ) B a \,b^{3}}{12 d}+\frac {15 \sin \left (2 d x +2 c \right ) B \,b^{4}}{64 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3} b}{d}+\frac {\sin \left (2 d x +2 c \right ) A a \,b^{3}}{d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b^{2}}{2 d}\)	\(404\)
norman	\(\frac {\left (2 A \,a^{3} b +\frac {3}{2} A a \,b^{3}+\frac {1}{2} B \,a^{4}+\frac {9}{4} B \,a^{2} b^{2}+\frac {5}{16} B \,b^{4}\right ) x +\left (2 A \,a^{3} b +\frac {3}{2} A a \,b^{3}+\frac {1}{2} B \,a^{4}+\frac {9}{4} B \,a^{2} b^{2}+\frac {5}{16} B \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (12 A \,a^{3} b +9 A a \,b^{3}+3 B \,a^{4}+\frac {27}{2} B \,a^{2} b^{2}+\frac {15}{8} B \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (12 A \,a^{3} b +9 A a \,b^{3}+3 B \,a^{4}+\frac {27}{2} B \,a^{2} b^{2}+\frac {15}{8} B \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (30 A \,a^{3} b +\frac {45}{2} A a \,b^{3}+\frac {15}{2} B \,a^{4}+\frac {135}{4} B \,a^{2} b^{2}+\frac {75}{16} B \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (30 A \,a^{3} b +\frac {45}{2} A a \,b^{3}+\frac {15}{2} B \,a^{4}+\frac {135}{4} B \,a^{2} b^{2}+\frac {75}{16} B \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (40 A \,a^{3} b +30 A a \,b^{3}+10 B \,a^{4}+45 B \,a^{2} b^{2}+\frac {25}{4} B \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {\left (16 a^{4} A -32 A \,a^{3} b +96 A \,a^{2} b^{2}-40 A a \,b^{3}+16 A \,b^{4}-8 B \,a^{4}+64 B \,a^{3} b -60 B \,a^{2} b^{2}+64 B a \,b^{3}-11 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (16 a^{4} A +32 A \,a^{3} b +96 A \,a^{2} b^{2}+40 A a \,b^{3}+16 A \,b^{4}+8 B \,a^{4}+64 B \,a^{3} b +60 B \,a^{2} b^{2}+64 B a \,b^{3}+11 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (240 a^{4} A -288 A \,a^{3} b +1056 A \,a^{2} b^{2}-168 A a \,b^{3}+112 A \,b^{4}-72 B \,a^{4}+704 B \,a^{3} b -252 B \,a^{2} b^{2}+448 B a \,b^{3}+5 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {\left (240 a^{4} A +288 A \,a^{3} b +1056 A \,a^{2} b^{2}+168 A a \,b^{3}+112 A \,b^{4}+72 B \,a^{4}+704 B \,a^{3} b +252 B \,a^{2} b^{2}+448 B a \,b^{3}-5 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {\left (400 a^{4} A -160 A \,a^{3} b +1440 A \,a^{2} b^{2}-40 A a \,b^{3}+208 A \,b^{4}-40 B \,a^{4}+960 B \,a^{3} b -60 B \,a^{2} b^{2}+832 B a \,b^{3}-75 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (400 a^{4} A +160 A \,a^{3} b +1440 A \,a^{2} b^{2}+40 A a \,b^{3}+208 A \,b^{4}+40 B \,a^{4}+960 B \,a^{3} b +60 B \,a^{2} b^{2}+832 B a \,b^{3}+75 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\)	\(870\)

Input:

int(cos(d*x+c)*(a+cos(d*x+c)*b)^4*(A+B*cos(d*x+c)),x,method=_RETURNVERBOSE 
)
 

Output:

1/5*(A*b^4+4*B*a*b^3)/d*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+(4* 
A*a*b^3+6*B*a^2*b^2)/d*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d 
*x+3/8*c)+1/3*(6*A*a^2*b^2+4*B*a^3*b)/d*(cos(d*x+c)^2+2)*sin(d*x+c)+(4*A*a 
^3*b+B*a^4)/d*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+B*b^4/d*(1/6*(cos( 
d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+1/d 
*sin(d*x+c)*a^4*A
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.75 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {15 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} d x + {\left (40 \, B b^{4} \cos \left (d x + c\right )^{5} + 240 \, A a^{4} + 640 \, B a^{3} b + 960 \, A a^{2} b^{2} + 512 \, B a b^{3} + 128 \, A b^{4} + 48 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 32 \, {\left (10 \, B a^{3} b + 15 \, A a^{2} b^{2} + 8 \, B a b^{3} + 2 \, A b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="fri 
cas")
 

Output:

1/240*(15*(8*B*a^4 + 32*A*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 5*B*b^4)*d*x 
 + (40*B*b^4*cos(d*x + c)^5 + 240*A*a^4 + 640*B*a^3*b + 960*A*a^2*b^2 + 51 
2*B*a*b^3 + 128*A*b^4 + 48*(4*B*a*b^3 + A*b^4)*cos(d*x + c)^4 + 10*(36*B*a 
^2*b^2 + 24*A*a*b^3 + 5*B*b^4)*cos(d*x + c)^3 + 32*(10*B*a^3*b + 15*A*a^2* 
b^2 + 8*B*a*b^3 + 2*A*b^4)*cos(d*x + c)^2 + 15*(8*B*a^4 + 32*A*a^3*b + 36* 
B*a^2*b^2 + 24*A*a*b^3 + 5*B*b^4)*cos(d*x + c))*sin(d*x + c))/d
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 811 vs. \(2 (335) = 670\).

Time = 0.46 (sec) , antiderivative size = 811, normalized size of antiderivative = 2.50 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx =\text {Too large to display} \] Input:

integrate(cos(d*x+c)*(a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)),x)
 

Output:

Piecewise((A*a**4*sin(c + d*x)/d + 2*A*a**3*b*x*sin(c + d*x)**2 + 2*A*a**3 
*b*x*cos(c + d*x)**2 + 2*A*a**3*b*sin(c + d*x)*cos(c + d*x)/d + 4*A*a**2*b 
**2*sin(c + d*x)**3/d + 6*A*a**2*b**2*sin(c + d*x)*cos(c + d*x)**2/d + 3*A 
*a*b**3*x*sin(c + d*x)**4/2 + 3*A*a*b**3*x*sin(c + d*x)**2*cos(c + d*x)**2 
 + 3*A*a*b**3*x*cos(c + d*x)**4/2 + 3*A*a*b**3*sin(c + d*x)**3*cos(c + d*x 
)/(2*d) + 5*A*a*b**3*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 8*A*b**4*sin(c + 
 d*x)**5/(15*d) + 4*A*b**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + A*b**4* 
sin(c + d*x)*cos(c + d*x)**4/d + B*a**4*x*sin(c + d*x)**2/2 + B*a**4*x*cos 
(c + d*x)**2/2 + B*a**4*sin(c + d*x)*cos(c + d*x)/(2*d) + 8*B*a**3*b*sin(c 
 + d*x)**3/(3*d) + 4*B*a**3*b*sin(c + d*x)*cos(c + d*x)**2/d + 9*B*a**2*b* 
*2*x*sin(c + d*x)**4/4 + 9*B*a**2*b**2*x*sin(c + d*x)**2*cos(c + d*x)**2/2 
 + 9*B*a**2*b**2*x*cos(c + d*x)**4/4 + 9*B*a**2*b**2*sin(c + d*x)**3*cos(c 
 + d*x)/(4*d) + 15*B*a**2*b**2*sin(c + d*x)*cos(c + d*x)**3/(4*d) + 32*B*a 
*b**3*sin(c + d*x)**5/(15*d) + 16*B*a*b**3*sin(c + d*x)**3*cos(c + d*x)**2 
/(3*d) + 4*B*a*b**3*sin(c + d*x)*cos(c + d*x)**4/d + 5*B*b**4*x*sin(c + d* 
x)**6/16 + 15*B*b**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*B*b**4*x*si 
n(c + d*x)**2*cos(c + d*x)**4/16 + 5*B*b**4*x*cos(c + d*x)**6/16 + 5*B*b** 
4*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*B*b**4*sin(c + d*x)**3*cos(c + d 
*x)**3/(6*d) + 11*B*b**4*sin(c + d*x)*cos(c + d*x)**5/(16*d), Ne(d, 0)), ( 
x*(A + B*cos(c))*(a + b*cos(c))**4*cos(c), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.94 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 960 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} b - 1920 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{2} + 120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} + 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a b^{3} + 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A b^{4} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{4} + 960 \, A a^{4} \sin \left (d x + c\right )}{960 \, d} \] Input:

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="max 
ima")
 

Output:

1/960*(240*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4 + 960*(2*d*x + 2*c + sin 
(2*d*x + 2*c))*A*a^3*b - 1280*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3*b - 
1920*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2*b^2 + 180*(12*d*x + 12*c + si 
n(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^2*b^2 + 120*(12*d*x + 12*c + sin( 
4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a*b^3 + 256*(3*sin(d*x + c)^5 - 10*si 
n(d*x + c)^3 + 15*sin(d*x + c))*B*a*b^3 + 64*(3*sin(d*x + c)^5 - 10*sin(d* 
x + c)^3 + 15*sin(d*x + c))*A*b^4 - 5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60* 
c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*B*b^4 + 960*A*a^4*sin(d*x + 
c))/d
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.81 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {B b^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {1}{16} \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} x + \frac {{\left (4 \, B a b^{3} + A b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (12 \, B a^{2} b^{2} + 8 \, A a b^{3} + 3 \, B b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 20 \, B a b^{3} + 5 \, A b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (16 \, B a^{4} + 64 \, A a^{3} b + 96 \, B a^{2} b^{2} + 64 \, A a b^{3} + 15 \, B b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (8 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 20 \, B a b^{3} + 5 \, A b^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \] Input:

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="gia 
c")
 

Output:

1/192*B*b^4*sin(6*d*x + 6*c)/d + 1/16*(8*B*a^4 + 32*A*a^3*b + 36*B*a^2*b^2 
 + 24*A*a*b^3 + 5*B*b^4)*x + 1/80*(4*B*a*b^3 + A*b^4)*sin(5*d*x + 5*c)/d + 
 1/64*(12*B*a^2*b^2 + 8*A*a*b^3 + 3*B*b^4)*sin(4*d*x + 4*c)/d + 1/48*(16*B 
*a^3*b + 24*A*a^2*b^2 + 20*B*a*b^3 + 5*A*b^4)*sin(3*d*x + 3*c)/d + 1/64*(1 
6*B*a^4 + 64*A*a^3*b + 96*B*a^2*b^2 + 64*A*a*b^3 + 15*B*b^4)*sin(2*d*x + 2 
*c)/d + 1/8*(8*A*a^4 + 24*B*a^3*b + 36*A*a^2*b^2 + 20*B*a*b^3 + 5*A*b^4)*s 
in(d*x + c)/d
 

Mupad [B] (verification not implemented)

Time = 42.90 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.24 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {B\,a^4\,x}{2}+\frac {5\,B\,b^4\,x}{16}+\frac {3\,A\,a\,b^3\,x}{2}+2\,A\,a^3\,b\,x+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {5\,A\,b^4\,\sin \left (c+d\,x\right )}{8\,d}+\frac {9\,B\,a^2\,b^2\,x}{4}+\frac {B\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,A\,b^4\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {A\,b^4\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {15\,B\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,B\,b^4\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {B\,b^4\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {A\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {A\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {A\,a\,b^3\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {9\,A\,a^2\,b^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {5\,B\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {B\,a\,b^3\,\sin \left (5\,c+5\,d\,x\right )}{20\,d}+\frac {A\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{2\,d}+\frac {3\,B\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {3\,B\,a^2\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {5\,B\,a\,b^3\,\sin \left (c+d\,x\right )}{2\,d}+\frac {3\,B\,a^3\,b\,\sin \left (c+d\,x\right )}{d} \] Input:

int(cos(c + d*x)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4,x)
 

Output:

(B*a^4*x)/2 + (5*B*b^4*x)/16 + (3*A*a*b^3*x)/2 + 2*A*a^3*b*x + (A*a^4*sin( 
c + d*x))/d + (5*A*b^4*sin(c + d*x))/(8*d) + (9*B*a^2*b^2*x)/4 + (B*a^4*si 
n(2*c + 2*d*x))/(4*d) + (5*A*b^4*sin(3*c + 3*d*x))/(48*d) + (A*b^4*sin(5*c 
 + 5*d*x))/(80*d) + (15*B*b^4*sin(2*c + 2*d*x))/(64*d) + (3*B*b^4*sin(4*c 
+ 4*d*x))/(64*d) + (B*b^4*sin(6*c + 6*d*x))/(192*d) + (A*a*b^3*sin(2*c + 2 
*d*x))/d + (A*a^3*b*sin(2*c + 2*d*x))/d + (A*a*b^3*sin(4*c + 4*d*x))/(8*d) 
 + (9*A*a^2*b^2*sin(c + d*x))/(2*d) + (5*B*a*b^3*sin(3*c + 3*d*x))/(12*d) 
+ (B*a^3*b*sin(3*c + 3*d*x))/(3*d) + (B*a*b^3*sin(5*c + 5*d*x))/(20*d) + ( 
A*a^2*b^2*sin(3*c + 3*d*x))/(2*d) + (3*B*a^2*b^2*sin(2*c + 2*d*x))/(2*d) + 
 (3*B*a^2*b^2*sin(4*c + 4*d*x))/(16*d) + (5*B*a*b^3*sin(c + d*x))/(2*d) + 
(3*B*a^3*b*sin(c + d*x))/d
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.70 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{5}-120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b^{3}-26 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{5}+120 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4} b +300 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{3}+33 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{5}+48 \sin \left (d x +c \right )^{5} a \,b^{4}-160 \sin \left (d x +c \right )^{3} a^{3} b^{2}-160 \sin \left (d x +c \right )^{3} a \,b^{4}+48 \sin \left (d x +c \right ) a^{5}+480 \sin \left (d x +c \right ) a^{3} b^{2}+240 \sin \left (d x +c \right ) a \,b^{4}+120 a^{4} b d x +180 a^{2} b^{3} d x +15 b^{5} d x}{48 d} \] Input:

int(cos(d*x+c)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),x)
 

Output:

(8*cos(c + d*x)*sin(c + d*x)**5*b**5 - 120*cos(c + d*x)*sin(c + d*x)**3*a* 
*2*b**3 - 26*cos(c + d*x)*sin(c + d*x)**3*b**5 + 120*cos(c + d*x)*sin(c + 
d*x)*a**4*b + 300*cos(c + d*x)*sin(c + d*x)*a**2*b**3 + 33*cos(c + d*x)*si 
n(c + d*x)*b**5 + 48*sin(c + d*x)**5*a*b**4 - 160*sin(c + d*x)**3*a**3*b** 
2 - 160*sin(c + d*x)**3*a*b**4 + 48*sin(c + d*x)*a**5 + 480*sin(c + d*x)*a 
**3*b**2 + 240*sin(c + d*x)*a*b**4 + 120*a**4*b*d*x + 180*a**2*b**3*d*x + 
15*b**5*d*x)/(48*d)