\(\int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx\) [231]

Optimal result

Integrand size = 31, antiderivative size = 269 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {1}{16} \left (8 a^3 A+18 a A b^2+18 a^2 b B+5 b^3 B\right ) x+\frac {\left (15 a^2 A b+4 A b^3+5 a^3 B+12 a b^2 B\right ) \sin (c+d x)}{5 d}+\frac {\left (8 a^3 A+18 a A b^2+18 a^2 b B+5 b^3 B\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \left (18 a A b+14 a^2 B+5 b^2 B\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {b^2 (3 A b+4 a B) \cos ^4(c+d x) \sin (c+d x)}{15 d}+\frac {b B \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}-\frac {\left (15 a^2 A b+4 A b^3+5 a^3 B+12 a b^2 B\right ) \sin ^3(c+d x)}{15 d} \] Output:

1/16*(8*A*a^3+18*A*a*b^2+18*B*a^2*b+5*B*b^3)*x+1/5*(15*A*a^2*b+4*A*b^3+5*B 
*a^3+12*B*a*b^2)*sin(d*x+c)/d+1/16*(8*A*a^3+18*A*a*b^2+18*B*a^2*b+5*B*b^3) 
*cos(d*x+c)*sin(d*x+c)/d+1/24*b*(18*A*a*b+14*B*a^2+5*B*b^2)*cos(d*x+c)^3*s 
in(d*x+c)/d+1/15*b^2*(3*A*b+4*B*a)*cos(d*x+c)^4*sin(d*x+c)/d+1/6*b*B*cos(d 
*x+c)^3*(a+b*cos(d*x+c))^2*sin(d*x+c)/d-1/15*(15*A*a^2*b+4*A*b^3+5*B*a^3+1 
2*B*a*b^2)*sin(d*x+c)^3/d
 

Mathematica [A] (verified)

Time = 2.16 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.07 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {480 a^3 A c+1080 a A b^2 c+1080 a^2 b B c+300 b^3 B c+480 a^3 A d x+1080 a A b^2 d x+1080 a^2 b B d x+300 b^3 B d x+120 \left (18 a^2 A b+5 A b^3+6 a^3 B+15 a b^2 B\right ) \sin (c+d x)+15 \left (16 a^3 A+48 a A b^2+48 a^2 b B+15 b^3 B\right ) \sin (2 (c+d x))+240 a^2 A b \sin (3 (c+d x))+100 A b^3 \sin (3 (c+d x))+80 a^3 B \sin (3 (c+d x))+300 a b^2 B \sin (3 (c+d x))+90 a A b^2 \sin (4 (c+d x))+90 a^2 b B \sin (4 (c+d x))+45 b^3 B \sin (4 (c+d x))+12 A b^3 \sin (5 (c+d x))+36 a b^2 B \sin (5 (c+d x))+5 b^3 B \sin (6 (c+d x))}{960 d} \] Input:

Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]
 

Output:

(480*a^3*A*c + 1080*a*A*b^2*c + 1080*a^2*b*B*c + 300*b^3*B*c + 480*a^3*A*d 
*x + 1080*a*A*b^2*d*x + 1080*a^2*b*B*d*x + 300*b^3*B*d*x + 120*(18*a^2*A*b 
 + 5*A*b^3 + 6*a^3*B + 15*a*b^2*B)*Sin[c + d*x] + 15*(16*a^3*A + 48*a*A*b^ 
2 + 48*a^2*b*B + 15*b^3*B)*Sin[2*(c + d*x)] + 240*a^2*A*b*Sin[3*(c + d*x)] 
 + 100*A*b^3*Sin[3*(c + d*x)] + 80*a^3*B*Sin[3*(c + d*x)] + 300*a*b^2*B*Si 
n[3*(c + d*x)] + 90*a*A*b^2*Sin[4*(c + d*x)] + 90*a^2*b*B*Sin[4*(c + d*x)] 
 + 45*b^3*B*Sin[4*(c + d*x)] + 12*A*b^3*Sin[5*(c + d*x)] + 36*a*b^2*B*Sin[ 
5*(c + d*x)] + 5*b^3*B*Sin[6*(c + d*x)])/(960*d)
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.86, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {3042, 3469, 3042, 3512, 3042, 3502, 27, 3042, 3227, 3042, 3113, 2009, 3115, 24}

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.

\(\displaystyle \int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\)

\(\Big \downarrow \) 3469

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3512

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \cos ^2(c+d x) \left (15 (2 a A+b B) a^2+5 b \left (14 B a^2+18 A b a+5 b^2 B\right ) \cos ^2(c+d x)+6 \left (5 B a^3+15 A b a^2+12 b^2 B a+4 A b^3\right ) \cos (c+d x)\right )dx+\frac {2 b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (15 (2 a A+b B) a^2+5 b \left (14 B a^2+18 A b a+5 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+6 \left (5 B a^3+15 A b a^2+12 b^2 B a+4 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int 3 \cos ^2(c+d x) \left (5 \left (8 A a^3+18 b B a^2+18 A b^2 a+5 b^3 B\right )+8 \left (5 B a^3+15 A b a^2+12 b^2 B a+4 A b^3\right ) \cos (c+d x)\right )dx+\frac {5 b \left (14 a^2 B+18 a A b+5 b^2 B\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \cos ^2(c+d x) \left (5 \left (8 A a^3+18 b B a^2+18 A b^2 a+5 b^3 B\right )+8 \left (5 B a^3+15 A b a^2+12 b^2 B a+4 A b^3\right ) \cos (c+d x)\right )dx+\frac {5 b \left (14 a^2 B+18 a A b+5 b^2 B\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (5 \left (8 A a^3+18 b B a^2+18 A b^2 a+5 b^3 B\right )+8 \left (5 B a^3+15 A b a^2+12 b^2 B a+4 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {5 b \left (14 a^2 B+18 a A b+5 b^2 B\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (8 \left (5 a^3 B+15 a^2 A b+12 a b^2 B+4 A b^3\right ) \int \cos ^3(c+d x)dx+5 \left (8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right ) \int \cos ^2(c+d x)dx\right )+\frac {5 b \left (14 a^2 B+18 a A b+5 b^2 B\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (5 \left (8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+8 \left (5 a^3 B+15 a^2 A b+12 a b^2 B+4 A b^3\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {5 b \left (14 a^2 B+18 a A b+5 b^2 B\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}\)

\(\Big \downarrow \) 3113

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (5 \left (8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {8 \left (5 a^3 B+15 a^2 A b+12 a b^2 B+4 A b^3\right ) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {5 b \left (14 a^2 B+18 a A b+5 b^2 B\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (5 \left (8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {8 \left (5 a^3 B+15 a^2 A b+12 a b^2 B+4 A b^3\right ) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {5 b \left (14 a^2 B+18 a A b+5 b^2 B\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (5 \left (8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right ) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {8 \left (5 a^3 B+15 a^2 A b+12 a b^2 B+4 A b^3\right ) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {5 b \left (14 a^2 B+18 a A b+5 b^2 B\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {5 b \left (14 a^2 B+18 a A b+5 b^2 B\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (5 \left (8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {8 \left (5 a^3 B+15 a^2 A b+12 a b^2 B+4 A b^3\right ) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )\right )+\frac {2 b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}\)

Input:

Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]
 

Output:

(b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((2*b^2*( 
3*A*b + 4*a*B)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) + ((5*b*(18*a*A*b + 14*a 
^2*B + 5*b^2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(5*(8*a^3*A + 18*a 
*A*b^2 + 18*a^2*b*B + 5*b^3*B)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - 
 (8*(15*a^2*A*b + 4*A*b^3 + 5*a^3*B + 12*a*b^2*B)*(-Sin[c + d*x] + Sin[c + 
 d*x]^3/3))/d))/4)/5)/6
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3227

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x 
_)]), x_Symbol] :> Simp[c   Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b   Int 
[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
 

rule 3469

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^( 
n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1))   Int[(a + b*Sin[e + 
 f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*( 
m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c - b*d*(m 
+ n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin 
[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c 
 - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !(IGt 
Q[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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]
 

rule 3512

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f 
_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si 
n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3))   Int[(a + b*Si 
n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + 
A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 
0] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]
 

Maple [A] (verified)

Time = 147.72 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.79

method	result	size
parts	\(\frac {\left (A \,b^{3}+3 B a \,b^{2}\right ) \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (3 A a \,b^{2}+3 B \,a^{2} b \right ) \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (3 A \,a^{2} b +a^{3} B \right ) \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3 d}+\frac {a^{3} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {b^{3} B \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(213\)
parallelrisch	\(\frac {\left (240 a^{3} A +720 A a \,b^{2}+720 B \,a^{2} b +225 b^{3} B \right ) \sin \left (2 d x +2 c \right )+\left (240 A \,a^{2} b +100 A \,b^{3}+80 a^{3} B +300 B a \,b^{2}\right ) \sin \left (3 d x +3 c \right )+\left (90 A a \,b^{2}+90 B \,a^{2} b +45 b^{3} B \right ) \sin \left (4 d x +4 c \right )+\left (12 A \,b^{3}+36 B a \,b^{2}\right ) \sin \left (5 d x +5 c \right )+5 b^{3} B \sin \left (6 d x +6 c \right )+\left (2160 A \,a^{2} b +600 A \,b^{3}+720 a^{3} B +1800 B a \,b^{2}\right ) \sin \left (d x +c \right )+480 x d \left (a^{3} A +\frac {9}{4} A a \,b^{2}+\frac {9}{4} B \,a^{2} b +\frac {5}{8} b^{3} B \right )}{960 d}\)	\(215\)
derivativedivides	\(\frac {a^{3} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{3} B \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+A \,a^{2} b \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )+3 B \,a^{2} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 A a \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 B a \,b^{2} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {A \,b^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+b^{3} B \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(270\)
default	\(\frac {a^{3} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{3} B \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+A \,a^{2} b \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )+3 B \,a^{2} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 A a \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 B a \,b^{2} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {A \,b^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+b^{3} B \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(270\)
risch	\(\frac {a^{3} A x}{2}+\frac {9 x A a \,b^{2}}{8}+\frac {9 x B \,a^{2} b}{8}+\frac {5 b^{3} B x}{16}+\frac {9 \sin \left (d x +c \right ) A \,a^{2} b}{4 d}+\frac {5 \sin \left (d x +c \right ) A \,b^{3}}{8 d}+\frac {3 a^{3} B \sin \left (d x +c \right )}{4 d}+\frac {15 \sin \left (d x +c \right ) B a \,b^{2}}{8 d}+\frac {b^{3} B \sin \left (6 d x +6 c \right )}{192 d}+\frac {\sin \left (5 d x +5 c \right ) A \,b^{3}}{80 d}+\frac {3 \sin \left (5 d x +5 c \right ) B a \,b^{2}}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) A a \,b^{2}}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) B \,a^{2} b}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) b^{3} B}{64 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{2} b}{4 d}+\frac {5 \sin \left (3 d x +3 c \right ) A \,b^{3}}{48 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3} B}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) B a \,b^{2}}{16 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} A}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) A a \,b^{2}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b}{4 d}+\frac {15 \sin \left (2 d x +2 c \right ) b^{3} B}{64 d}\)	\(353\)
norman	\(\frac {\left (\frac {1}{2} a^{3} A +\frac {9}{8} A a \,b^{2}+\frac {9}{8} B \,a^{2} b +\frac {5}{16} b^{3} B \right ) x +\left (3 a^{3} A +\frac {27}{4} A a \,b^{2}+\frac {27}{4} B \,a^{2} b +\frac {15}{8} b^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (3 a^{3} A +\frac {27}{4} A a \,b^{2}+\frac {27}{4} B \,a^{2} b +\frac {15}{8} b^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (10 a^{3} A +\frac {45}{2} A a \,b^{2}+\frac {45}{2} B \,a^{2} b +\frac {25}{4} b^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {1}{2} a^{3} A +\frac {9}{8} A a \,b^{2}+\frac {9}{8} B \,a^{2} b +\frac {5}{16} b^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {15}{2} a^{3} A +\frac {135}{8} A a \,b^{2}+\frac {135}{8} B \,a^{2} b +\frac {75}{16} b^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {15}{2} a^{3} A +\frac {135}{8} A a \,b^{2}+\frac {135}{8} B \,a^{2} b +\frac {75}{16} b^{3} B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {\left (8 a^{3} A -48 A \,a^{2} b +30 A a \,b^{2}-16 A \,b^{3}-16 a^{3} B +30 B \,a^{2} b -48 B a \,b^{2}+11 b^{3} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (8 a^{3} A +48 A \,a^{2} b +30 A a \,b^{2}+16 A \,b^{3}+16 a^{3} B +30 B \,a^{2} b +48 B a \,b^{2}+11 b^{3} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {\left (40 a^{3} A -720 A \,a^{2} b +30 A a \,b^{2}-208 A \,b^{3}-240 a^{3} B +30 B \,a^{2} b -624 B a \,b^{2}+75 b^{3} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (40 a^{3} A +720 A \,a^{2} b +30 A a \,b^{2}+208 A \,b^{3}+240 a^{3} B +30 B \,a^{2} b +624 B a \,b^{2}+75 b^{3} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}-\frac {\left (72 a^{3} A -528 A \,a^{2} b +126 A a \,b^{2}-112 A \,b^{3}-176 a^{3} B +126 B \,a^{2} b -336 B a \,b^{2}-5 b^{3} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {\left (72 a^{3} A +528 A \,a^{2} b +126 A a \,b^{2}+112 A \,b^{3}+176 a^{3} B +126 B \,a^{2} b +336 B a \,b^{2}-5 b^{3} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\)	\(699\)
orering	\(\text {Expression too large to display}\)	\(6488\)

Input:

int(cos(d*x+c)^2*(a+cos(d*x+c)*b)^3*(A+B*cos(d*x+c)),x,method=_RETURNVERBO 
SE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.78 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {15 \, {\left (8 \, A a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2} + 5 \, B b^{3}\right )} d x + {\left (40 \, B b^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{4} + 160 \, B a^{3} + 480 \, A a^{2} b + 384 \, B a b^{2} + 128 \, A b^{3} + 10 \, {\left (18 \, B a^{2} b + 18 \, A a b^{2} + 5 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (5 \, B a^{3} + 15 \, A a^{2} b + 12 \, B a b^{2} + 4 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (8 \, A a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2} + 5 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)),x, algorithm="f 
ricas")
 

Output:

1/240*(15*(8*A*a^3 + 18*B*a^2*b + 18*A*a*b^2 + 5*B*b^3)*d*x + (40*B*b^3*co 
s(d*x + c)^5 + 48*(3*B*a*b^2 + A*b^3)*cos(d*x + c)^4 + 160*B*a^3 + 480*A*a 
^2*b + 384*B*a*b^2 + 128*A*b^3 + 10*(18*B*a^2*b + 18*A*a*b^2 + 5*B*b^3)*co 
s(d*x + c)^3 + 16*(5*B*a^3 + 15*A*a^2*b + 12*B*a*b^2 + 4*A*b^3)*cos(d*x + 
c)^2 + 15*(8*A*a^3 + 18*B*a^2*b + 18*A*a*b^2 + 5*B*b^3)*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. 721 vs. \(2 (277) = 554\).

Time = 0.45 (sec) , antiderivative size = 721, normalized size of antiderivative = 2.68 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx =\text {Too large to display} \] Input:

integrate(cos(d*x+c)**2*(a+b*cos(d*x+c))**3*(A+B*cos(d*x+c)),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.99 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{2} + 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a b^{2} + 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A b^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{3}}{960 \, d} \] Input:

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)),x, algorithm="m 
axima")
 

Output:

1/960*(240*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^3 - 320*(sin(d*x + c)^3 - 
3*sin(d*x + c))*B*a^3 - 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2*b + 90 
*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^2*b + 90*(12* 
d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a*b^2 + 192*(3*sin(d 
*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a*b^2 + 64*(3*sin(d*x + 
 c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*b^3 - 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^3)/d
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {B b^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {1}{16} \, {\left (8 \, A a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2} + 5 \, B b^{3}\right )} x + \frac {{\left (3 \, B a b^{2} + A b^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {3 \, {\left (2 \, B a^{2} b + 2 \, A a b^{2} + B b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (4 \, B a^{3} + 12 \, A a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (16 \, A a^{3} + 48 \, B a^{2} b + 48 \, A a b^{2} + 15 \, B b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (6 \, B a^{3} + 18 \, A a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \] Input:

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)),x, algorithm="g 
iac")
 

Output:

1/192*B*b^3*sin(6*d*x + 6*c)/d + 1/16*(8*A*a^3 + 18*B*a^2*b + 18*A*a*b^2 + 
 5*B*b^3)*x + 1/80*(3*B*a*b^2 + A*b^3)*sin(5*d*x + 5*c)/d + 3/64*(2*B*a^2* 
b + 2*A*a*b^2 + B*b^3)*sin(4*d*x + 4*c)/d + 1/48*(4*B*a^3 + 12*A*a^2*b + 1 
5*B*a*b^2 + 5*A*b^3)*sin(3*d*x + 3*c)/d + 1/64*(16*A*a^3 + 48*B*a^2*b + 48 
*A*a*b^2 + 15*B*b^3)*sin(2*d*x + 2*c)/d + 1/8*(6*B*a^3 + 18*A*a^2*b + 15*B 
*a*b^2 + 5*A*b^3)*sin(d*x + c)/d
 

Mupad [B] (verification not implemented)

Time = 42.94 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.31 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {A\,a^3\,x}{2}+\frac {5\,B\,b^3\,x}{16}+\frac {9\,A\,a\,b^2\,x}{8}+\frac {9\,B\,a^2\,b\,x}{8}+\frac {5\,A\,b^3\,\sin \left (c+d\,x\right )}{8\,d}+\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,A\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {A\,b^3\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {15\,B\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,B\,b^3\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {B\,b^3\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {3\,A\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {3\,A\,a\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,B\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,B\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{16\,d}+\frac {3\,B\,a^2\,b\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,B\,a\,b^2\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {9\,A\,a^2\,b\,\sin \left (c+d\,x\right )}{4\,d}+\frac {15\,B\,a\,b^2\,\sin \left (c+d\,x\right )}{8\,d} \] Input:

int(cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3,x)
 

Output:

(A*a^3*x)/2 + (5*B*b^3*x)/16 + (9*A*a*b^2*x)/8 + (9*B*a^2*b*x)/8 + (5*A*b^ 
3*sin(c + d*x))/(8*d) + (3*B*a^3*sin(c + d*x))/(4*d) + (A*a^3*sin(2*c + 2* 
d*x))/(4*d) + (5*A*b^3*sin(3*c + 3*d*x))/(48*d) + (B*a^3*sin(3*c + 3*d*x)) 
/(12*d) + (A*b^3*sin(5*c + 5*d*x))/(80*d) + (15*B*b^3*sin(2*c + 2*d*x))/(6 
4*d) + (3*B*b^3*sin(4*c + 4*d*x))/(64*d) + (B*b^3*sin(6*c + 6*d*x))/(192*d 
) + (3*A*a*b^2*sin(2*c + 2*d*x))/(4*d) + (A*a^2*b*sin(3*c + 3*d*x))/(4*d) 
+ (3*A*a*b^2*sin(4*c + 4*d*x))/(32*d) + (3*B*a^2*b*sin(2*c + 2*d*x))/(4*d) 
 + (5*B*a*b^2*sin(3*c + 3*d*x))/(16*d) + (3*B*a^2*b*sin(4*c + 4*d*x))/(32* 
d) + (3*B*a*b^2*sin(5*c + 5*d*x))/(80*d) + (9*A*a^2*b*sin(c + d*x))/(4*d) 
+ (15*B*a*b^2*sin(c + d*x))/(8*d)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.78 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{4}-360 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b^{2}-130 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{4}+120 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4}+900 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{2}+165 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{4}+192 \sin \left (d x +c \right )^{5} a \,b^{3}-320 \sin \left (d x +c \right )^{3} a^{3} b -640 \sin \left (d x +c \right )^{3} a \,b^{3}+960 \sin \left (d x +c \right ) a^{3} b +960 \sin \left (d x +c \right ) a \,b^{3}+120 a^{4} d x +540 a^{2} b^{2} d x +75 b^{4} d x}{240 d} \] Input:

int(cos(d*x+c)^2*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)),x)
 

Output:

(40*cos(c + d*x)*sin(c + d*x)**5*b**4 - 360*cos(c + d*x)*sin(c + d*x)**3*a 
**2*b**2 - 130*cos(c + d*x)*sin(c + d*x)**3*b**4 + 120*cos(c + d*x)*sin(c 
+ d*x)*a**4 + 900*cos(c + d*x)*sin(c + d*x)*a**2*b**2 + 165*cos(c + d*x)*s 
in(c + d*x)*b**4 + 192*sin(c + d*x)**5*a*b**3 - 320*sin(c + d*x)**3*a**3*b 
 - 640*sin(c + d*x)**3*a*b**3 + 960*sin(c + d*x)*a**3*b + 960*sin(c + d*x) 
*a*b**3 + 120*a**4*d*x + 540*a**2*b**2*d*x + 75*b**4*d*x)/(240*d)