\(\int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx\) [12]

Optimal result

Integrand size = 29, antiderivative size = 129 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {1}{8} a^2 (8 A+7 B) x+\frac {a^2 (8 A+7 B) \sin (c+d x)}{6 d}+\frac {a^2 (8 A+7 B) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 A-B) (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d} \] Output:

1/8*a^2*(8*A+7*B)*x+1/6*a^2*(8*A+7*B)*sin(d*x+c)/d+1/24*a^2*(8*A+7*B)*cos( 
d*x+c)*sin(d*x+c)/d+1/12*(4*A-B)*(a+a*cos(d*x+c))^2*sin(d*x+c)/d+1/4*B*(a+ 
a*cos(d*x+c))^3*sin(d*x+c)/a/d
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.67 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {a^2 (84 B c+96 A d x+84 B d x+24 (7 A+6 B) \sin (c+d x)+48 (A+B) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+16 B \sin (3 (c+d x))+3 B \sin (4 (c+d x)))}{96 d} \] Input:

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

Output:

(a^2*(84*B*c + 96*A*d*x + 84*B*d*x + 24*(7*A + 6*B)*Sin[c + d*x] + 48*(A + 
 B)*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + 16*B*Sin[3*(c + d*x)] + 3*B* 
Sin[4*(c + d*x)]))/(96*d)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {3042, 3447, 3042, 3502, 3042, 3230, 3042, 3123}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3447

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\int (\cos (c+d x) a+a)^2 (3 a B+a (4 A-B) \cos (c+d x))dx}{4 a}+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3230

\(\displaystyle \frac {\frac {1}{3} a (8 A+7 B) \int (\cos (c+d x) a+a)^2dx+\frac {a (4 A-B) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}}{4 a}+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3123

\(\displaystyle \frac {\frac {1}{3} a (8 A+7 B) \left (\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {3 a^2 x}{2}\right )+\frac {a (4 A-B) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}}{4 a}+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}\)

Input:

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

Output:

(B*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d) + ((a*(4*A - B)*(a + a*Cos 
[c + d*x])^2*Sin[c + d*x])/(3*d) + (a*(8*A + 7*B)*((3*a^2*x)/2 + (2*a^2*Si 
n[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)))/3)/(4*a)
 

Defintions of rubi rules used

rule 3042

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

rule 3123

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(2*a^2 + b^ 
2)*(x/2), x] + (-Simp[2*a*b*(Cos[c + d*x]/d), x] - Simp[b^2*Cos[c + d*x]*(S 
in[c + d*x]/(2*d)), x]) /; FreeQ[{a, b, c, d}, x]
 

rule 3230

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)]
 

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 = 19.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.57

method	result	size
parallelrisch	\(\frac {a^{2} \left (\frac {\left (A +B \right ) \sin \left (2 d x +2 c \right )}{2}+\frac {\left (\frac {A}{2}+B \right ) \sin \left (3 d x +3 c \right )}{6}+\frac {\sin \left (4 d x +4 c \right ) B}{32}+\frac {\left (\frac {7 A}{2}+3 B \right ) \sin \left (d x +c \right )}{2}+d x \left (A +\frac {7 B}{8}\right )\right )}{d}\)	\(74\)
parts	\(\frac {\left (a^{2} A +2 a^{2} B \right ) \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (2 a^{2} A +a^{2} B \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\sin \left (d x +c \right ) a^{2} A}{d}+\frac {a^{2} B \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(128\)
risch	\(a^{2} x A +\frac {7 a^{2} B x}{8}+\frac {7 \sin \left (d x +c \right ) a^{2} A}{4 d}+\frac {3 \sin \left (d x +c \right ) a^{2} B}{2 d}+\frac {\sin \left (4 d x +4 c \right ) a^{2} B}{32 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} A}{12 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} B}{6 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} A}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} B}{2 d}\)	\(135\)
derivativedivides	\(\frac {\frac {a^{2} A \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+a^{2} B \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 a^{2} B \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+a^{2} A \sin \left (d x +c \right )+a^{2} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(154\)
default	\(\frac {\frac {a^{2} A \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+a^{2} B \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 a^{2} B \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+a^{2} A \sin \left (d x +c \right )+a^{2} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(154\)
norman	\(\frac {\frac {a^{2} \left (8 A +7 B \right ) x}{8}+\frac {11 a^{2} \left (8 A +7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}+\frac {a^{2} \left (8 A +7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {a^{2} \left (8 A +7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {3 a^{2} \left (8 A +7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {a^{2} \left (8 A +7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}+\frac {a^{2} \left (8 A +7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}+\frac {a^{2} \left (24 A +25 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{2} \left (136 A +83 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\)	\(229\)
orering	\(\text {Expression too large to display}\)	\(2112\)

Input:

int(cos(d*x+c)*(a+a*cos(d*x+c))^2*(A+B*cos(d*x+c)),x,method=_RETURNVERBOSE 
)
 

Output:

a^2*(1/2*(A+B)*sin(2*d*x+2*c)+1/6*(1/2*A+B)*sin(3*d*x+3*c)+1/32*sin(4*d*x+ 
4*c)*B+1/2*(7/2*A+3*B)*sin(d*x+c)+d*x*(A+7/8*B))/d
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {3 \, {\left (8 \, A + 7 \, B\right )} a^{2} d x + {\left (6 \, B a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (8 \, A + 7 \, B\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (5 \, A + 4 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{24 \, d} \] Input:

integrate(cos(d*x+c)*(a+a*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorithm="fri 
cas")
 

Output:

1/24*(3*(8*A + 7*B)*a^2*d*x + (6*B*a^2*cos(d*x + c)^3 + 8*(A + 2*B)*a^2*co 
s(d*x + c)^2 + 3*(8*A + 7*B)*a^2*cos(d*x + c) + 8*(5*A + 4*B)*a^2)*sin(d*x 
 + c))/d
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (112) = 224\).

Time = 0.20 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.62 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\begin {cases} A a^{2} x \sin ^{2}{\left (c + d x \right )} + A a^{2} x \cos ^{2}{\left (c + d x \right )} + \frac {2 A a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {A a^{2} \sin {\left (c + d x \right )}}{d} + \frac {3 B a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {B a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {B a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {4 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 B a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {2 B a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {B a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{2} \cos {\left (c \right )} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((A*a**2*x*sin(c + d*x)**2 + A*a**2*x*cos(c + d*x)**2 + 2*A*a**2* 
sin(c + d*x)**3/(3*d) + A*a**2*sin(c + d*x)*cos(c + d*x)**2/d + A*a**2*sin 
(c + d*x)*cos(c + d*x)/d + A*a**2*sin(c + d*x)/d + 3*B*a**2*x*sin(c + d*x) 
**4/8 + 3*B*a**2*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + B*a**2*x*sin(c + d* 
x)**2/2 + 3*B*a**2*x*cos(c + d*x)**4/8 + B*a**2*x*cos(c + d*x)**2/2 + 3*B* 
a**2*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 4*B*a**2*sin(c + d*x)**3/(3*d) + 
 5*B*a**2*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 2*B*a**2*sin(c + d*x)*cos(c 
 + d*x)**2/d + B*a**2*sin(c + d*x)*cos(c + d*x)/(2*d), Ne(d, 0)), (x*(A + 
B*cos(c))*(a*cos(c) + a)**2*cos(c), True))
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.12 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=-\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 48 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 96 \, A a^{2} \sin \left (d x + c\right )}{96 \, d} \] Input:

integrate(cos(d*x+c)*(a+a*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorithm="max 
ima")
 

Output:

-1/96*(32*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2 - 48*(2*d*x + 2*c + sin( 
2*d*x + 2*c))*A*a^2 + 64*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^2 - 3*(12*d 
*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^2 - 24*(2*d*x + 2*c 
 + sin(2*d*x + 2*c))*B*a^2 - 96*A*a^2*sin(d*x + c))/d
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.85 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {B a^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {1}{8} \, {\left (8 \, A a^{2} + 7 \, B a^{2}\right )} x + \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (A a^{2} + B a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac {{\left (7 \, A a^{2} + 6 \, B a^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \] Input:

integrate(cos(d*x+c)*(a+a*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorithm="gia 
c")
 

Output:

1/32*B*a^2*sin(4*d*x + 4*c)/d + 1/8*(8*A*a^2 + 7*B*a^2)*x + 1/12*(A*a^2 + 
2*B*a^2)*sin(3*d*x + 3*c)/d + 1/2*(A*a^2 + B*a^2)*sin(2*d*x + 2*c)/d + 1/4 
*(7*A*a^2 + 6*B*a^2)*sin(d*x + c)/d
 

Mupad [B] (verification not implemented)

Time = 41.06 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.04 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=A\,a^2\,x+\frac {7\,B\,a^2\,x}{8}+\frac {7\,A\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,B\,a^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {A\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {B\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {B\,a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \] Input:

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

Output:

A*a^2*x + (7*B*a^2*x)/8 + (7*A*a^2*sin(c + d*x))/(4*d) + (3*B*a^2*sin(c + 
d*x))/(2*d) + (A*a^2*sin(2*c + 2*d*x))/(2*d) + (A*a^2*sin(3*c + 3*d*x))/(1 
2*d) + (B*a^2*sin(2*c + 2*d*x))/(2*d) + (B*a^2*sin(3*c + 3*d*x))/(6*d) + ( 
B*a^2*sin(4*c + 4*d*x))/(32*d)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {a^{2} \left (-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +24 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +27 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b -8 \sin \left (d x +c \right )^{3} a -16 \sin \left (d x +c \right )^{3} b +48 \sin \left (d x +c \right ) a +48 \sin \left (d x +c \right ) b +24 a d x +21 b d x \right )}{24 d} \] Input:

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

Output:

(a**2*( - 6*cos(c + d*x)*sin(c + d*x)**3*b + 24*cos(c + d*x)*sin(c + d*x)* 
a + 27*cos(c + d*x)*sin(c + d*x)*b - 8*sin(c + d*x)**3*a - 16*sin(c + d*x) 
**3*b + 48*sin(c + d*x)*a + 48*sin(c + d*x)*b + 24*a*d*x + 21*b*d*x))/(24* 
d)