\(\int (a+b \cos (c+d x))^2 (B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [779]

Optimal result

Integrand size = 32, antiderivative size = 170 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} \left (8 a b B+4 a^2 C+3 b^2 C\right ) x+\frac {\left (4 a^2 b B+4 b^3 B-a^3 C+8 a b^2 C\right ) \sin (c+d x)}{6 b d}+\frac {\left (8 a b B-2 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 b B-a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 b d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 b d} \] Output:

1/8*(8*B*a*b+4*C*a^2+3*C*b^2)*x+1/6*(4*B*a^2*b+4*B*b^3-C*a^3+8*C*a*b^2)*si 
n(d*x+c)/b/d+1/24*(8*B*a*b-2*C*a^2+9*C*b^2)*cos(d*x+c)*sin(d*x+c)/d+1/12*( 
4*B*b-C*a)*(a+b*cos(d*x+c))^2*sin(d*x+c)/b/d+1/4*C*(a+b*cos(d*x+c))^3*sin( 
d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 2.48 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.69 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {12 \left (8 a b B+4 a^2 C+3 b^2 C\right ) (c+d x)+24 \left (4 a^2 B+3 b^2 B+6 a b C\right ) \sin (c+d x)+24 \left (2 a b B+a^2 C+b^2 C\right ) \sin (2 (c+d x))+8 b (b B+2 a C) \sin (3 (c+d x))+3 b^2 C \sin (4 (c+d x))}{96 d} \] Input:

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

Output:

(12*(8*a*b*B + 4*a^2*C + 3*b^2*C)*(c + d*x) + 24*(4*a^2*B + 3*b^2*B + 6*a* 
b*C)*Sin[c + d*x] + 24*(2*a*b*B + a^2*C + b^2*C)*Sin[2*(c + d*x)] + 8*b*(b 
*B + 2*a*C)*Sin[3*(c + d*x)] + 3*b^2*C*Sin[4*(c + d*x)])/(96*d)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3042, 3502, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {1}{3} \int (a+b \cos (c+d x)) \left (b (8 b B+7 a C)+\left (-2 C a^2+8 b B a+9 b^2 C\right ) \cos (c+d x)\right )dx+\frac {(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}}{4 b}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b (8 b B+7 a C)+\left (-2 C a^2+8 b B a+9 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}}{4 b}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}\)

\(\Big \downarrow \) 3213

\(\displaystyle \frac {\frac {1}{3} \left (\frac {b \left (-2 a^2 C+8 a b B+9 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {3}{2} b x \left (4 a^2 C+8 a b B+3 b^2 C\right )+\frac {2 \left (a^3 (-C)+4 a^2 b B+8 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{d}\right )+\frac {(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}}{4 b}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}\)

Input:

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

Output:

(C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d) + (((4*b*B - a*C)*(a + b*C 
os[c + d*x])^2*Sin[c + d*x])/(3*d) + ((3*b*(8*a*b*B + 4*a^2*C + 3*b^2*C)*x 
)/2 + (2*(4*a^2*b*B + 4*b^3*B - a^3*C + 8*a*b^2*C)*Sin[c + d*x])/d + (b*(8 
*a*b*B - 2*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d))/3)/(4*b)
 

Defintions of rubi rules used

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 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 = 8.48 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.69

method	result	size
parallelrisch	\(\frac {24 \left (2 B a b +a^{2} C +b^{2} C \right ) \sin \left (2 d x +2 c \right )+8 \left (B \,b^{2}+2 a b C \right ) \sin \left (3 d x +3 c \right )+3 b^{2} C \sin \left (4 d x +4 c \right )+24 \left (4 B \,a^{2}+3 B \,b^{2}+6 a b C \right ) \sin \left (d x +c \right )+96 x \left (B a b +\frac {1}{2} a^{2} C +\frac {3}{8} b^{2} C \right ) d}{96 d}\)	\(118\)
parts	\(\frac {\left (B \,b^{2}+2 a b C \right ) \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (2 B a b +a^{2} C \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\sin \left (d x +c \right ) B \,a^{2}}{d}+\frac {b^{2} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(126\)
derivativedivides	\(\frac {a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{2} \sin \left (d x +c \right )+\frac {2 a b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 B a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+b^{2} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(152\)
default	\(\frac {a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{2} \sin \left (d x +c \right )+\frac {2 a b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 B a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+b^{2} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(152\)
risch	\(x B a b +\frac {a^{2} C x}{2}+\frac {3 b^{2} C x}{8}+\frac {\sin \left (d x +c \right ) B \,a^{2}}{d}+\frac {3 \sin \left (d x +c \right ) B \,b^{2}}{4 d}+\frac {3 \sin \left (d x +c \right ) a b C}{2 d}+\frac {b^{2} C \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B \,b^{2}}{12 d}+\frac {\sin \left (3 d x +3 c \right ) a b C}{6 d}+\frac {\sin \left (2 d x +2 c \right ) B a b}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{4 d}+\frac {\sin \left (2 d x +2 c \right ) b^{2} C}{4 d}\)	\(170\)
norman	\(\frac {\left (B a b +\frac {1}{2} a^{2} C +\frac {3}{8} b^{2} C \right ) x +\left (B a b +\frac {1}{2} a^{2} C +\frac {3}{8} b^{2} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (4 B a b +2 a^{2} C +\frac {3}{2} b^{2} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (4 B a b +2 a^{2} C +\frac {3}{2} b^{2} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (6 B a b +3 a^{2} C +\frac {9}{4} b^{2} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {\left (8 B \,a^{2}-8 B a b +8 B \,b^{2}-4 a^{2} C +16 a b C -5 b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {\left (8 B \,a^{2}+8 B a b +8 B \,b^{2}+4 a^{2} C +16 a b C +5 b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (72 B \,a^{2}-24 B a b +40 B \,b^{2}-12 a^{2} C +80 a b C +9 b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}+\frac {\left (72 B \,a^{2}+24 B a b +40 B \,b^{2}+12 a^{2} C +80 a b C -9 b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\)	\(362\)
orering	\(\text {Expression too large to display}\)	\(4095\)

Input:

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

Output:

1/96*(24*(2*B*a*b+C*a^2+C*b^2)*sin(2*d*x+2*c)+8*(B*b^2+2*C*a*b)*sin(3*d*x+ 
3*c)+3*b^2*C*sin(4*d*x+4*c)+24*(4*B*a^2+3*B*b^2+6*C*a*b)*sin(d*x+c)+96*x*( 
B*a*b+1/2*a^2*C+3/8*b^2*C)*d)/d
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.67 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (4 \, C a^{2} + 8 \, B a b + 3 \, C b^{2}\right )} d x + {\left (6 \, C b^{2} \cos \left (d x + c\right )^{3} + 24 \, B a^{2} + 32 \, C a b + 16 \, B b^{2} + 8 \, {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, C a^{2} + 8 \, B a b + 3 \, C b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \] Input:

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

Output:

1/24*(3*(4*C*a^2 + 8*B*a*b + 3*C*b^2)*d*x + (6*C*b^2*cos(d*x + c)^3 + 24*B 
*a^2 + 32*C*a*b + 16*B*b^2 + 8*(2*C*a*b + B*b^2)*cos(d*x + c)^2 + 3*(4*C*a 
^2 + 8*B*a*b + 3*C*b^2)*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. 340 vs. \(2 (162) = 324\).

Time = 0.18 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.00 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {B a^{2} \sin {\left (c + d x \right )}}{d} + B a b x \sin ^{2}{\left (c + d x \right )} + B a b x \cos ^{2}{\left (c + d x \right )} + \frac {B a b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {2 B b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B b^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {C a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 C a b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 C a b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C b^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C b^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C b^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right )^{2} \left (B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.84 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 48 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b - 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{2} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{2} + 96 \, B a^{2} \sin \left (d x + c\right )}{96 \, d} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.73 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C b^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {1}{8} \, {\left (4 \, C a^{2} + 8 \, B a b + 3 \, C b^{2}\right )} x + \frac {{\left (2 \, C a b + B b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (C a^{2} + 2 \, B a b + C b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, B a^{2} + 6 \, C a b + 3 \, B b^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C\,a^2\,x}{2}+\frac {3\,C\,b^2\,x}{8}+\frac {B\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {3\,B\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+B\,a\,b\,x+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,C\,a\,b\,\sin \left (c+d\,x\right )}{2\,d}+\frac {B\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {C\,a\,b\,\sin \left (3\,c+3\,d\,x\right )}{6\,d} \] Input:

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

Output:

(C*a^2*x)/2 + (3*C*b^2*x)/8 + (B*a^2*sin(c + d*x))/d + (3*B*b^2*sin(c + d* 
x))/(4*d) + B*a*b*x + (C*a^2*sin(2*c + 2*d*x))/(4*d) + (B*b^2*sin(3*c + 3* 
d*x))/(12*d) + (C*b^2*sin(2*c + 2*d*x))/(4*d) + (C*b^2*sin(4*c + 4*d*x))/( 
32*d) + (3*C*a*b*sin(c + d*x))/(2*d) + (B*a*b*sin(2*c + 2*d*x))/(2*d) + (C 
*a*b*sin(3*c + 3*d*x))/(6*d)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.96 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{2} c +12 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} c +24 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{2}+15 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{2} c -16 \sin \left (d x +c \right )^{3} a b c -8 \sin \left (d x +c \right )^{3} b^{3}+24 \sin \left (d x +c \right ) a^{2} b +48 \sin \left (d x +c \right ) a b c +24 \sin \left (d x +c \right ) b^{3}+12 a^{2} c d x +24 a \,b^{2} d x +9 b^{2} c d x}{24 d} \] Input:

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

Output:

( - 6*cos(c + d*x)*sin(c + d*x)**3*b**2*c + 12*cos(c + d*x)*sin(c + d*x)*a 
**2*c + 24*cos(c + d*x)*sin(c + d*x)*a*b**2 + 15*cos(c + d*x)*sin(c + d*x) 
*b**2*c - 16*sin(c + d*x)**3*a*b*c - 8*sin(c + d*x)**3*b**3 + 24*sin(c + d 
*x)*a**2*b + 48*sin(c + d*x)*a*b*c + 24*sin(c + d*x)*b**3 + 12*a**2*c*d*x 
+ 24*a*b**2*d*x + 9*b**2*c*d*x)/(24*d)