Integrand size = 37, antiderivative size = 128 \[ \int \cos (c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} (4 A b+4 a B+3 b C) x+\frac {(3 a A+2 b B+2 a C) \sin (c+d x)}{3 d}+\frac {(4 A b+4 a B+3 b C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {(b B+a C) \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {b C \cos ^3(c+d x) \sin (c+d x)}{4 d} \] Output:
1/8*(4*A*b+4*B*a+3*C*b)*x+1/3*(3*A*a+2*B*b+2*C*a)*sin(d*x+c)/d+1/8*(4*A*b+ 4*B*a+3*C*b)*cos(d*x+c)*sin(d*x+c)/d+1/3*(B*b+C*a)*cos(d*x+c)^2*sin(d*x+c) /d+1/4*b*C*cos(d*x+c)^3*sin(d*x+c)/d
Time = 0.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.92 \[ \int \cos (c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {48 A b c+48 a B c+36 b c C+48 A b d x+48 a B d x+36 b C d x+24 (4 a A+3 b B+3 a C) \sin (c+d x)+24 (A b+a B+b C) \sin (2 (c+d x))+8 b B \sin (3 (c+d x))+8 a C \sin (3 (c+d x))+3 b C \sin (4 (c+d x))}{96 d} \] Input:
Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Output:
(48*A*b*c + 48*a*B*c + 36*b*c*C + 48*A*b*d*x + 48*a*B*d*x + 36*b*C*d*x + 2 4*(4*a*A + 3*b*B + 3*a*C)*Sin[c + d*x] + 24*(A*b + a*B + b*C)*Sin[2*(c + d *x)] + 8*b*B*Sin[3*(c + d*x)] + 8*a*C*Sin[3*(c + d*x)] + 3*b*C*Sin[4*(c + d*x)])/(96*d)
Time = 0.50 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {3042, 3512, 3042, 3502, 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.
| \(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{4} \int \cos (c+d x) \left (4 (b B+a C) \cos ^2(c+d x)+(4 A b+3 C b+4 a B) \cos (c+d x)+4 a A\right )dx+\frac {b C \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (4 (b B+a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(4 A b+3 C b+4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+4 a A\right )dx+\frac {b C \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \cos (c+d x) (4 (3 a A+2 b B+2 a C)+3 (4 A b+3 C b+4 a B) \cos (c+d x))dx+\frac {4 (a C+b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {b C \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (4 (3 a A+2 b B+2 a C)+3 (4 A b+3 C b+4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {4 (a C+b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {b C \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {4 \sin (c+d x) (3 a A+2 a C+2 b B)}{d}+\frac {3 \sin (c+d x) \cos (c+d x) (4 a B+4 A b+3 b C)}{2 d}+\frac {3}{2} x (4 a B+4 A b+3 b C)\right )+\frac {4 (a C+b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {b C \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
Input:
Int[Cos[c + d*x]*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x] ^2),x]
Output:
(b*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + ((4*(b*B + a*C)*Cos[c + d*x]^2*S in[c + d*x])/(3*d) + ((3*(4*A*b + 4*a*B + 3*b*C)*x)/2 + (4*(3*a*A + 2*b*B + 2*a*C)*Sin[c + d*x])/d + (3*(4*A*b + 4*a*B + 3*b*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d))/3)/4
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]
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]
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]
Time = 4.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.73
| method | result | size |
| parallelrisch | \(\frac {24 \left (\left (A +C \right ) b +B a \right ) \sin \left (2 d x +2 c \right )+8 \left (B b +C a \right ) \sin \left (3 d x +3 c \right )+3 C b \sin \left (4 d x +4 c \right )+24 \left (3 B b +4 a \left (A +\frac {3 C}{4}\right )\right ) \sin \left (d x +c \right )+48 \left (\left (A +\frac {3 C}{4}\right ) b +B a \right ) x d}{96 d}\) | \(94\) |
| parts | \(\frac {\left (A b +B a \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (B b +C a \right ) \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3 d}+\frac {C b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\sin \left (d x +c \right ) a A}{d}\) | \(114\) |
| derivativedivides | \(\frac {C b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {C a \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+A b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )}{d}\) | \(141\) |
| default | \(\frac {C b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {C a \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+A b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )}{d}\) | \(141\) |
| risch | \(\frac {x A b}{2}+\frac {x B a}{2}+\frac {3 b C x}{8}+\frac {\sin \left (d x +c \right ) a A}{d}+\frac {3 \sin \left (d x +c \right ) B b}{4 d}+\frac {3 a C \sin \left (d x +c \right )}{4 d}+\frac {C b \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B b}{12 d}+\frac {\sin \left (3 d x +3 c \right ) C a}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A b}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B a}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C b}{4 d}\) | \(151\) |
| norman | \(\frac {\left (\frac {1}{2} A b +\frac {1}{2} B a +\frac {3}{8} C b \right ) x +\left (2 A b +2 B a +\frac {3}{2} C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (2 A b +2 B a +\frac {3}{2} C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (3 A b +3 B a +\frac {9}{4} C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {1}{2} A b +\frac {1}{2} B a +\frac {3}{8} C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {\left (8 a A -4 A b -4 B a +8 B b +8 C a -5 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {\left (8 a A +4 A b +4 B a +8 B b +8 C a +5 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (72 a A -12 A b -12 B a +40 B b +40 C a +9 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}+\frac {\left (72 a A +12 A b +12 B a +40 B b +40 C a -9 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\) | \(299\) |
| orering | \(\text {Expression too large to display}\) | \(4752\) |
Input:
int(cos(d*x+c)*(a+b*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,method=_ RETURNVERBOSE)
Output:
1/96*(24*((A+C)*b+B*a)*sin(2*d*x+2*c)+8*(B*b+C*a)*sin(3*d*x+3*c)+3*C*b*sin (4*d*x+4*c)+24*(3*B*b+4*a*(A+3/4*C))*sin(d*x+c)+48*((A+3/4*C)*b+B*a)*x*d)/ d
Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.76 \[ \int \cos (c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (4 \, B a + {\left (4 \, A + 3 \, C\right )} b\right )} d x + {\left (6 \, C b \cos \left (d x + c\right )^{3} + 8 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, A + 2 \, C\right )} a + 16 \, B b + 3 \, {\left (4 \, B a + {\left (4 \, A + 3 \, C\right )} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, a lgorithm="fricas")
Output:
1/24*(3*(4*B*a + (4*A + 3*C)*b)*d*x + (6*C*b*cos(d*x + c)^3 + 8*(C*a + B*b )*cos(d*x + c)^2 + 8*(3*A + 2*C)*a + 16*B*b + 3*(4*B*a + (4*A + 3*C)*b)*co s(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (124) = 248\).
Time = 0.18 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.50 \[ \int \cos (c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {A a \sin {\left (c + d x \right )}}{d} + \frac {A b x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A b x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {B a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {B a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {B a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 B b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 C a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right ) \left (A + B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \cos {\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)
Output:
Piecewise((A*a*sin(c + d*x)/d + A*b*x*sin(c + d*x)**2/2 + A*b*x*cos(c + d* x)**2/2 + A*b*sin(c + d*x)*cos(c + d*x)/(2*d) + B*a*x*sin(c + d*x)**2/2 + B*a*x*cos(c + d*x)**2/2 + B*a*sin(c + d*x)*cos(c + d*x)/(2*d) + 2*B*b*sin( c + d*x)**3/(3*d) + B*b*sin(c + d*x)*cos(c + d*x)**2/d + 2*C*a*sin(c + d*x )**3/(3*d) + C*a*sin(c + d*x)*cos(c + d*x)**2/d + 3*C*b*x*sin(c + d*x)**4/ 8 + 3*C*b*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*C*b*x*cos(c + d*x)**4/8 + 3*C*b*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 5*C*b*sin(c + d*x)*cos(c + d* x)**3/(8*d), Ne(d, 0)), (x*(a + b*cos(c))*(A + B*cos(c) + C*cos(c)**2)*cos (c), True))
Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.03 \[ \int \cos (c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b + 96 \, A a \sin \left (d x + c\right )}{96 \, d} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, a lgorithm="maxima")
Output:
1/96*(24*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a - 32*(sin(d*x + c)^3 - 3*sin (d*x + c))*C*a + 24*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*b - 32*(sin(d*x + c )^3 - 3*sin(d*x + c))*B*b + 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2* d*x + 2*c))*C*b + 96*A*a*sin(d*x + c))/d
Time = 0.13 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int \cos (c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} \, {\left (4 \, B a + 4 \, A b + 3 \, C b\right )} x + \frac {C b \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (C a + B b\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (B a + A b + C b\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A a + 3 \, C a + 3 \, B b\right )} \sin \left (d x + c\right )}{4 \, d} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, a lgorithm="giac")
Output:
1/8*(4*B*a + 4*A*b + 3*C*b)*x + 1/32*C*b*sin(4*d*x + 4*c)/d + 1/12*(C*a + B*b)*sin(3*d*x + 3*c)/d + 1/4*(B*a + A*b + C*b)*sin(2*d*x + 2*c)/d + 1/4*( 4*A*a + 3*C*a + 3*B*b)*sin(d*x + c)/d
Time = 0.86 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.17 \[ \int \cos (c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {A\,b\,x}{2}+\frac {B\,a\,x}{2}+\frac {3\,C\,b\,x}{8}+\frac {A\,a\,\sin \left (c+d\,x\right )}{d}+\frac {3\,B\,b\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,C\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {A\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \] Input:
int(cos(c + d*x)*(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x) ^2),x)
Output:
(A*b*x)/2 + (B*a*x)/2 + (3*C*b*x)/8 + (A*a*sin(c + d*x))/d + (3*B*b*sin(c + d*x))/(4*d) + (3*C*a*sin(c + d*x))/(4*d) + (A*b*sin(2*c + 2*d*x))/(4*d) + (B*a*sin(2*c + 2*d*x))/(4*d) + (B*b*sin(3*c + 3*d*x))/(12*d) + (C*a*sin( 3*c + 3*d*x))/(12*d) + (C*b*sin(2*c + 2*d*x))/(4*d) + (C*b*sin(4*c + 4*d*x ))/(32*d)
Time = 0.15 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98 \[ \int \cos (c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b c +24 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b +15 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b c -8 \sin \left (d x +c \right )^{3} a c -8 \sin \left (d x +c \right )^{3} b^{2}+24 \sin \left (d x +c \right ) a^{2}+24 \sin \left (d x +c \right ) a c +24 \sin \left (d x +c \right ) b^{2}+24 a b d x +9 b c d x}{24 d} \] Input:
int(cos(d*x+c)*(a+b*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
Output:
( - 6*cos(c + d*x)*sin(c + d*x)**3*b*c + 24*cos(c + d*x)*sin(c + d*x)*a*b + 15*cos(c + d*x)*sin(c + d*x)*b*c - 8*sin(c + d*x)**3*a*c - 8*sin(c + d*x )**3*b**2 + 24*sin(c + d*x)*a**2 + 24*sin(c + d*x)*a*c + 24*sin(c + d*x)*b **2 + 24*a*b*d*x + 9*b*c*d*x)/(24*d)