\(\int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx\) [240]

Optimal result

Integrand size = 31, antiderivative size = 366 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {1}{16} \left (8 a^4 A+36 a^2 A b^2+5 A b^4+24 a^3 b B+20 a b^3 B\right ) x+\frac {\left (140 a^3 A b+112 a A b^3+35 a^4 B+168 a^2 b^2 B+24 b^4 B\right ) \sin (c+d x)}{35 d}+\frac {\left (8 a^4 A+36 a^2 A b^2+5 A b^4+24 a^3 b B+20 a b^3 B\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \left (224 a^2 A b+35 A b^3+104 a^3 B+140 a b^2 B\right ) \cos ^3(c+d x) \sin (c+d x)}{168 d}+\frac {b^2 \left (49 a A b+31 a^2 B+18 b^2 B\right ) \cos ^4(c+d x) \sin (c+d x)}{105 d}+\frac {b (7 A b+10 a B) \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{42 d}+\frac {b B \cos ^3(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d}-\frac {\left (140 a^3 A b+112 a A b^3+35 a^4 B+168 a^2 b^2 B+24 b^4 B\right ) \sin ^3(c+d x)}{105 d} \]

[Out]

1/16*(8*A*a^4+36*A*a^2*b^2+5*A*b^4+24*B*a^3*b+20*B*a*b^3)*x+1/35*(140*A*a^3*b+112*A*a*b^3+35*B*a^4+168*B*a^2*b
^2+24*B*b^4)*sin(d*x+c)/d+1/16*(8*A*a^4+36*A*a^2*b^2+5*A*b^4+24*B*a^3*b+20*B*a*b^3)*cos(d*x+c)*sin(d*x+c)/d+1/
168*b*(224*A*a^2*b+35*A*b^3+104*B*a^3+140*B*a*b^2)*cos(d*x+c)^3*sin(d*x+c)/d+1/105*b^2*(49*A*a*b+31*B*a^2+18*B
*b^2)*cos(d*x+c)^4*sin(d*x+c)/d+1/42*b*(7*A*b+10*B*a)*cos(d*x+c)^3*(a+b*cos(d*x+c))^2*sin(d*x+c)/d+1/7*b*B*cos
(d*x+c)^3*(a+b*cos(d*x+c))^3*sin(d*x+c)/d-1/105*(140*A*a^3*b+112*A*a*b^3+35*B*a^4+168*B*a^2*b^2+24*B*b^4)*sin(
d*x+c)^3/d

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {3069, 3128, 3112, 3102, 2827, 2715, 8, 2713} \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {b^2 \left (31 a^2 B+49 a A b+18 b^2 B\right ) \sin (c+d x) \cos ^4(c+d x)}{105 d}+\frac {b \left (104 a^3 B+224 a^2 A b+140 a b^2 B+35 A b^3\right ) \sin (c+d x) \cos ^3(c+d x)}{168 d}-\frac {\left (35 a^4 B+140 a^3 A b+168 a^2 b^2 B+112 a A b^3+24 b^4 B\right ) \sin ^3(c+d x)}{105 d}+\frac {\left (35 a^4 B+140 a^3 A b+168 a^2 b^2 B+112 a A b^3+24 b^4 B\right ) \sin (c+d x)}{35 d}+\frac {\left (8 a^4 A+24 a^3 b B+36 a^2 A b^2+20 a b^3 B+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} x \left (8 a^4 A+24 a^3 b B+36 a^2 A b^2+20 a b^3 B+5 A b^4\right )+\frac {b (10 a B+7 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{42 d}+\frac {b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^3}{7 d} \]

[In]

Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]

[Out]

((8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b*B + 20*a*b^3*B)*x)/16 + ((140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B +
168*a^2*b^2*B + 24*b^4*B)*Sin[c + d*x])/(35*d) + ((8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b*B + 20*a*b^3*B
)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b*(224*a^2*A*b + 35*A*b^3 + 104*a^3*B + 140*a*b^2*B)*Cos[c + d*x]^3*Sin
[c + d*x])/(168*d) + (b^2*(49*a*A*b + 31*a^2*B + 18*b^2*B)*Cos[c + d*x]^4*Sin[c + d*x])/(105*d) + (b*(7*A*b +
10*a*B)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(42*d) + (b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^
3*Sin[c + d*x])/(7*d) - ((140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B + 168*a^2*b^2*B + 24*b^4*B)*Sin[c + d*x]^3)/(10
5*d)

Rule 8

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

Rule 2713

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

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2827

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

Rule 3069

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-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] + Dist[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] &&  !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[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 3112

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*Sin[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[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]

Rule 3128

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

Rubi steps \begin{align*} \text {integral}& = \frac {b B \cos ^3(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {1}{7} \int \cos ^2(c+d x) (a+b \cos (c+d x))^2 \left (a (7 a A+3 b B)+\left (6 b^2 B+7 a (2 A b+a B)\right ) \cos (c+d x)+b (7 A b+10 a B) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {b (7 A b+10 a B) \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{42 d}+\frac {b B \cos ^3(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {1}{42} \int \cos ^2(c+d x) (a+b \cos (c+d x)) \left (3 a \left (14 a^2 A+7 A b^2+16 a b B\right )+\left (126 a^2 A b+35 A b^3+42 a^3 B+104 a b^2 B\right ) \cos (c+d x)+2 b \left (49 a A b+31 a^2 B+18 b^2 B\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {b^2 \left (49 a A b+31 a^2 B+18 b^2 B\right ) \cos ^4(c+d x) \sin (c+d x)}{105 d}+\frac {b (7 A b+10 a B) \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{42 d}+\frac {b B \cos ^3(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {1}{210} \int \cos ^2(c+d x) \left (15 a^2 \left (14 a^2 A+7 A b^2+16 a b B\right )+6 \left (140 a^3 A b+112 a A b^3+35 a^4 B+168 a^2 b^2 B+24 b^4 B\right ) \cos (c+d x)+5 b \left (224 a^2 A b+35 A b^3+104 a^3 B+140 a b^2 B\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {b \left (224 a^2 A b+35 A b^3+104 a^3 B+140 a b^2 B\right ) \cos ^3(c+d x) \sin (c+d x)}{168 d}+\frac {b^2 \left (49 a A b+31 a^2 B+18 b^2 B\right ) \cos ^4(c+d x) \sin (c+d x)}{105 d}+\frac {b (7 A b+10 a B) \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{42 d}+\frac {b B \cos ^3(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {1}{840} \int \cos ^2(c+d x) \left (105 \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 )+24 \left (140 a^3 A b+112 a A b^3+35 a^4 B+168 a^2 b^2 B+24 b^4 B\right ) \cos (c+d x)\right ) \, dx \\ & = \frac {b \left (224 a^2 A b+35 A b^3+104 a^3 B+140 a b^2 B\right ) \cos ^3(c+d x) \sin (c+d x)}{168 d}+\frac {b^2 \left (49 a A b+31 a^2 B+18 b^2 B\right ) \cos ^4(c+d x) \sin (c+d x)}{105 d}+\frac {b (7 A b+10 a B) \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{42 d}+\frac {b B \cos ^3(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {1}{8} \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 ) \int \cos ^2(c+d x) \, dx+\frac {1}{35} \left (140 a^3 A b+112 a A b^3+35 a^4 B+168 a^2 b^2 B+24 b^4 B\right ) \int \cos ^3(c+d x) \, dx \\ & = \frac {\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 ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \left (224 a^2 A b+35 A b^3+104 a^3 B+140 a b^2 B\right ) \cos ^3(c+d x) \sin (c+d x)}{168 d}+\frac {b^2 \left (49 a A b+31 a^2 B+18 b^2 B\right ) \cos ^4(c+d x) \sin (c+d x)}{105 d}+\frac {b (7 A b+10 a B) \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{42 d}+\frac {b B \cos ^3(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {1}{16} \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 ) \int 1 \, dx-\frac {\left (140 a^3 A b+112 a A b^3+35 a^4 B+168 a^2 b^2 B+24 b^4 B\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d} \\ & = \frac {1}{16} \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 ) x+\frac {\left (140 a^3 A b+112 a A b^3+35 a^4 B+168 a^2 b^2 B+24 b^4 B\right ) \sin (c+d x)}{35 d}+\frac {\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 ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \left (224 a^2 A b+35 A b^3+104 a^3 B+140 a b^2 B\right ) \cos ^3(c+d x) \sin (c+d x)}{168 d}+\frac {b^2 \left (49 a A b+31 a^2 B+18 b^2 B\right ) \cos ^4(c+d x) \sin (c+d x)}{105 d}+\frac {b (7 A b+10 a B) \cos ^3(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{42 d}+\frac {b B \cos ^3(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d}-\frac {\left (140 a^3 A b+112 a A b^3+35 a^4 B+168 a^2 b^2 B+24 b^4 B\right ) \sin ^3(c+d x)}{105 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.89 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.11 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {3360 a^4 A c+15120 a^2 A b^2 c+2100 A b^4 c+10080 a^3 b B c+8400 a b^3 B c+3360 a^4 A d x+15120 a^2 A b^2 d x+2100 A b^4 d x+10080 a^3 b B d x+8400 a b^3 B d x+105 \left (192 a^3 A b+160 a A b^3+48 a^4 B+240 a^2 b^2 B+35 b^4 B\right ) \sin (c+d x)+105 \left (16 a^4 A+96 a^2 A b^2+15 A b^4+64 a^3 b B+60 a b^3 B\right ) \sin (2 (c+d x))+2240 a^3 A b \sin (3 (c+d x))+2800 a A b^3 \sin (3 (c+d x))+560 a^4 B \sin (3 (c+d x))+4200 a^2 b^2 B \sin (3 (c+d x))+735 b^4 B \sin (3 (c+d x))+1260 a^2 A b^2 \sin (4 (c+d x))+315 A b^4 \sin (4 (c+d x))+840 a^3 b B \sin (4 (c+d x))+1260 a b^3 B \sin (4 (c+d x))+336 a A b^3 \sin (5 (c+d x))+504 a^2 b^2 B \sin (5 (c+d x))+147 b^4 B \sin (5 (c+d x))+35 A b^4 \sin (6 (c+d x))+140 a b^3 B \sin (6 (c+d x))+15 b^4 B \sin (7 (c+d x))}{6720 d} \]

[In]

Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]

[Out]

(3360*a^4*A*c + 15120*a^2*A*b^2*c + 2100*A*b^4*c + 10080*a^3*b*B*c + 8400*a*b^3*B*c + 3360*a^4*A*d*x + 15120*a
^2*A*b^2*d*x + 2100*A*b^4*d*x + 10080*a^3*b*B*d*x + 8400*a*b^3*B*d*x + 105*(192*a^3*A*b + 160*a*A*b^3 + 48*a^4
*B + 240*a^2*b^2*B + 35*b^4*B)*Sin[c + d*x] + 105*(16*a^4*A + 96*a^2*A*b^2 + 15*A*b^4 + 64*a^3*b*B + 60*a*b^3*
B)*Sin[2*(c + d*x)] + 2240*a^3*A*b*Sin[3*(c + d*x)] + 2800*a*A*b^3*Sin[3*(c + d*x)] + 560*a^4*B*Sin[3*(c + d*x
)] + 4200*a^2*b^2*B*Sin[3*(c + d*x)] + 735*b^4*B*Sin[3*(c + d*x)] + 1260*a^2*A*b^2*Sin[4*(c + d*x)] + 315*A*b^
4*Sin[4*(c + d*x)] + 840*a^3*b*B*Sin[4*(c + d*x)] + 1260*a*b^3*B*Sin[4*(c + d*x)] + 336*a*A*b^3*Sin[5*(c + d*x
)] + 504*a^2*b^2*B*Sin[5*(c + d*x)] + 147*b^4*B*Sin[5*(c + d*x)] + 35*A*b^4*Sin[6*(c + d*x)] + 140*a*b^3*B*Sin
[6*(c + d*x)] + 15*b^4*B*Sin[7*(c + d*x)])/(6720*d)

Maple [A] (verified)

Time = 6.26 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.75

method	result	size
parts	\(\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {B \,b^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}\)	\(273\)
parallelrisch	\(\frac {\left (1680 a^{4} A +10080 A \,a^{2} b^{2}+1575 A \,b^{4}+6720 B \,a^{3} b +6300 B a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (2240 A \,a^{3} b +2800 A a \,b^{3}+560 B \,a^{4}+4200 B \,a^{2} b^{2}+735 B \,b^{4}\right ) \sin \left (3 d x +3 c \right )+1260 \left (A \,a^{2} b +\frac {1}{4} A \,b^{3}+\frac {2}{3} B \,a^{3}+B a \,b^{2}\right ) b \sin \left (4 d x +4 c \right )+\left (336 A a \,b^{3}+504 B \,a^{2} b^{2}+147 B \,b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (35 A \,b^{4}+140 B a \,b^{3}\right ) \sin \left (6 d x +6 c \right )+15 B \,b^{4} \sin \left (7 d x +7 c \right )+\left (20160 A \,a^{3} b +16800 A a \,b^{3}+5040 B \,a^{4}+25200 B \,a^{2} b^{2}+3675 B \,b^{4}\right ) \sin \left (d x +c \right )+3360 x \left (a^{4} A +\frac {9}{2} A \,a^{2} b^{2}+\frac {5}{8} A \,b^{4}+3 B \,a^{3} b +\frac {5}{2} B a \,b^{3}\right ) d}{6720 d}\)	\(290\)
derivativedivides	\(\frac {\frac {B \,b^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+A \,b^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 B a \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 A a \,b^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {6 B \,a^{2} b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+6 A \,a^{2} b^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 B \,a^{3} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 A \,a^{3} b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(368\)
default	\(\frac {\frac {B \,b^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+A \,b^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 B a \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 A a \,b^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {6 B \,a^{2} b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+6 A \,a^{2} b^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 B \,a^{3} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 A \,a^{3} b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(368\)
risch	\(\frac {7 \sin \left (5 d x +5 c \right ) B \,b^{4}}{320 d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,b^{4}}{64 d}+\frac {7 \sin \left (3 d x +3 c \right ) B \,b^{4}}{64 d}+\frac {15 \sin \left (2 d x +2 c \right ) B a \,b^{3}}{16 d}+\frac {15 \sin \left (d x +c \right ) B \,a^{2} b^{2}}{4 d}+\frac {\sin \left (6 d x +6 c \right ) B a \,b^{3}}{48 d}+\frac {\sin \left (5 d x +5 c \right ) A a \,b^{3}}{20 d}+\frac {3 \sin \left (5 d x +5 c \right ) B \,a^{2} b^{2}}{40 d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{2} b^{2}}{16 d}+\frac {5 x A \,b^{4}}{16}+\frac {B \,b^{4} \sin \left (7 d x +7 c \right )}{448 d}+\frac {15 \sin \left (2 d x +2 c \right ) A \,b^{4}}{64 d}+\frac {3 \sin \left (d x +c \right ) A \,a^{3} b}{d}+\frac {5 \sin \left (d x +c \right ) A a \,b^{3}}{2 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{3} b}{8 d}+\frac {3 \sin \left (4 d x +4 c \right ) B a \,b^{3}}{16 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{3} b}{3 d}+\frac {5 \sin \left (3 d x +3 c \right ) A a \,b^{3}}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) B \,a^{2} b^{2}}{8 d}+\frac {3 \sin \left (2 d x +2 c \right ) A \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3} b}{d}+\frac {9 x A \,a^{2} b^{2}}{4}+\frac {3 x B \,a^{3} b}{2}+\frac {5 x B a \,b^{3}}{4}+\frac {3 \sin \left (d x +c \right ) B \,a^{4}}{4 d}+\frac {35 \sin \left (d x +c \right ) B \,b^{4}}{64 d}+\frac {\sin \left (6 d x +6 c \right ) A \,b^{4}}{192 d}+\frac {a^{4} x A}{2}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{4}}{12 d}+\frac {\sin \left (2 d x +2 c \right ) a^{4} A}{4 d}\)	\(501\)
norman	\(\text {Expression too large to display}\)	\(971\)

[In]

int(cos(d*x+c)^2*(a+cos(d*x+c)*b)^4*(A+B*cos(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

(A*b^4+4*B*a*b^3)/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)+1/5*(4*A*
a*b^3+6*B*a^2*b^2)/d*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+(6*A*a^2*b^2+4*B*a^3*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*(4*A*a^3*b+B*a^4)/d*(2+cos(d*x+c)^2)*sin(d*x+c)+a^4*A/d*(1/2*
cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+1/7*B*b^4/d*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x
+c)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.79 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {105 \, {\left (8 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 20 \, B a b^{3} + 5 \, A b^{4}\right )} d x + {\left (240 \, B b^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{5} + 1120 \, B a^{4} + 4480 \, A a^{3} b + 5376 \, B a^{2} b^{2} + 3584 \, A a b^{3} + 768 \, B b^{4} + 96 \, {\left (21 \, B a^{2} b^{2} + 14 \, A a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 20 \, B a b^{3} + 5 \, A b^{4}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (35 \, B a^{4} + 140 \, A a^{3} b + 168 \, B a^{2} b^{2} + 112 \, A a b^{3} + 24 \, B b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (8 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 20 \, B a b^{3} + 5 \, A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, d} \]

[In]

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/1680*(105*(8*A*a^4 + 24*B*a^3*b + 36*A*a^2*b^2 + 20*B*a*b^3 + 5*A*b^4)*d*x + (240*B*b^4*cos(d*x + c)^6 + 280
*(4*B*a*b^3 + A*b^4)*cos(d*x + c)^5 + 1120*B*a^4 + 4480*A*a^3*b + 5376*B*a^2*b^2 + 3584*A*a*b^3 + 768*B*b^4 +
96*(21*B*a^2*b^2 + 14*A*a*b^3 + 3*B*b^4)*cos(d*x + c)^4 + 70*(24*B*a^3*b + 36*A*a^2*b^2 + 20*B*a*b^3 + 5*A*b^4
)*cos(d*x + c)^3 + 16*(35*B*a^4 + 140*A*a^3*b + 168*B*a^2*b^2 + 112*A*a*b^3 + 24*B*b^4)*cos(d*x + c)^2 + 105*(
8*A*a^4 + 24*B*a^3*b + 36*A*a^2*b^2 + 20*B*a*b^3 + 5*A*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. 1017 vs. \(2 (391) = 782\).

Time = 0.59 (sec) , antiderivative size = 1017, normalized size of antiderivative = 2.78 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)**2*(a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)),x)

[Out]

Piecewise((A*a**4*x*sin(c + d*x)**2/2 + A*a**4*x*cos(c + d*x)**2/2 + A*a**4*sin(c + d*x)*cos(c + d*x)/(2*d) +
8*A*a**3*b*sin(c + d*x)**3/(3*d) + 4*A*a**3*b*sin(c + d*x)*cos(c + d*x)**2/d + 9*A*a**2*b**2*x*sin(c + d*x)**4
/4 + 9*A*a**2*b**2*x*sin(c + d*x)**2*cos(c + d*x)**2/2 + 9*A*a**2*b**2*x*cos(c + d*x)**4/4 + 9*A*a**2*b**2*sin
(c + d*x)**3*cos(c + d*x)/(4*d) + 15*A*a**2*b**2*sin(c + d*x)*cos(c + d*x)**3/(4*d) + 32*A*a*b**3*sin(c + d*x)
**5/(15*d) + 16*A*a*b**3*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 4*A*a*b**3*sin(c + d*x)*cos(c + d*x)**4/d + 5
*A*b**4*x*sin(c + d*x)**6/16 + 15*A*b**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*A*b**4*x*sin(c + d*x)**2*co
s(c + d*x)**4/16 + 5*A*b**4*x*cos(c + d*x)**6/16 + 5*A*b**4*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*A*b**4*sin
(c + d*x)**3*cos(c + d*x)**3/(6*d) + 11*A*b**4*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 2*B*a**4*sin(c + d*x)**3/
(3*d) + B*a**4*sin(c + d*x)*cos(c + d*x)**2/d + 3*B*a**3*b*x*sin(c + d*x)**4/2 + 3*B*a**3*b*x*sin(c + d*x)**2*
cos(c + d*x)**2 + 3*B*a**3*b*x*cos(c + d*x)**4/2 + 3*B*a**3*b*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 5*B*a**3*b*
sin(c + d*x)*cos(c + d*x)**3/(2*d) + 16*B*a**2*b**2*sin(c + d*x)**5/(5*d) + 8*B*a**2*b**2*sin(c + d*x)**3*cos(
c + d*x)**2/d + 6*B*a**2*b**2*sin(c + d*x)*cos(c + d*x)**4/d + 5*B*a*b**3*x*sin(c + d*x)**6/4 + 15*B*a*b**3*x*
sin(c + d*x)**4*cos(c + d*x)**2/4 + 15*B*a*b**3*x*sin(c + d*x)**2*cos(c + d*x)**4/4 + 5*B*a*b**3*x*cos(c + d*x
)**6/4 + 5*B*a*b**3*sin(c + d*x)**5*cos(c + d*x)/(4*d) + 10*B*a*b**3*sin(c + d*x)**3*cos(c + d*x)**3/(3*d) + 1
1*B*a*b**3*sin(c + d*x)*cos(c + d*x)**5/(4*d) + 16*B*b**4*sin(c + d*x)**7/(35*d) + 8*B*b**4*sin(c + d*x)**5*co
s(c + d*x)**2/(5*d) + 2*B*b**4*sin(c + d*x)**3*cos(c + d*x)**4/d + B*b**4*sin(c + d*x)*cos(c + d*x)**6/d, Ne(d
, 0)), (x*(A + B*cos(c))*(a + b*cos(c))**4*cos(c)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 8960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} b + 840 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} b + 1260 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} + 2688 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{2} b^{2} + 1792 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a b^{3} - 140 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{3} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{4} - 192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} B b^{4}}{6720 \, d} \]

[In]

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="maxima")

[Out]

1/6720*(1680*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 - 2240*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^4 - 8960*(sin
(d*x + c)^3 - 3*sin(d*x + c))*A*a^3*b + 840*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^3*b +
1260*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^2*b^2 + 2688*(3*sin(d*x + c)^5 - 10*sin(d*x +
c)^3 + 15*sin(d*x + c))*B*a^2*b^2 + 1792*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a*b^3 - 1
40*(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*a*b^3 - 35*(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))*A*b^4 - 192*(5*sin(d*x + c)^7 - 21*sin(d
*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*B*b^4)/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {B b^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {1}{16} \, {\left (8 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 20 \, B a b^{3} + 5 \, A b^{4}\right )} x + \frac {{\left (4 \, B a b^{3} + A b^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 7 \, B b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (8 \, B a^{3} b + 12 \, A a^{2} b^{2} + 12 \, B a b^{3} + 3 \, A b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (16 \, B a^{4} + 64 \, A a^{3} b + 120 \, B a^{2} b^{2} + 80 \, A a b^{3} + 21 \, B b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (16 \, A a^{4} + 64 \, B a^{3} b + 96 \, A a^{2} b^{2} + 60 \, B a b^{3} + 15 \, A b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (48 \, B a^{4} + 192 \, A a^{3} b + 240 \, B a^{2} b^{2} + 160 \, A a b^{3} + 35 \, B b^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]

[In]

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="giac")

[Out]

1/448*B*b^4*sin(7*d*x + 7*c)/d + 1/16*(8*A*a^4 + 24*B*a^3*b + 36*A*a^2*b^2 + 20*B*a*b^3 + 5*A*b^4)*x + 1/192*(
4*B*a*b^3 + A*b^4)*sin(6*d*x + 6*c)/d + 1/320*(24*B*a^2*b^2 + 16*A*a*b^3 + 7*B*b^4)*sin(5*d*x + 5*c)/d + 1/64*
(8*B*a^3*b + 12*A*a^2*b^2 + 12*B*a*b^3 + 3*A*b^4)*sin(4*d*x + 4*c)/d + 1/192*(16*B*a^4 + 64*A*a^3*b + 120*B*a^
2*b^2 + 80*A*a*b^3 + 21*B*b^4)*sin(3*d*x + 3*c)/d + 1/64*(16*A*a^4 + 64*B*a^3*b + 96*A*a^2*b^2 + 60*B*a*b^3 +
15*A*b^4)*sin(2*d*x + 2*c)/d + 1/64*(48*B*a^4 + 192*A*a^3*b + 240*B*a^2*b^2 + 160*A*a*b^3 + 35*B*b^4)*sin(d*x
+ c)/d

Mupad [B] (verification not implemented)

Time = 2.91 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.19 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {420\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )+\frac {1575\,A\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4}+140\,B\,a^4\,\sin \left (3\,c+3\,d\,x\right )+\frac {315\,A\,b^4\,\sin \left (4\,c+4\,d\,x\right )}{4}+\frac {35\,A\,b^4\,\sin \left (6\,c+6\,d\,x\right )}{4}+\frac {735\,B\,b^4\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {147\,B\,b^4\,\sin \left (5\,c+5\,d\,x\right )}{4}+\frac {15\,B\,b^4\,\sin \left (7\,c+7\,d\,x\right )}{4}+1260\,B\,a^4\,\sin \left (c+d\,x\right )+\frac {3675\,B\,b^4\,\sin \left (c+d\,x\right )}{4}+4200\,A\,a\,b^3\,\sin \left (c+d\,x\right )+5040\,A\,a^3\,b\,\sin \left (c+d\,x\right )+840\,A\,a^4\,d\,x+525\,A\,b^4\,d\,x+700\,A\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )+560\,A\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )+84\,A\,a\,b^3\,\sin \left (5\,c+5\,d\,x\right )+1575\,B\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )+1680\,B\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )+315\,B\,a\,b^3\,\sin \left (4\,c+4\,d\,x\right )+210\,B\,a^3\,b\,\sin \left (4\,c+4\,d\,x\right )+35\,B\,a\,b^3\,\sin \left (6\,c+6\,d\,x\right )+6300\,B\,a^2\,b^2\,\sin \left (c+d\,x\right )+2520\,A\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )+315\,A\,a^2\,b^2\,\sin \left (4\,c+4\,d\,x\right )+1050\,B\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )+126\,B\,a^2\,b^2\,\sin \left (5\,c+5\,d\,x\right )+2100\,B\,a\,b^3\,d\,x+2520\,B\,a^3\,b\,d\,x+3780\,A\,a^2\,b^2\,d\,x}{1680\,d} \]

[In]

int(cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4,x)

[Out]

(420*A*a^4*sin(2*c + 2*d*x) + (1575*A*b^4*sin(2*c + 2*d*x))/4 + 140*B*a^4*sin(3*c + 3*d*x) + (315*A*b^4*sin(4*
c + 4*d*x))/4 + (35*A*b^4*sin(6*c + 6*d*x))/4 + (735*B*b^4*sin(3*c + 3*d*x))/4 + (147*B*b^4*sin(5*c + 5*d*x))/
4 + (15*B*b^4*sin(7*c + 7*d*x))/4 + 1260*B*a^4*sin(c + d*x) + (3675*B*b^4*sin(c + d*x))/4 + 4200*A*a*b^3*sin(c
+ d*x) + 5040*A*a^3*b*sin(c + d*x) + 840*A*a^4*d*x + 525*A*b^4*d*x + 700*A*a*b^3*sin(3*c + 3*d*x) + 560*A*a^3
*b*sin(3*c + 3*d*x) + 84*A*a*b^3*sin(5*c + 5*d*x) + 1575*B*a*b^3*sin(2*c + 2*d*x) + 1680*B*a^3*b*sin(2*c + 2*d
*x) + 315*B*a*b^3*sin(4*c + 4*d*x) + 210*B*a^3*b*sin(4*c + 4*d*x) + 35*B*a*b^3*sin(6*c + 6*d*x) + 6300*B*a^2*b
^2*sin(c + d*x) + 2520*A*a^2*b^2*sin(2*c + 2*d*x) + 315*A*a^2*b^2*sin(4*c + 4*d*x) + 1050*B*a^2*b^2*sin(3*c +
3*d*x) + 126*B*a^2*b^2*sin(5*c + 5*d*x) + 2100*B*a*b^3*d*x + 2520*B*a^3*b*d*x + 3780*A*a^2*b^2*d*x)/(1680*d)