3.964 \(\int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=445 \[ \frac{\sin (c+d x) \left (84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+280 a^3 b B+224 a b^3 B+8 b^4 (7 A+6 C)\right )}{105 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right ) (a+b \cos (c+d x))^2}{70 d}+\frac{b \sin (c+d x) \cos ^3(c+d x) \left (336 a^2 b B+24 a^3 C+4 a b^2 (126 A+103 C)+175 b^3 B\right )}{840 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (3 a^2 b^2 (63 A+50 C)+91 a^3 b B+4 a^4 C+112 a b^3 B+4 b^4 (7 A+6 C)\right )}{105 d}+\frac{\sin (c+d x) \cos (c+d x) \left (8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right )}{16 d}+\frac{1}{16} x \left (8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right )+\frac{(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{42 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d} \]

[Out]

((8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*x)/16 + ((280*a^3*b*B + 224*a*
b^3*B + 35*a^4*(3*A + 2*C) + 84*a^2*b^2*(5*A + 4*C) + 8*b^4*(7*A + 6*C))*Sin[c + d*x])/(105*d) + ((8*a^4*B + 3
6*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((91*a^
3*b*B + 112*a*b^3*B + 4*a^4*C + 4*b^4*(7*A + 6*C) + 3*a^2*b^2*(63*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(105
*d) + (b*(336*a^2*b*B + 175*b^3*B + 24*a^3*C + 4*a*b^2*(126*A + 103*C))*Cos[c + d*x]^3*Sin[c + d*x])/(840*d) +
((14*A*b^2 + 21*a*b*B + 4*a^2*C + 12*b^2*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(70*d) + ((7*
b*B + 4*a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(42*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x
])^4*Sin[c + d*x])/(7*d)

________________________________________________________________________________________

Rubi [A]  time = 1.04241, antiderivative size = 445, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3049, 3033, 3023, 2734} \[ \frac{\sin (c+d x) \left (84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+280 a^3 b B+224 a b^3 B+8 b^4 (7 A+6 C)\right )}{105 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right ) (a+b \cos (c+d x))^2}{70 d}+\frac{b \sin (c+d x) \cos ^3(c+d x) \left (336 a^2 b B+24 a^3 C+4 a b^2 (126 A+103 C)+175 b^3 B\right )}{840 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (3 a^2 b^2 (63 A+50 C)+91 a^3 b B+4 a^4 C+112 a b^3 B+4 b^4 (7 A+6 C)\right )}{105 d}+\frac{\sin (c+d x) \cos (c+d x) \left (8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right )}{16 d}+\frac{1}{16} x \left (8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right )+\frac{(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{42 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*x)/16 + ((280*a^3*b*B + 224*a*
b^3*B + 35*a^4*(3*A + 2*C) + 84*a^2*b^2*(5*A + 4*C) + 8*b^4*(7*A + 6*C))*Sin[c + d*x])/(105*d) + ((8*a^4*B + 3
6*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((91*a^
3*b*B + 112*a*b^3*B + 4*a^4*C + 4*b^4*(7*A + 6*C) + 3*a^2*b^2*(63*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(105
*d) + (b*(336*a^2*b*B + 175*b^3*B + 24*a^3*C + 4*a*b^2*(126*A + 103*C))*Cos[c + d*x]^3*Sin[c + d*x])/(840*d) +
((14*A*b^2 + 21*a*b*B + 4*a^2*C + 12*b^2*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(70*d) + ((7*
b*B + 4*a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(42*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x
])^4*Sin[c + d*x])/(7*d)

Rule 3049

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])))

Rule 3033

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 3023

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 2734

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)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{1}{7} \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (a (7 A+2 C)+(7 A b+7 a B+6 b C) \cos (c+d x)+(7 b B+4 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{(7 b B+4 a C) \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{42 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{1}{42} \int \cos (c+d x) (a+b \cos (c+d x))^2 \left (2 a (21 a A+7 b B+10 a C)+\left (84 a A b+42 a^2 B+35 b^2 B+68 a b C\right ) \cos (c+d x)+3 \left (14 A b^2+21 a b B+4 a^2 C+12 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{\left (14 A b^2+21 a b B+4 a^2 C+12 b^2 C\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{70 d}+\frac{(7 b B+4 a C) \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{42 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{1}{210} \int \cos (c+d x) (a+b \cos (c+d x)) \left (2 a \left (98 a b B+6 b^2 (7 A+6 C)+a^2 (105 A+62 C)\right )+\left (210 a^3 B+497 a b^2 B+24 b^3 (7 A+6 C)+a^2 (630 A b+488 b C)\right ) \cos (c+d x)+\left (336 a^2 b B+175 b^3 B+24 a^3 C+4 a b^2 (126 A+103 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{b \left (336 a^2 b B+175 b^3 B+24 a^3 C+4 a b^2 (126 A+103 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac{\left (14 A b^2+21 a b B+4 a^2 C+12 b^2 C\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{70 d}+\frac{(7 b B+4 a C) \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{42 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{1}{840} \int \cos (c+d x) \left (8 a^2 \left (98 a b B+6 b^2 (7 A+6 C)+a^2 (105 A+62 C)\right )+105 \left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \cos (c+d x)+24 \left (91 a^3 b B+112 a b^3 B+4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{\left (91 a^3 b B+112 a b^3 B+4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{105 d}+\frac{b \left (336 a^2 b B+175 b^3 B+24 a^3 C+4 a b^2 (126 A+103 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac{\left (14 A b^2+21 a b B+4 a^2 C+12 b^2 C\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{70 d}+\frac{(7 b B+4 a C) \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{42 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{\int \cos (c+d x) \left (24 \left (280 a^3 b B+224 a b^3 B+35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right )+315 \left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \cos (c+d x)\right ) \, dx}{2520}\\ &=\frac{1}{16} \left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) x+\frac{\left (280 a^3 b B+224 a b^3 B+35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x)}{105 d}+\frac{\left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{\left (91 a^3 b B+112 a b^3 B+4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{105 d}+\frac{b \left (336 a^2 b B+175 b^3 B+24 a^3 C+4 a b^2 (126 A+103 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac{\left (14 A b^2+21 a b B+4 a^2 C+12 b^2 C\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{70 d}+\frac{(7 b B+4 a C) \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{42 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 1.42587, size = 528, normalized size = 1.19 \[ \frac{105 \sin (c+d x) \left (48 a^2 b^2 (6 A+5 C)+16 a^4 (4 A+3 C)+192 a^3 b B+160 a b^3 B+5 b^4 (8 A+7 C)\right )+105 \sin (2 (c+d x)) \left (64 a^3 b (A+C)+96 a^2 b^2 B+16 a^4 B+4 a b^3 (16 A+15 C)+15 b^4 B\right )+3360 a^2 A b^2 \sin (3 (c+d x))+13440 a^3 A b c+13440 a^3 A b d x+1260 a^2 b^2 B \sin (4 (c+d x))+15120 a^2 b^2 B c+15120 a^2 b^2 B d x+4200 a^2 b^2 C \sin (3 (c+d x))+504 a^2 b^2 C \sin (5 (c+d x))+2240 a^3 b B \sin (3 (c+d x))+840 a^3 b C \sin (4 (c+d x))+10080 a^3 b c C+10080 a^3 b C d x+3360 a^4 B c+3360 a^4 B d x+560 a^4 C \sin (3 (c+d x))+840 a A b^3 \sin (4 (c+d x))+10080 a A b^3 c+10080 a A b^3 d x+2800 a b^3 B \sin (3 (c+d x))+336 a b^3 B \sin (5 (c+d x))+1260 a b^3 C \sin (4 (c+d x))+140 a b^3 C \sin (6 (c+d x))+8400 a b^3 c C+8400 a b^3 C d x+700 A b^4 \sin (3 (c+d x))+84 A b^4 \sin (5 (c+d x))+315 b^4 B \sin (4 (c+d x))+35 b^4 B \sin (6 (c+d x))+2100 b^4 B c+2100 b^4 B d x+735 b^4 C \sin (3 (c+d x))+147 b^4 C \sin (5 (c+d x))+15 b^4 C \sin (7 (c+d x))}{6720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13440*a^3*A*b*c + 10080*a*A*b^3*c + 3360*a^4*B*c + 15120*a^2*b^2*B*c + 2100*b^4*B*c + 10080*a^3*b*c*C + 8400*
a*b^3*c*C + 13440*a^3*A*b*d*x + 10080*a*A*b^3*d*x + 3360*a^4*B*d*x + 15120*a^2*b^2*B*d*x + 2100*b^4*B*d*x + 10
080*a^3*b*C*d*x + 8400*a*b^3*C*d*x + 105*(192*a^3*b*B + 160*a*b^3*B + 16*a^4*(4*A + 3*C) + 48*a^2*b^2*(6*A + 5
*C) + 5*b^4*(8*A + 7*C))*Sin[c + d*x] + 105*(16*a^4*B + 96*a^2*b^2*B + 15*b^4*B + 64*a^3*b*(A + C) + 4*a*b^3*(
16*A + 15*C))*Sin[2*(c + d*x)] + 3360*a^2*A*b^2*Sin[3*(c + d*x)] + 700*A*b^4*Sin[3*(c + d*x)] + 2240*a^3*b*B*S
in[3*(c + d*x)] + 2800*a*b^3*B*Sin[3*(c + d*x)] + 560*a^4*C*Sin[3*(c + d*x)] + 4200*a^2*b^2*C*Sin[3*(c + d*x)]
+ 735*b^4*C*Sin[3*(c + d*x)] + 840*a*A*b^3*Sin[4*(c + d*x)] + 1260*a^2*b^2*B*Sin[4*(c + d*x)] + 315*b^4*B*Sin
[4*(c + d*x)] + 840*a^3*b*C*Sin[4*(c + d*x)] + 1260*a*b^3*C*Sin[4*(c + d*x)] + 84*A*b^4*Sin[5*(c + d*x)] + 336
*a*b^3*B*Sin[5*(c + d*x)] + 504*a^2*b^2*C*Sin[5*(c + d*x)] + 147*b^4*C*Sin[5*(c + d*x)] + 35*b^4*B*Sin[6*(c +
d*x)] + 140*a*b^3*C*Sin[6*(c + d*x)] + 15*b^4*C*Sin[7*(c + d*x)])/(6720*d)

________________________________________________________________________________________

Maple [A]  time = 0.024, size = 505, normalized size = 1.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/5*A*b^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+b^4*B*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*c
os(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+1/7*C*b^4*(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)+4*a*A*b^3*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4/5*a*b^3*B*(8/3+cos(d*x+c)^4+4/3*c
os(d*x+c)^2)*sin(d*x+c)+4*C*a*b^3*(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+5/1
6*c)+2*a^2*A*b^2*(2+cos(d*x+c)^2)*sin(d*x+c)+6*a^2*b^2*B*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x
+3/8*c)+6/5*a^2*b^2*C*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4*A*a^3*b*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*
d*x+1/2*c)+4/3*a^3*b*B*(2+cos(d*x+c)^2)*sin(d*x+c)+4*a^3*b*C*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8
*d*x+3/8*c)+A*a^4*sin(d*x+c)+a^4*B*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+1/3*a^4*C*(2+cos(d*x+c)^2)*sin(d*
x+c))

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Maxima [A]  time = 1.04286, size = 672, normalized size = 1.51 \begin{align*} \frac{1680 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 2240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 6720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 8960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B 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 )} C a^{3} b - 13440 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 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 )} B 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 )} C a^{2} b^{2} + 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 )} A a b^{3} + 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 )} B 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 )} C a b^{3} + 448 \,{\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^{4} - 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 )} B 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 )} C b^{4} + 6720 \, A a^{4} \sin \left (d x + c\right )}{6720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6720*(1680*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4 - 2240*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 + 6720*(2*d
*x + 2*c + sin(2*d*x + 2*c))*A*a^3*b - 8960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3*b + 840*(12*d*x + 12*c + s
in(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^3*b - 13440*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2*b^2 + 1260*(12*d
*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^2*b^2 + 2688*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15
*sin(d*x + c))*C*a^2*b^2 + 840*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a*b^3 + 1792*(3*sin(d
*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a*b^3 - 140*(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))*C*a*b^3 + 448*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*b^
4 - 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))*B*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))*C*b^4 + 6720*A*a^4*sin(d*x + c))/d

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Fricas [A]  time = 2.08945, size = 852, normalized size = 1.91 \begin{align*} \frac{105 \,{\left (8 \, B a^{4} + 8 \,{\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \,{\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} d x +{\left (240 \, C b^{4} \cos \left (d x + c\right )^{6} + 280 \,{\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + 560 \,{\left (3 \, A + 2 \, C\right )} a^{4} + 4480 \, B a^{3} b + 1344 \,{\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 3584 \, B a b^{3} + 128 \,{\left (7 \, A + 6 \, C\right )} b^{4} + 48 \,{\left (42 \, C a^{2} b^{2} + 28 \, B a b^{3} +{\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \,{\left (24 \, C a^{3} b + 36 \, B a^{2} b^{2} + 4 \,{\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 16 \,{\left (35 \, C a^{4} + 140 \, B a^{3} b + 42 \,{\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 112 \, B a b^{3} + 4 \,{\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \,{\left (8 \, B a^{4} + 8 \,{\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \,{\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(105*(8*B*a^4 + 8*(4*A + 3*C)*a^3*b + 36*B*a^2*b^2 + 4*(6*A + 5*C)*a*b^3 + 5*B*b^4)*d*x + (240*C*b^4*co
s(d*x + c)^6 + 280*(4*C*a*b^3 + B*b^4)*cos(d*x + c)^5 + 560*(3*A + 2*C)*a^4 + 4480*B*a^3*b + 1344*(5*A + 4*C)*
a^2*b^2 + 3584*B*a*b^3 + 128*(7*A + 6*C)*b^4 + 48*(42*C*a^2*b^2 + 28*B*a*b^3 + (7*A + 6*C)*b^4)*cos(d*x + c)^4
+ 70*(24*C*a^3*b + 36*B*a^2*b^2 + 4*(6*A + 5*C)*a*b^3 + 5*B*b^4)*cos(d*x + c)^3 + 16*(35*C*a^4 + 140*B*a^3*b
+ 42*(5*A + 4*C)*a^2*b^2 + 112*B*a*b^3 + 4*(7*A + 6*C)*b^4)*cos(d*x + c)^2 + 105*(8*B*a^4 + 8*(4*A + 3*C)*a^3*
b + 36*B*a^2*b^2 + 4*(6*A + 5*C)*a*b^3 + 5*B*b^4)*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 13.1221, size = 1334, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21344, size = 527, normalized size = 1.18 \begin{align*} \frac{C b^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{1}{16} \,{\left (8 \, B a^{4} + 32 \, A a^{3} b + 24 \, C a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 20 \, C a b^{3} + 5 \, B b^{4}\right )} x + \frac{{\left (4 \, C a b^{3} + B b^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{{\left (24 \, C a^{2} b^{2} + 16 \, B a b^{3} + 4 \, A b^{4} + 7 \, C b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (8 \, C a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3} + 12 \, C a b^{3} + 3 \, B b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (16 \, C a^{4} + 64 \, B a^{3} b + 96 \, A a^{2} b^{2} + 120 \, C a^{2} b^{2} + 80 \, B a b^{3} + 20 \, A b^{4} + 21 \, C b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{{\left (16 \, B a^{4} + 64 \, A a^{3} b + 64 \, C a^{3} b + 96 \, B a^{2} b^{2} + 64 \, A a b^{3} + 60 \, C a b^{3} + 15 \, B b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (64 \, A a^{4} + 48 \, C a^{4} + 192 \, B a^{3} b + 288 \, A a^{2} b^{2} + 240 \, C a^{2} b^{2} + 160 \, B a b^{3} + 40 \, A b^{4} + 35 \, C b^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/448*C*b^4*sin(7*d*x + 7*c)/d + 1/16*(8*B*a^4 + 32*A*a^3*b + 24*C*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 20*C*a*
b^3 + 5*B*b^4)*x + 1/192*(4*C*a*b^3 + B*b^4)*sin(6*d*x + 6*c)/d + 1/320*(24*C*a^2*b^2 + 16*B*a*b^3 + 4*A*b^4 +
7*C*b^4)*sin(5*d*x + 5*c)/d + 1/64*(8*C*a^3*b + 12*B*a^2*b^2 + 8*A*a*b^3 + 12*C*a*b^3 + 3*B*b^4)*sin(4*d*x +
4*c)/d + 1/192*(16*C*a^4 + 64*B*a^3*b + 96*A*a^2*b^2 + 120*C*a^2*b^2 + 80*B*a*b^3 + 20*A*b^4 + 21*C*b^4)*sin(3
*d*x + 3*c)/d + 1/64*(16*B*a^4 + 64*A*a^3*b + 64*C*a^3*b + 96*B*a^2*b^2 + 64*A*a*b^3 + 60*C*a*b^3 + 15*B*b^4)*
sin(2*d*x + 2*c)/d + 1/64*(64*A*a^4 + 48*C*a^4 + 192*B*a^3*b + 288*A*a^2*b^2 + 240*C*a^2*b^2 + 160*B*a*b^3 + 4
0*A*b^4 + 35*C*b^4)*sin(d*x + c)/d