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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.837696, antiderivative size = 366, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {2990, 3049, 3033, 3023, 2748, 2635, 8, 2633} \[ -\frac{\left (140 a^3 A b+168 a^2 b^2 B+35 a^4 B+112 a A b^3+24 b^4 B\right ) \sin ^3(c+d x)}{105 d}+\frac{\left (140 a^3 A b+168 a^2 b^2 B+35 a^4 B+112 a A b^3+24 b^4 B\right ) \sin (c+d x)}{35 d}+\frac{b^2 \left (31 a^2 B+49 a A b+18 b^2 B\right ) \sin (c+d x) \cos ^4(c+d x)}{105 d}+\frac{b \left (224 a^2 A b+104 a^3 B+140 a b^2 B+35 A b^3\right ) \sin (c+d x) \cos ^3(c+d x)}{168 d}+\frac{\left (36 a^2 A b^2+8 a^4 A+24 a^3 b B+20 a b^3 B+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left (36 a^2 A b^2+8 a^4 A+24 a^3 b B+20 a b^3 B+5 A b^4\right )+\frac{b (10 a B+7 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{42 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^3}{7 d} \]

Antiderivative was successfully verified.

[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 2990

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] /; F
reeQ[{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 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 2748

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 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 2633

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]

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx &=\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 ) \operatorname{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]  time = 0.864315, size = 408, normalized size = 1.11 \[ \frac{105 \left (192 a^3 A b+240 a^2 b^2 B+48 a^4 B+160 a A b^3+35 b^4 B\right ) \sin (c+d x)+105 \left (96 a^2 A b^2+16 a^4 A+64 a^3 b B+60 a b^3 B+15 A b^4\right ) \sin (2 (c+d x))+1260 a^2 A b^2 \sin (4 (c+d x))+15120 a^2 A b^2 c+15120 a^2 A b^2 d x+2240 a^3 A b \sin (3 (c+d x))+3360 a^4 A c+3360 a^4 A d x+4200 a^2 b^2 B \sin (3 (c+d x))+504 a^2 b^2 B \sin (5 (c+d x))+840 a^3 b B \sin (4 (c+d x))+10080 a^3 b B c+10080 a^3 b B d x+560 a^4 B \sin (3 (c+d x))+2800 a A b^3 \sin (3 (c+d x))+336 a A b^3 \sin (5 (c+d x))+1260 a b^3 B \sin (4 (c+d x))+140 a b^3 B \sin (6 (c+d x))+8400 a b^3 B c+8400 a b^3 B d x+315 A b^4 \sin (4 (c+d x))+35 A b^4 \sin (6 (c+d x))+2100 A b^4 c+2100 A b^4 d x+735 b^4 B \sin (3 (c+d x))+147 b^4 B \sin (5 (c+d x))+15 b^4 B \sin (7 (c+d x))}{6720 d} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.047, size = 368, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( A{a}^{4} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{\frac{{a}^{4}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{\frac{4\,A{a}^{3}b \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+4\,B{a}^{3}b \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +6\,A{a}^{2}{b}^{2} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{6\,B{a}^{2}{b}^{2}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+{\frac{4\,Aa{b}^{3}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+4\,Ba{b}^{3} \left ( 1/6\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +A{b}^{4} \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +{\frac{B{b}^{4}\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17136, size = 494, normalized size = 1.35 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.69899, size = 717, normalized size = 1.96 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 11.8799, size = 1017, normalized size = 2.78 \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)**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))

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Giac [A]  time = 1.46736, size = 423, normalized size = 1.16 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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