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

Optimal. Leaf size=243 \[ -\frac{2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d}+\frac{4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac{a^4 (56 A+49 B+44 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{27 a^4 (56 A+49 B+44 C) \sin (c+d x) \cos (c+d x)}{560 d}+\frac{1}{16} a^4 x (56 A+49 B+44 C)+\frac{(42 A-7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{210 d}+\frac{(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{42 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d} \]

[Out]

(a^4*(56*A + 49*B + 44*C)*x)/16 + (4*a^4*(56*A + 49*B + 44*C)*Sin[c + d*x])/(35*d) + (27*a^4*(56*A + 49*B + 44
*C)*Cos[c + d*x]*Sin[c + d*x])/(560*d) + (a^4*(56*A + 49*B + 44*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + ((42
*A - 7*B + 8*C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(210*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c
+ d*x])/(7*d) + ((7*B + 4*C)*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(42*a*d) - (2*a^4*(56*A + 49*B + 44*C)*Sin[c
+ d*x]^3)/(105*d)

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Rubi [A]  time = 0.454472, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3045, 2968, 3023, 2751, 2645, 2637, 2635, 8, 2633} \[ -\frac{2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d}+\frac{4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac{a^4 (56 A+49 B+44 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{27 a^4 (56 A+49 B+44 C) \sin (c+d x) \cos (c+d x)}{560 d}+\frac{1}{16} a^4 x (56 A+49 B+44 C)+\frac{(42 A-7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{210 d}+\frac{(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{42 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(56*A + 49*B + 44*C)*x)/16 + (4*a^4*(56*A + 49*B + 44*C)*Sin[c + d*x])/(35*d) + (27*a^4*(56*A + 49*B + 44
*C)*Cos[c + d*x]*Sin[c + d*x])/(560*d) + (a^4*(56*A + 49*B + 44*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + ((42
*A - 7*B + 8*C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(210*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c
+ d*x])/(7*d) + ((7*B + 4*C)*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(42*a*d) - (2*a^4*(56*A + 49*B + 44*C)*Sin[c
+ d*x]^3)/(105*d)

Rule 3045

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/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x
])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*
d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

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 2751

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

Rule 2645

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTrig[(a + b*sin[c + d*x])^n, x], x] /;
FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, 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 (c+d x) (a+a \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+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{\int \cos (c+d x) (a+a \cos (c+d x))^4 (a (7 A+2 C)+a (7 B+4 C) \cos (c+d x)) \, dx}{7 a}\\ &=\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{\int (a+a \cos (c+d x))^4 \left (a (7 A+2 C) \cos (c+d x)+a (7 B+4 C) \cos ^2(c+d x)\right ) \, dx}{7 a}\\ &=\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}+\frac{\int (a+a \cos (c+d x))^4 \left (5 a^2 (7 B+4 C)+a^2 (42 A-7 B+8 C) \cos (c+d x)\right ) \, dx}{42 a^2}\\ &=\frac{(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}+\frac{1}{70} (56 A+49 B+44 C) \int (a+a \cos (c+d x))^4 \, dx\\ &=\frac{(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}+\frac{1}{70} (56 A+49 B+44 C) \int \left (a^4+4 a^4 \cos (c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^4(c+d x)\right ) \, dx\\ &=\frac{1}{70} a^4 (56 A+49 B+44 C) x+\frac{(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}+\frac{1}{70} \left (a^4 (56 A+49 B+44 C)\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{35} \left (2 a^4 (56 A+49 B+44 C)\right ) \int \cos (c+d x) \, dx+\frac{1}{35} \left (2 a^4 (56 A+49 B+44 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{35} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{1}{70} a^4 (56 A+49 B+44 C) x+\frac{2 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac{3 a^4 (56 A+49 B+44 C) \cos (c+d x) \sin (c+d x)}{70 d}+\frac{a^4 (56 A+49 B+44 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}+\frac{1}{280} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{70} \left (3 a^4 (56 A+49 B+44 C)\right ) \int 1 \, dx-\frac{\left (2 a^4 (56 A+49 B+44 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac{2}{35} a^4 (56 A+49 B+44 C) x+\frac{4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac{27 a^4 (56 A+49 B+44 C) \cos (c+d x) \sin (c+d x)}{560 d}+\frac{a^4 (56 A+49 B+44 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}-\frac{2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d}+\frac{1}{560} \left (3 a^4 (56 A+49 B+44 C)\right ) \int 1 \, dx\\ &=\frac{1}{16} a^4 (56 A+49 B+44 C) x+\frac{4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac{27 a^4 (56 A+49 B+44 C) \cos (c+d x) \sin (c+d x)}{560 d}+\frac{a^4 (56 A+49 B+44 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}-\frac{2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d}\\ \end{align*}

Mathematica [A]  time = 0.940435, size = 204, normalized size = 0.84 \[ \frac{a^4 (105 (392 A+352 B+323 C) \sin (c+d x)+105 (128 A+127 B+124 C) \sin (2 (c+d x))+4060 A \sin (3 (c+d x))+840 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+23520 A d x+5040 B \sin (3 (c+d x))+1575 B \sin (4 (c+d x))+336 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+20580 B c+20580 B d x+5495 C \sin (3 (c+d x))+2100 C \sin (4 (c+d x))+651 C \sin (5 (c+d x))+140 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+11760 c C+18480 C d x)}{6720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(20580*B*c + 11760*c*C + 23520*A*d*x + 20580*B*d*x + 18480*C*d*x + 105*(392*A + 352*B + 323*C)*Sin[c + d*
x] + 105*(128*A + 127*B + 124*C)*Sin[2*(c + d*x)] + 4060*A*Sin[3*(c + d*x)] + 5040*B*Sin[3*(c + d*x)] + 5495*C
*Sin[3*(c + d*x)] + 840*A*Sin[4*(c + d*x)] + 1575*B*Sin[4*(c + d*x)] + 2100*C*Sin[4*(c + d*x)] + 84*A*Sin[5*(c
+ d*x)] + 336*B*Sin[5*(c + d*x)] + 651*C*Sin[5*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 140*C*Sin[6*(c + d*x)] +
15*C*Sin[7*(c + d*x)]))/(6720*d)

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Maple [B]  time = 0.033, size = 490, normalized size = 2. \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+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)

[Out]

1/d*(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)+4*
A*a^4*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+4/3*a^4*B*(2+cos(d*x+c)^2)*sin(d*x+c)+4*a^4*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)+2*A*a^4*(2+cos(d*x+c)^2)*sin(d*x+c)+6*a^4*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^4*C*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4*A*a^4*(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^4*B*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+
c)+4*a^4*C*(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*A*a^4*(8/3+cos
(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+a^4*B*(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*a^4*C*(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 [B]  time = 1.0405, size = 652, normalized size = 2.68 \begin{align*} \frac{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 a^{4} - 13440 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 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^{4} + 6720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 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^{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 a^{4} - 8960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 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^{4} + 1680 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{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 a^{4} + 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^{4} - 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^{4} - 2240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 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^{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+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

1/6720*(448*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 - 13440*(sin(d*x + c)^3 - 3*sin(d*x
+ c))*A*a^4 + 840*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 6720*(2*d*x + 2*c + sin(2*d
*x + 2*c))*A*a^4 + 1792*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^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*a^4 - 8960*(sin(d*x + c)^3 - 3*sin(d*x + c))
*B*a^4 + 1260*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 + 1680*(2*d*x + 2*c + sin(2*d*x +
2*c))*B*a^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*a^4 + 2688*(3
*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^4 - 140*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*s
in(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*C*a^4 - 2240*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 + 840*(12*d*x + 12
*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4 + 6720*A*a^4*sin(d*x + c))/d

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Fricas [A]  time = 2.15902, size = 454, normalized size = 1.87 \begin{align*} \frac{105 \,{\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} d x +{\left (240 \, C a^{4} \cos \left (d x + c\right )^{6} + 280 \,{\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 48 \,{\left (7 \, A + 28 \, B + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \,{\left (24 \, A + 41 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \,{\left (238 \, A + 252 \, B + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \,{\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \,{\left (581 \, A + 504 \, B + 454 \, C\right )} a^{4}\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+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/1680*(105*(56*A + 49*B + 44*C)*a^4*d*x + (240*C*a^4*cos(d*x + c)^6 + 280*(B + 4*C)*a^4*cos(d*x + c)^5 + 48*(
7*A + 28*B + 48*C)*a^4*cos(d*x + c)^4 + 70*(24*A + 41*B + 44*C)*a^4*cos(d*x + c)^3 + 16*(238*A + 252*B + 227*C
)*a^4*cos(d*x + c)^2 + 105*(56*A + 49*B + 44*C)*a^4*cos(d*x + c) + 16*(581*A + 504*B + 454*C)*a^4)*sin(d*x + c
))/d

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Sympy [A]  time = 11.4942, size = 1258, normalized size = 5.18 \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+a*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)

[Out]

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

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Giac [A]  time = 1.25046, size = 309, normalized size = 1.27 \begin{align*} \frac{C a^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{1}{16} \,{\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, C a^{4}\right )} x + \frac{{\left (B a^{4} + 4 \, C a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{{\left (4 \, A a^{4} + 16 \, B a^{4} + 31 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (8 \, A a^{4} + 15 \, B a^{4} + 20 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (116 \, A a^{4} + 144 \, B a^{4} + 157 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{{\left (128 \, A a^{4} + 127 \, B a^{4} + 124 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (392 \, A a^{4} + 352 \, B a^{4} + 323 \, C a^{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+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/448*C*a^4*sin(7*d*x + 7*c)/d + 1/16*(56*A*a^4 + 49*B*a^4 + 44*C*a^4)*x + 1/192*(B*a^4 + 4*C*a^4)*sin(6*d*x +
6*c)/d + 1/320*(4*A*a^4 + 16*B*a^4 + 31*C*a^4)*sin(5*d*x + 5*c)/d + 1/64*(8*A*a^4 + 15*B*a^4 + 20*C*a^4)*sin(
4*d*x + 4*c)/d + 1/192*(116*A*a^4 + 144*B*a^4 + 157*C*a^4)*sin(3*d*x + 3*c)/d + 1/64*(128*A*a^4 + 127*B*a^4 +
124*C*a^4)*sin(2*d*x + 2*c)/d + 1/64*(392*A*a^4 + 352*B*a^4 + 323*C*a^4)*sin(d*x + c)/d