3.319 \(\int \cos (c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=207 \[ -\frac{a^3 (30 A+26 B+23 C) \sin ^3(c+d x)}{120 d}+\frac{a^3 (30 A+26 B+23 C) \sin (c+d x)}{10 d}+\frac{3 a^3 (30 A+26 B+23 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{1}{16} a^3 x (30 A+26 B+23 C)+\frac{(30 A-6 B+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{120 d}+\frac{(2 B+C) \sin (c+d x) (a \cos (c+d x)+a)^4}{10 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d} \]

[Out]

(a^3*(30*A + 26*B + 23*C)*x)/16 + (a^3*(30*A + 26*B + 23*C)*Sin[c + d*x])/(10*d) + (3*a^3*(30*A + 26*B + 23*C)
*Cos[c + d*x]*Sin[c + d*x])/(80*d) + ((30*A - 6*B + 7*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(120*d) + (C*Cos
[c + d*x]^2*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(6*d) + ((2*B + C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(10*a
*d) - (a^3*(30*A + 26*B + 23*C)*Sin[c + d*x]^3)/(120*d)

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Rubi [A]  time = 0.396215, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -\frac{a^3 (30 A+26 B+23 C) \sin ^3(c+d x)}{120 d}+\frac{a^3 (30 A+26 B+23 C) \sin (c+d x)}{10 d}+\frac{3 a^3 (30 A+26 B+23 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{1}{16} a^3 x (30 A+26 B+23 C)+\frac{(30 A-6 B+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{120 d}+\frac{(2 B+C) \sin (c+d x) (a \cos (c+d x)+a)^4}{10 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(a^3*(30*A + 26*B + 23*C)*x)/16 + (a^3*(30*A + 26*B + 23*C)*Sin[c + d*x])/(10*d) + (3*a^3*(30*A + 26*B + 23*C)
*Cos[c + d*x]*Sin[c + d*x])/(80*d) + ((30*A - 6*B + 7*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(120*d) + (C*Cos
[c + d*x]^2*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(6*d) + ((2*B + C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(10*a
*d) - (a^3*(30*A + 26*B + 23*C)*Sin[c + d*x]^3)/(120*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))^3 \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))^3 \sin (c+d x)}{6 d}+\frac{\int \cos (c+d x) (a+a \cos (c+d x))^3 (2 a (3 A+C)+3 a (2 B+C) \cos (c+d x)) \, dx}{6 a}\\ &=\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac{\int (a+a \cos (c+d x))^3 \left (2 a (3 A+C) \cos (c+d x)+3 a (2 B+C) \cos ^2(c+d x)\right ) \, dx}{6 a}\\ &=\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(2 B+C) (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}+\frac{\int (a+a \cos (c+d x))^3 \left (12 a^2 (2 B+C)+a^2 (30 A-6 B+7 C) \cos (c+d x)\right ) \, dx}{30 a^2}\\ &=\frac{(30 A-6 B+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(2 B+C) (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}+\frac{1}{40} (30 A+26 B+23 C) \int (a+a \cos (c+d x))^3 \, dx\\ &=\frac{(30 A-6 B+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(2 B+C) (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}+\frac{1}{40} (30 A+26 B+23 C) \int \left (a^3+3 a^3 \cos (c+d x)+3 a^3 \cos ^2(c+d x)+a^3 \cos ^3(c+d x)\right ) \, dx\\ &=\frac{1}{40} a^3 (30 A+26 B+23 C) x+\frac{(30 A-6 B+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(2 B+C) (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}+\frac{1}{40} \left (a^3 (30 A+26 B+23 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{40} \left (3 a^3 (30 A+26 B+23 C)\right ) \int \cos (c+d x) \, dx+\frac{1}{40} \left (3 a^3 (30 A+26 B+23 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{1}{40} a^3 (30 A+26 B+23 C) x+\frac{3 a^3 (30 A+26 B+23 C) \sin (c+d x)}{40 d}+\frac{3 a^3 (30 A+26 B+23 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac{(30 A-6 B+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(2 B+C) (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}+\frac{1}{80} \left (3 a^3 (30 A+26 B+23 C)\right ) \int 1 \, dx-\frac{\left (a^3 (30 A+26 B+23 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{40 d}\\ &=\frac{1}{16} a^3 (30 A+26 B+23 C) x+\frac{a^3 (30 A+26 B+23 C) \sin (c+d x)}{10 d}+\frac{3 a^3 (30 A+26 B+23 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac{(30 A-6 B+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(2 B+C) (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}-\frac{a^3 (30 A+26 B+23 C) \sin ^3(c+d x)}{120 d}\\ \end{align*}

Mathematica [A]  time = 0.629833, size = 171, normalized size = 0.83 \[ \frac{a^3 (120 (26 A+23 B+21 C) \sin (c+d x)+15 (64 A+64 B+63 C) \sin (2 (c+d x))+240 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+1800 A d x+340 B \sin (3 (c+d x))+90 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+1560 B c+1560 B d x+380 C \sin (3 (c+d x))+135 C \sin (4 (c+d x))+36 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+900 c C+1380 C d x)}{960 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(a^3*(1560*B*c + 900*c*C + 1800*A*d*x + 1560*B*d*x + 1380*C*d*x + 120*(26*A + 23*B + 21*C)*Sin[c + d*x] + 15*(
64*A + 64*B + 63*C)*Sin[2*(c + d*x)] + 240*A*Sin[3*(c + d*x)] + 340*B*Sin[3*(c + d*x)] + 380*C*Sin[3*(c + d*x)
] + 30*A*Sin[4*(c + d*x)] + 90*B*Sin[4*(c + d*x)] + 135*C*Sin[4*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 36*C*Sin[
5*(c + d*x)] + 5*C*Sin[6*(c + d*x)]))/(960*d)

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Maple [A]  time = 0.03, size = 364, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ( A{a}^{3}\sin \left ( dx+c \right ) +{a}^{3}B \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{\frac{{a}^{3}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+3\,A{a}^{3} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{a}^{3}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,{a}^{3}C \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 ) +A{a}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,{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 ) +{\frac{3\,{a}^{3}C\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 ) }+A{a}^{3} \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{{a}^{3}B\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 ) }+{a}^{3}C \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 ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)

[Out]

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

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Maxima [A]  time = 1.02349, size = 478, normalized size = 2.31 \begin{align*} -\frac{960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 30 \,{\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^{3} - 720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 64 \,{\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^{3} + 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 90 \,{\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} - 240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 192 \,{\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^{3} + 5 \,{\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^{3} + 320 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 90 \,{\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} - 960 \, A a^{3} \sin \left (d x + c\right )}{960 \, 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))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/960*(960*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 - 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c
))*A*a^3 - 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^3 - 64*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x
+ c))*B*a^3 + 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3 - 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x
+ 2*c))*B*a^3 - 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^3 - 192*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*s
in(d*x + c))*C*a^3 + 5*(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^3
+ 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 - 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*
a^3 - 960*A*a^3*sin(d*x + c))/d

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Fricas [A]  time = 1.96045, size = 378, normalized size = 1.83 \begin{align*} \frac{15 \,{\left (30 \, A + 26 \, B + 23 \, C\right )} a^{3} d x +{\left (40 \, C a^{3} \cos \left (d x + c\right )^{5} + 48 \,{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 10 \,{\left (6 \, A + 18 \, B + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \,{\left (15 \, A + 19 \, B + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \,{\left (30 \, A + 26 \, B + 23 \, C\right )} a^{3} \cos \left (d x + c\right ) + 16 \,{\left (45 \, A + 38 \, B + 34 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{240 \, 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))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/240*(15*(30*A + 26*B + 23*C)*a^3*d*x + (40*C*a^3*cos(d*x + c)^5 + 48*(B + 3*C)*a^3*cos(d*x + c)^4 + 10*(6*A
+ 18*B + 23*C)*a^3*cos(d*x + c)^3 + 16*(15*A + 19*B + 17*C)*a^3*cos(d*x + c)^2 + 15*(30*A + 26*B + 23*C)*a^3*c
os(d*x + c) + 16*(45*A + 38*B + 34*C)*a^3)*sin(d*x + c))/d

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Sympy [A]  time = 6.3139, size = 932, normalized size = 4.5 \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))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)

[Out]

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

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Giac [A]  time = 1.18807, size = 265, normalized size = 1.28 \begin{align*} \frac{C a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{1}{16} \,{\left (30 \, A a^{3} + 26 \, B a^{3} + 23 \, C a^{3}\right )} x + \frac{{\left (B a^{3} + 3 \, C a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{{\left (2 \, A a^{3} + 6 \, B a^{3} + 9 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (12 \, A a^{3} + 17 \, B a^{3} + 19 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (64 \, A a^{3} + 64 \, B a^{3} + 63 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (26 \, A a^{3} + 23 \, B a^{3} + 21 \, C a^{3}\right )} \sin \left (d x + c\right )}{8 \, 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))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/192*C*a^3*sin(6*d*x + 6*c)/d + 1/16*(30*A*a^3 + 26*B*a^3 + 23*C*a^3)*x + 1/80*(B*a^3 + 3*C*a^3)*sin(5*d*x +
5*c)/d + 1/64*(2*A*a^3 + 6*B*a^3 + 9*C*a^3)*sin(4*d*x + 4*c)/d + 1/48*(12*A*a^3 + 17*B*a^3 + 19*C*a^3)*sin(3*d
*x + 3*c)/d + 1/64*(64*A*a^3 + 64*B*a^3 + 63*C*a^3)*sin(2*d*x + 2*c)/d + 1/8*(26*A*a^3 + 23*B*a^3 + 21*C*a^3)*
sin(d*x + c)/d