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

Optimal. Leaf size=200 \[ -\frac{2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac{a^4 (10 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+8 B+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (10 A+8 B+7 C)+\frac{(6 B-C) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d} \]

[Out]

(7*a^4*(10*A + 8*B + 7*C)*x)/16 + (4*a^4*(10*A + 8*B + 7*C)*Sin[c + d*x])/(5*d) + (27*a^4*(10*A + 8*B + 7*C)*C
os[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(10*A + 8*B + 7*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((6*B - C)*(a
+ a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d) - (2*a^4*(10*A + 8
*B + 7*C)*Sin[c + d*x]^3)/(15*d)

________________________________________________________________________________________

Rubi [A]  time = 0.279667, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {3023, 2751, 2645, 2637, 2635, 8, 2633} \[ -\frac{2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac{a^4 (10 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+8 B+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (10 A+8 B+7 C)+\frac{(6 B-C) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*a^4*(10*A + 8*B + 7*C)*x)/16 + (4*a^4*(10*A + 8*B + 7*C)*Sin[c + d*x])/(5*d) + (27*a^4*(10*A + 8*B + 7*C)*C
os[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(10*A + 8*B + 7*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((6*B - C)*(a
+ a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d) - (2*a^4*(10*A + 8
*B + 7*C)*Sin[c + d*x]^3)/(15*d)

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 (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac{\int (a+a \cos (c+d x))^4 (a (6 A+5 C)+a (6 B-C) \cos (c+d x)) \, dx}{6 a}\\ &=\frac{(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac{C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac{1}{10} (10 A+8 B+7 C) \int (a+a \cos (c+d x))^4 \, dx\\ &=\frac{(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac{C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac{1}{10} (10 A+8 B+7 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}{10} a^4 (10 A+8 B+7 C) x+\frac{(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac{C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac{1}{10} \left (a^4 (10 A+8 B+7 C)\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{5} \left (2 a^4 (10 A+8 B+7 C)\right ) \int \cos (c+d x) \, dx+\frac{1}{5} \left (2 a^4 (10 A+8 B+7 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{5} \left (3 a^4 (10 A+8 B+7 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{1}{10} a^4 (10 A+8 B+7 C) x+\frac{2 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac{3 a^4 (10 A+8 B+7 C) \cos (c+d x) \sin (c+d x)}{10 d}+\frac{a^4 (10 A+8 B+7 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac{C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac{1}{40} \left (3 a^4 (10 A+8 B+7 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{10} \left (3 a^4 (10 A+8 B+7 C)\right ) \int 1 \, dx-\frac{\left (2 a^4 (10 A+8 B+7 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac{2}{5} a^4 (10 A+8 B+7 C) x+\frac{4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac{27 a^4 (10 A+8 B+7 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac{a^4 (10 A+8 B+7 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac{C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}-\frac{2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d}+\frac{1}{80} \left (3 a^4 (10 A+8 B+7 C)\right ) \int 1 \, dx\\ &=\frac{7}{16} a^4 (10 A+8 B+7 C) x+\frac{4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac{27 a^4 (10 A+8 B+7 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac{a^4 (10 A+8 B+7 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac{C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}-\frac{2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d}\\ \end{align*}

Mathematica [A]  time = 0.534772, size = 163, normalized size = 0.82 \[ \frac{a^4 (120 (56 A+49 B+44 C) \sin (c+d x)+15 (112 A+128 B+127 C) \sin (2 (c+d x))+320 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+4200 A d x+580 B \sin (3 (c+d x))+120 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+3360 B d x+720 C \sin (3 (c+d x))+225 C \sin (4 (c+d x))+48 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+2940 C d x)}{960 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(4200*A*d*x + 3360*B*d*x + 2940*C*d*x + 120*(56*A + 49*B + 44*C)*Sin[c + d*x] + 15*(112*A + 128*B + 127*C
)*Sin[2*(c + d*x)] + 320*A*Sin[3*(c + d*x)] + 580*B*Sin[3*(c + d*x)] + 720*C*Sin[3*(c + d*x)] + 30*A*Sin[4*(c
+ d*x)] + 120*B*Sin[4*(c + d*x)] + 225*C*Sin[4*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 48*C*Sin[5*(c + d*x)] + 5*
C*Sin[6*(c + d*x)]))/(960*d)

________________________________________________________________________________________

Maple [B]  time = 0.028, size = 416, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ({a}^{4}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 ) +{\frac{{a}^{4}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 ) }+{\frac{4\,{a}^{4}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}^{4} \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 ) +4\,{a}^{4}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}^{4}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 ) +{\frac{4\,A{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,{a}^{4}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +{\frac{4\,{a}^{4}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+6\,A{a}^{4} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +4\,{a}^{4}B \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{a}^{4}C \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +4\,A{a}^{4}\sin \left ( dx+c \right ) +{a}^{4}B\sin \left ( dx+c \right ) +A{a}^{4} \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(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^4*B*(8/3+cos
(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4/5*a^4*C*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+A*a^4*(1/4*(co
s(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4*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*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)+4/3*A*a^4*(2+cos(d*x+c)^2)*si
n(d*x+c)+2*a^4*B*(2+cos(d*x+c)^2)*sin(d*x+c)+4/3*a^4*C*(2+cos(d*x+c)^2)*sin(d*x+c)+6*A*a^4*(1/2*cos(d*x+c)*sin
(d*x+c)+1/2*d*x+1/2*c)+4*a^4*B*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+a^4*C*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*
d*x+1/2*c)+4*A*a^4*sin(d*x+c)+a^4*B*sin(d*x+c)+A*a^4*(d*x+c))

________________________________________________________________________________________

Maxima [B]  time = 1.04867, size = 540, normalized size = 2.7 \begin{align*} -\frac{1280 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 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^{4} - 1440 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 960 \,{\left (d x + c\right )} A a^{4} - 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^{4} + 1920 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 120 \,{\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} - 960 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 256 \,{\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} + 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^{4} + 1280 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 180 \,{\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} - 240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 3840 \, A a^{4} \sin \left (d x + c\right ) - 960 \, B a^{4} \sin \left (d x + c\right )}{960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/960*(1280*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 - 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*
c))*A*a^4 - 1440*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 - 960*(d*x + c)*A*a^4 - 64*(3*sin(d*x + c)^5 - 10*sin(
d*x + c)^3 + 15*sin(d*x + c))*B*a^4 + 1920*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^4 - 120*(12*d*x + 12*c + sin(
4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 - 960*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4 - 256*(3*sin(d*x + c)^5
- 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^4 + 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^4 + 1280*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 - 180*(12*d*x + 12*c + sin(4*d*x + 4
*c) + 8*sin(2*d*x + 2*c))*C*a^4 - 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^4 - 3840*A*a^4*sin(d*x + c) - 960*B
*a^4*sin(d*x + c))/d

________________________________________________________________________________________

Fricas [A]  time = 1.92585, size = 378, normalized size = 1.89 \begin{align*} \frac{105 \,{\left (10 \, A + 8 \, B + 7 \, C\right )} a^{4} d x +{\left (40 \, C a^{4} \cos \left (d x + c\right )^{5} + 48 \,{\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 10 \,{\left (6 \, A + 24 \, B + 41 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 32 \,{\left (10 \, A + 17 \, B + 18 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \,{\left (54 \, A + 56 \, B + 49 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \,{\left (100 \, A + 83 \, B + 72 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(105*(10*A + 8*B + 7*C)*a^4*d*x + (40*C*a^4*cos(d*x + c)^5 + 48*(B + 4*C)*a^4*cos(d*x + c)^4 + 10*(6*A +
24*B + 41*C)*a^4*cos(d*x + c)^3 + 32*(10*A + 17*B + 18*C)*a^4*cos(d*x + c)^2 + 15*(54*A + 56*B + 49*C)*a^4*co
s(d*x + c) + 16*(100*A + 83*B + 72*C)*a^4)*sin(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 6.61921, size = 1005, normalized size = 5.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)**4/8 + 3*A*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*A*a**4*x*sin(c + d*
x)**2 + 3*A*a**4*x*cos(c + d*x)**4/8 + 3*A*a**4*x*cos(c + d*x)**2 + A*a**4*x + 3*A*a**4*sin(c + d*x)**3*cos(c
+ d*x)/(8*d) + 8*A*a**4*sin(c + d*x)**3/(3*d) + 5*A*a**4*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 4*A*a**4*sin(c +
d*x)*cos(c + d*x)**2/d + 3*A*a**4*sin(c + d*x)*cos(c + d*x)/d + 4*A*a**4*sin(c + d*x)/d + 3*B*a**4*x*sin(c +
d*x)**4/2 + 3*B*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2 + 2*B*a**4*x*sin(c + d*x)**2 + 3*B*a**4*x*cos(c + d*x)*
*4/2 + 2*B*a**4*x*cos(c + d*x)**2 + 8*B*a**4*sin(c + d*x)**5/(15*d) + 4*B*a**4*sin(c + d*x)**3*cos(c + d*x)**2
/(3*d) + 3*B*a**4*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 4*B*a**4*sin(c + d*x)**3/d + B*a**4*sin(c + d*x)*cos(c
+ d*x)**4/d + 5*B*a**4*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 6*B*a**4*sin(c + d*x)*cos(c + d*x)**2/d + 2*B*a**4
*sin(c + d*x)*cos(c + d*x)/d + B*a**4*sin(c + d*x)/d + 5*C*a**4*x*sin(c + d*x)**6/16 + 15*C*a**4*x*sin(c + d*x
)**4*cos(c + d*x)**2/16 + 9*C*a**4*x*sin(c + d*x)**4/4 + 15*C*a**4*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 9*C*
a**4*x*sin(c + d*x)**2*cos(c + d*x)**2/2 + C*a**4*x*sin(c + d*x)**2/2 + 5*C*a**4*x*cos(c + d*x)**6/16 + 9*C*a*
*4*x*cos(c + d*x)**4/4 + C*a**4*x*cos(c + d*x)**2/2 + 5*C*a**4*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 32*C*a**4
*sin(c + d*x)**5/(15*d) + 5*C*a**4*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 16*C*a**4*sin(c + d*x)**3*cos(c + d
*x)**2/(3*d) + 9*C*a**4*sin(c + d*x)**3*cos(c + d*x)/(4*d) + 8*C*a**4*sin(c + d*x)**3/(3*d) + 11*C*a**4*sin(c
+ d*x)*cos(c + d*x)**5/(16*d) + 4*C*a**4*sin(c + d*x)*cos(c + d*x)**4/d + 15*C*a**4*sin(c + d*x)*cos(c + d*x)*
*3/(4*d) + 4*C*a**4*sin(c + d*x)*cos(c + d*x)**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), True))

________________________________________________________________________________________

Giac [A]  time = 1.29142, size = 265, normalized size = 1.32 \begin{align*} \frac{C a^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{7}{16} \,{\left (10 \, A a^{4} + 8 \, B a^{4} + 7 \, C a^{4}\right )} x + \frac{{\left (B a^{4} + 4 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{{\left (2 \, A a^{4} + 8 \, B a^{4} + 15 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (16 \, A a^{4} + 29 \, B a^{4} + 36 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (112 \, A a^{4} + 128 \, B a^{4} + 127 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, C a^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*C*a^4*sin(6*d*x + 6*c)/d + 7/16*(10*A*a^4 + 8*B*a^4 + 7*C*a^4)*x + 1/80*(B*a^4 + 4*C*a^4)*sin(5*d*x + 5*
c)/d + 1/64*(2*A*a^4 + 8*B*a^4 + 15*C*a^4)*sin(4*d*x + 4*c)/d + 1/48*(16*A*a^4 + 29*B*a^4 + 36*C*a^4)*sin(3*d*
x + 3*c)/d + 1/64*(112*A*a^4 + 128*B*a^4 + 127*C*a^4)*sin(2*d*x + 2*c)/d + 1/8*(56*A*a^4 + 49*B*a^4 + 44*C*a^4
)*sin(d*x + c)/d