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

Optimal. Leaf size=219 \[ -\frac{8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d}+\frac{16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac{a^4 (14 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{70 d}+\frac{27 a^4 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{140 d}+\frac{1}{4} a^4 x (14 A+11 C)+\frac{(21 A+4 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{105 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{21 a d} \]

[Out]

(a^4*(14*A + 11*C)*x)/4 + (16*a^4*(14*A + 11*C)*Sin[c + d*x])/(35*d) + (27*a^4*(14*A + 11*C)*Cos[c + d*x]*Sin[
c + d*x])/(140*d) + (a^4*(14*A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(70*d) + ((21*A + 4*C)*(a + a*Cos[c + d*x]
)^4*Sin[c + d*x])/(105*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c +
d*x])^5*Sin[c + d*x])/(21*a*d) - (8*a^4*(14*A + 11*C)*Sin[c + d*x]^3)/(105*d)

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Rubi [A]  time = 0.412092, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {3046, 2968, 3023, 2751, 2645, 2637, 2635, 8, 2633} \[ -\frac{8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d}+\frac{16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac{a^4 (14 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{70 d}+\frac{27 a^4 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{140 d}+\frac{1}{4} a^4 x (14 A+11 C)+\frac{(21 A+4 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{105 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{21 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(14*A + 11*C)*x)/4 + (16*a^4*(14*A + 11*C)*Sin[c + d*x])/(35*d) + (27*a^4*(14*A + 11*C)*Cos[c + d*x]*Sin[
c + d*x])/(140*d) + (a^4*(14*A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(70*d) + ((21*A + 4*C)*(a + a*Cos[c + d*x]
)^4*Sin[c + d*x])/(105*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c +
d*x])^5*Sin[c + d*x])/(21*a*d) - (8*a^4*(14*A + 11*C)*Sin[c + d*x]^3)/(105*d)

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b
, c, d, e, f, A, 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+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)+4 a 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)+4 a 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{2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}+\frac{\int (a+a \cos (c+d x))^4 \left (20 a^2 C+2 a^2 (21 A+4 C) \cos (c+d x)\right ) \, dx}{42 a^2}\\ &=\frac{(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}+\frac{1}{35} (2 (14 A+11 C)) \int (a+a \cos (c+d x))^4 \, dx\\ &=\frac{(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}+\frac{1}{35} (2 (14 A+11 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{2}{35} a^4 (14 A+11 C) x+\frac{(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}+\frac{1}{35} \left (2 a^4 (14 A+11 C)\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{35} \left (8 a^4 (14 A+11 C)\right ) \int \cos (c+d x) \, dx+\frac{1}{35} \left (8 a^4 (14 A+11 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{35} \left (12 a^4 (14 A+11 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{2}{35} a^4 (14 A+11 C) x+\frac{8 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac{6 a^4 (14 A+11 C) \cos (c+d x) \sin (c+d x)}{35 d}+\frac{a^4 (14 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{70 d}+\frac{(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}+\frac{1}{70} \left (3 a^4 (14 A+11 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{35} \left (6 a^4 (14 A+11 C)\right ) \int 1 \, dx-\frac{\left (8 a^4 (14 A+11 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac{8}{35} a^4 (14 A+11 C) x+\frac{16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac{27 a^4 (14 A+11 C) \cos (c+d x) \sin (c+d x)}{140 d}+\frac{a^4 (14 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{70 d}+\frac{(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}-\frac{8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d}+\frac{1}{140} \left (3 a^4 (14 A+11 C)\right ) \int 1 \, dx\\ &=\frac{1}{4} a^4 (14 A+11 C) x+\frac{16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac{27 a^4 (14 A+11 C) \cos (c+d x) \sin (c+d x)}{140 d}+\frac{a^4 (14 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{70 d}+\frac{(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac{C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac{2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}-\frac{8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d}\\ \end{align*}

Mathematica [A]  time = 0.574277, size = 145, normalized size = 0.66 \[ \frac{a^4 (105 (392 A+323 C) \sin (c+d x)+420 (32 A+31 C) \sin (2 (c+d x))+4060 A \sin (3 (c+d x))+840 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+23520 A d x+5495 C \sin (3 (c+d x))+2100 C \sin (4 (c+d x))+651 C \sin (5 (c+d x))+140 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+11760 c C+18480 C d x)}{6720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(11760*c*C + 23520*A*d*x + 18480*C*d*x + 105*(392*A + 323*C)*Sin[c + d*x] + 420*(32*A + 31*C)*Sin[2*(c +
d*x)] + 4060*A*Sin[3*(c + d*x)] + 5495*C*Sin[3*(c + d*x)] + 840*A*Sin[4*(c + d*x)] + 2100*C*Sin[4*(c + d*x)] +
84*A*Sin[5*(c + d*x)] + 651*C*Sin[5*(c + d*x)] + 140*C*Sin[6*(c + d*x)] + 15*C*Sin[7*(c + d*x)]))/(6720*d)

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Maple [A]  time = 0.032, size = 322, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({\frac{A{a}^{4}\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{{a}^{4}C\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 ) }+4\,A{a}^{4} \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 ) +4\,{a}^{4}C \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 ) +2\,A{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +{\frac{6\,{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 ) }+4\,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}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}^{4}\sin \left ( dx+c \right ) +{\frac{{a}^{4}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \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))^4*(A+C*cos(d*x+c)^2),x)

[Out]

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

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Maxima [A]  time = 1.02724, size = 431, normalized size = 1.97 \begin{align*} \frac{112 \,{\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} - 3360 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 210 \,{\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} + 1680 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 48 \,{\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} + 672 \,{\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} - 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 )} C a^{4} - 560 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 210 \,{\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} + 1680 \, A a^{4} \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+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

1/1680*(112*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 - 3360*(sin(d*x + c)^3 - 3*sin(d*x
+ c))*A*a^4 + 210*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 1680*(2*d*x + 2*c + sin(2*d*
x + 2*c))*A*a^4 - 48*(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 + 672*
(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*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))*C*a^4 - 560*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 + 210*(12*d*x + 12
*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4 + 1680*A*a^4*sin(d*x + c))/d

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Fricas [A]  time = 1.46895, size = 377, normalized size = 1.72 \begin{align*} \frac{105 \,{\left (14 \, A + 11 \, C\right )} a^{4} d x +{\left (60 \, C a^{4} \cos \left (d x + c\right )^{6} + 280 \, C a^{4} \cos \left (d x + c\right )^{5} + 12 \,{\left (7 \, A + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \,{\left (6 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 4 \,{\left (238 \, A + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \,{\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right ) + 4 \,{\left (581 \, A + 454 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{420 \, 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+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/420*(105*(14*A + 11*C)*a^4*d*x + (60*C*a^4*cos(d*x + c)^6 + 280*C*a^4*cos(d*x + c)^5 + 12*(7*A + 48*C)*a^4*c
os(d*x + c)^4 + 70*(6*A + 11*C)*a^4*cos(d*x + c)^3 + 4*(238*A + 227*C)*a^4*cos(d*x + c)^2 + 105*(14*A + 11*C)*
a^4*cos(d*x + c) + 4*(581*A + 454*C)*a^4)*sin(d*x + c))/d

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Sympy [A]  time = 11.4555, size = 799, normalized size = 3.65 \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+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*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 + C*cos(c)**2)*(a*cos(c) +
a)**4*cos(c), True))

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Giac [A]  time = 1.23046, size = 250, normalized size = 1.14 \begin{align*} \frac{C a^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{C a^{4} \sin \left (6 \, d x + 6 \, c\right )}{48 \, d} + \frac{1}{4} \,{\left (14 \, A a^{4} + 11 \, C a^{4}\right )} x + \frac{{\left (4 \, A a^{4} + 31 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (2 \, A a^{4} + 5 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac{{\left (116 \, A a^{4} + 157 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{{\left (32 \, A a^{4} + 31 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{16 \, d} + \frac{{\left (392 \, A 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+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/448*C*a^4*sin(7*d*x + 7*c)/d + 1/48*C*a^4*sin(6*d*x + 6*c)/d + 1/4*(14*A*a^4 + 11*C*a^4)*x + 1/320*(4*A*a^4
+ 31*C*a^4)*sin(5*d*x + 5*c)/d + 1/16*(2*A*a^4 + 5*C*a^4)*sin(4*d*x + 4*c)/d + 1/192*(116*A*a^4 + 157*C*a^4)*s
in(3*d*x + 3*c)/d + 1/16*(32*A*a^4 + 31*C*a^4)*sin(2*d*x + 2*c)/d + 1/64*(392*A*a^4 + 323*C*a^4)*sin(d*x + c)/
d