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

Optimal. Leaf size=185 \[ -\frac{2 a^4 (8 A+7 B) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (8 A+7 B) \sin (c+d x)}{5 d}+\frac{a^4 (8 A+7 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (8 A+7 B) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (8 A+7 B)+\frac{(6 A-B) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.303942, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2968, 3023, 2751, 2645, 2637, 2635, 8, 2633} \[ -\frac{2 a^4 (8 A+7 B) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (8 A+7 B) \sin (c+d x)}{5 d}+\frac{a^4 (8 A+7 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (8 A+7 B) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (8 A+7 B)+\frac{(6 A-B) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.490434, size = 134, normalized size = 0.72 \[ \frac{a^4 (120 (49 A+44 B) \sin (c+d x)+15 (128 A+127 B) \sin (2 (c+d x))+580 A \sin (3 (c+d x))+120 A \sin (4 (c+d x))+12 A \sin (5 (c+d x))+3360 A d x+720 B \sin (3 (c+d x))+225 B \sin (4 (c+d x))+48 B \sin (5 (c+d x))+5 B \sin (6 (c+d x))+2940 B c+2940 B d x)}{960 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.059, size = 306, normalized size = 1.7 \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 ) }+{a}^{4}B \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 ) +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 ) +{\frac{4\,{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 ) }+2\,A{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +6\,{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 ) +4\,A{a}^{4} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{\frac{4\,{a}^{4}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+A{a}^{4}\sin \left ( dx+c \right ) +{a}^{4}B \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*(a+cos(d*x+c)*a)^4*(A+B*cos(d*x+c)),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)+a^4*B*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*c
os(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*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)+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)+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)+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))

________________________________________________________________________________________

Maxima [A]  time = 1.01919, size = 401, normalized size = 2.17 \begin{align*} \frac{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 )} A a^{4} - 1920 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A 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 )} A a^{4} + 960 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A 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 )} B 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 )} B a^{4} - 1280 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B 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 )} B a^{4} + 240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 960 \, A a^{4} \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))^4*(A+B*cos(d*x+c)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.46569, size = 329, normalized size = 1.78 \begin{align*} \frac{105 \,{\left (8 \, A + 7 \, B\right )} a^{4} d x +{\left (40 \, B a^{4} \cos \left (d x + c\right )^{5} + 48 \,{\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 10 \,{\left (24 \, A + 41 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 32 \,{\left (17 \, A + 18 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \,{\left (8 \, A + 7 \, B\right )} a^{4} \cos \left (d x + c\right ) + 16 \,{\left (83 \, A + 72 \, B\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(cos(d*x+c)*(a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/240*(105*(8*A + 7*B)*a^4*d*x + (40*B*a^4*cos(d*x + c)^5 + 48*(A + 4*B)*a^4*cos(d*x + c)^4 + 10*(24*A + 41*B)
*a^4*cos(d*x + c)^3 + 32*(17*A + 18*B)*a^4*cos(d*x + c)^2 + 105*(8*A + 7*B)*a^4*cos(d*x + c) + 16*(83*A + 72*B
)*a^4)*sin(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 6.80233, size = 765, normalized size = 4.14 \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)),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), Ne(d, 0)), (x*(A + B*cos(c))*(a*cos(c) + a)**4*cos(c), True))

________________________________________________________________________________________

Giac [A]  time = 1.23271, size = 224, normalized size = 1.21 \begin{align*} \frac{B a^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{7}{16} \,{\left (8 \, A a^{4} + 7 \, B a^{4}\right )} x + \frac{{\left (A a^{4} + 4 \, B a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{{\left (8 \, A a^{4} + 15 \, B a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (29 \, A a^{4} + 36 \, B a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (128 \, A a^{4} + 127 \, B a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (49 \, A a^{4} + 44 \, B 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(cos(d*x+c)*(a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="giac")

[Out]

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