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

Optimal. Leaf size=325 \[ \frac{\left (224 a^2 A b^3+24 a^4 A b+121 a^3 b^2 B-4 a^5 B+128 a b^4 B+32 A b^5\right ) \sin (c+d x)}{60 b d}+\frac{\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}+\frac{\left (24 a^2 A b-4 a^3 B+53 a b^2 B+32 A b^3\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}+\frac{\left (48 a^3 A b+178 a^2 b^2 B-8 a^4 B+232 a A b^3+75 b^4 B\right ) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left (32 a^3 A b+36 a^2 b^2 B+8 a^4 B+24 a A b^3+5 b^4 B\right )+\frac{(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d} \]

[Out]

((32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*x)/16 + ((24*a^4*A*b + 224*a^2*A*b^3 + 32*A*b^5
- 4*a^5*B + 121*a^3*b^2*B + 128*a*b^4*B)*Sin[c + d*x])/(60*b*d) + ((48*a^3*A*b + 232*a*A*b^3 - 8*a^4*B + 178*a
^2*b^2*B + 75*b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(240*d) + ((24*a^2*A*b + 32*A*b^3 - 4*a^3*B + 53*a*b^2*B)*(a +
b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) + ((24*a*A*b - 4*a^2*B + 25*b^2*B)*(a + b*Cos[c + d*x])^3*Sin[c + d
*x])/(120*b*d) + ((6*A*b - a*B)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (B*(a + b*Cos[c + d*x])^5*Sin[
c + d*x])/(6*b*d)

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Rubi [A]  time = 0.509253, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2968, 3023, 2753, 2734} \[ \frac{\left (224 a^2 A b^3+24 a^4 A b+121 a^3 b^2 B-4 a^5 B+128 a b^4 B+32 A b^5\right ) \sin (c+d x)}{60 b d}+\frac{\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}+\frac{\left (24 a^2 A b-4 a^3 B+53 a b^2 B+32 A b^3\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}+\frac{\left (48 a^3 A b+178 a^2 b^2 B-8 a^4 B+232 a A b^3+75 b^4 B\right ) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left (32 a^3 A b+36 a^2 b^2 B+8 a^4 B+24 a A b^3+5 b^4 B\right )+\frac{(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*x)/16 + ((24*a^4*A*b + 224*a^2*A*b^3 + 32*A*b^5
- 4*a^5*B + 121*a^3*b^2*B + 128*a*b^4*B)*Sin[c + d*x])/(60*b*d) + ((48*a^3*A*b + 232*a*A*b^3 - 8*a^4*B + 178*a
^2*b^2*B + 75*b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(240*d) + ((24*a^2*A*b + 32*A*b^3 - 4*a^3*B + 53*a*b^2*B)*(a +
b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) + ((24*a*A*b - 4*a^2*B + 25*b^2*B)*(a + b*Cos[c + d*x])^3*Sin[c + d
*x])/(120*b*d) + ((6*A*b - a*B)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (B*(a + b*Cos[c + d*x])^5*Sin[
c + d*x])/(6*b*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 2753

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[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2734

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

Rubi steps

\begin{align*} \int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx &=\int (a+b \cos (c+d x))^4 \left (A \cos (c+d x)+B \cos ^2(c+d x)\right ) \, dx\\ &=\frac{B (a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d}+\frac{\int (a+b \cos (c+d x))^4 (5 b B+(6 A b-a B) \cos (c+d x)) \, dx}{6 b}\\ &=\frac{(6 A b-a B) (a+b \cos (c+d x))^4 \sin (c+d x)}{30 b d}+\frac{B (a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d}+\frac{\int (a+b \cos (c+d x))^3 \left (3 b (8 A b+7 a B)+\left (24 a A b-4 a^2 B+25 b^2 B\right ) \cos (c+d x)\right ) \, dx}{30 b}\\ &=\frac{\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \cos (c+d x))^3 \sin (c+d x)}{120 b d}+\frac{(6 A b-a B) (a+b \cos (c+d x))^4 \sin (c+d x)}{30 b d}+\frac{B (a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d}+\frac{\int (a+b \cos (c+d x))^2 \left (3 b \left (56 a A b+24 a^2 B+25 b^2 B\right )+3 \left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) \cos (c+d x)\right ) \, dx}{120 b}\\ &=\frac{\left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{120 b d}+\frac{\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \cos (c+d x))^3 \sin (c+d x)}{120 b d}+\frac{(6 A b-a B) (a+b \cos (c+d x))^4 \sin (c+d x)}{30 b d}+\frac{B (a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d}+\frac{\int (a+b \cos (c+d x)) \left (3 b \left (216 a^2 A b+64 A b^3+64 a^3 B+181 a b^2 B\right )+3 \left (48 a^3 A b+232 a A b^3-8 a^4 B+178 a^2 b^2 B+75 b^4 B\right ) \cos (c+d x)\right ) \, dx}{360 b}\\ &=\frac{1}{16} \left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right ) x+\frac{\left (24 a^4 A b+224 a^2 A b^3+32 A b^5-4 a^5 B+121 a^3 b^2 B+128 a b^4 B\right ) \sin (c+d x)}{60 b d}+\frac{\left (48 a^3 A b+232 a A b^3-8 a^4 B+178 a^2 b^2 B+75 b^4 B\right ) \cos (c+d x) \sin (c+d x)}{240 d}+\frac{\left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{120 b d}+\frac{\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \cos (c+d x))^3 \sin (c+d x)}{120 b d}+\frac{(6 A b-a B) (a+b \cos (c+d x))^4 \sin (c+d x)}{30 b d}+\frac{B (a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d}\\ \end{align*}

Mathematica [A]  time = 1.05533, size = 333, normalized size = 1.02 \[ \frac{120 \left (36 a^2 A b^2+8 a^4 A+24 a^3 b B+20 a b^3 B+5 A b^4\right ) \sin (c+d x)+15 \left (64 a^3 A b+96 a^2 b^2 B+16 a^4 B+64 a A b^3+15 b^4 B\right ) \sin (2 (c+d x))+480 a^2 A b^2 \sin (3 (c+d x))+1920 a^3 A b c+1920 a^3 A b d x+180 a^2 b^2 B \sin (4 (c+d x))+2160 a^2 b^2 B c+2160 a^2 b^2 B d x+320 a^3 b B \sin (3 (c+d x))+480 a^4 B c+480 a^4 B d x+120 a A b^3 \sin (4 (c+d x))+1440 a A b^3 c+1440 a A b^3 d x+400 a b^3 B \sin (3 (c+d x))+48 a b^3 B \sin (5 (c+d x))+100 A b^4 \sin (3 (c+d x))+12 A b^4 \sin (5 (c+d x))+45 b^4 B \sin (4 (c+d x))+5 b^4 B \sin (6 (c+d x))+300 b^4 B c+300 b^4 B d x}{960 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1920*a^3*A*b*c + 1440*a*A*b^3*c + 480*a^4*B*c + 2160*a^2*b^2*B*c + 300*b^4*B*c + 1920*a^3*A*b*d*x + 1440*a*A*
b^3*d*x + 480*a^4*B*d*x + 2160*a^2*b^2*B*d*x + 300*b^4*B*d*x + 120*(8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*
b*B + 20*a*b^3*B)*Sin[c + d*x] + 15*(64*a^3*A*b + 64*a*A*b^3 + 16*a^4*B + 96*a^2*b^2*B + 15*b^4*B)*Sin[2*(c +
d*x)] + 480*a^2*A*b^2*Sin[3*(c + d*x)] + 100*A*b^4*Sin[3*(c + d*x)] + 320*a^3*b*B*Sin[3*(c + d*x)] + 400*a*b^3
*B*Sin[3*(c + d*x)] + 120*a*A*b^3*Sin[4*(c + d*x)] + 180*a^2*b^2*B*Sin[4*(c + d*x)] + 45*b^4*B*Sin[4*(c + d*x)
] + 12*A*b^4*Sin[5*(c + d*x)] + 48*a*b^3*B*Sin[5*(c + d*x)] + 5*b^4*B*Sin[6*(c + d*x)])/(960*d)

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Maple [A]  time = 0.084, size = 316, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( 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 ) +4\,A{a}^{3}b \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{\frac{4\,B{a}^{3}b \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,A{a}^{2}{b}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +6\,B{a}^{2}{b}^{2} \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\,Aa{b}^{3} \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\,Ba{b}^{3}\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{b}^{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 ) }+B{b}^{4} \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+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),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)+4*A*a^3*b*(1/2*cos(d*x+c)*sin(d*x+c)+1/2
*d*x+1/2*c)+4/3*B*a^3*b*(2+cos(d*x+c)^2)*sin(d*x+c)+2*A*a^2*b^2*(2+cos(d*x+c)^2)*sin(d*x+c)+6*B*a^2*b^2*(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*b^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)+4/5*B*a*b^3*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+1/5*A*b^4*(8/3+cos(d*x+c)^4+4/3*cos
(d*x+c)^2)*sin(d*x+c)+B*b^4*(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 = 1.04622, size = 414, normalized size = 1.27 \begin{align*} \frac{240 \,{\left (2 \, d x + 2 \, c + \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 )} A a^{3} b - 1280 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} b - 1920 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 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^{2} b^{2} + 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 b^{3} + 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 b^{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 )} A b^{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 b^{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+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.57297, size = 587, normalized size = 1.81 \begin{align*} \frac{15 \,{\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} d x +{\left (40 \, B b^{4} \cos \left (d x + c\right )^{5} + 240 \, A a^{4} + 640 \, B a^{3} b + 960 \, A a^{2} b^{2} + 512 \, B a b^{3} + 128 \, A b^{4} + 48 \,{\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 10 \,{\left (36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 32 \,{\left (10 \, B a^{3} b + 15 \, A a^{2} b^{2} + 8 \, B a b^{3} + 2 \, A b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )\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+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/240*(15*(8*B*a^4 + 32*A*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 5*B*b^4)*d*x + (40*B*b^4*cos(d*x + c)^5 + 240*A*
a^4 + 640*B*a^3*b + 960*A*a^2*b^2 + 512*B*a*b^3 + 128*A*b^4 + 48*(4*B*a*b^3 + A*b^4)*cos(d*x + c)^4 + 10*(36*B
*a^2*b^2 + 24*A*a*b^3 + 5*B*b^4)*cos(d*x + c)^3 + 32*(10*B*a^3*b + 15*A*a^2*b^2 + 8*B*a*b^3 + 2*A*b^4)*cos(d*x
+ c)^2 + 15*(8*B*a^4 + 32*A*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 5*B*b^4)*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 6.86899, size = 811, normalized size = 2.5 \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+b*cos(d*x+c))**4*(A+B*cos(d*x+c)),x)

[Out]

Piecewise((A*a**4*sin(c + d*x)/d + 2*A*a**3*b*x*sin(c + d*x)**2 + 2*A*a**3*b*x*cos(c + d*x)**2 + 2*A*a**3*b*si
n(c + d*x)*cos(c + d*x)/d + 4*A*a**2*b**2*sin(c + d*x)**3/d + 6*A*a**2*b**2*sin(c + d*x)*cos(c + d*x)**2/d + 3
*A*a*b**3*x*sin(c + d*x)**4/2 + 3*A*a*b**3*x*sin(c + d*x)**2*cos(c + d*x)**2 + 3*A*a*b**3*x*cos(c + d*x)**4/2
+ 3*A*a*b**3*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 5*A*a*b**3*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 8*A*b**4*sin
(c + d*x)**5/(15*d) + 4*A*b**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + A*b**4*sin(c + d*x)*cos(c + d*x)**4/d +
B*a**4*x*sin(c + d*x)**2/2 + B*a**4*x*cos(c + d*x)**2/2 + B*a**4*sin(c + d*x)*cos(c + d*x)/(2*d) + 8*B*a**3*b
*sin(c + d*x)**3/(3*d) + 4*B*a**3*b*sin(c + d*x)*cos(c + d*x)**2/d + 9*B*a**2*b**2*x*sin(c + d*x)**4/4 + 9*B*a
**2*b**2*x*sin(c + d*x)**2*cos(c + d*x)**2/2 + 9*B*a**2*b**2*x*cos(c + d*x)**4/4 + 9*B*a**2*b**2*sin(c + d*x)*
*3*cos(c + d*x)/(4*d) + 15*B*a**2*b**2*sin(c + d*x)*cos(c + d*x)**3/(4*d) + 32*B*a*b**3*sin(c + d*x)**5/(15*d)
+ 16*B*a*b**3*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 4*B*a*b**3*sin(c + d*x)*cos(c + d*x)**4/d + 5*B*b**4*x*
sin(c + d*x)**6/16 + 15*B*b**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*B*b**4*x*sin(c + d*x)**2*cos(c + d*x)
**4/16 + 5*B*b**4*x*cos(c + d*x)**6/16 + 5*B*b**4*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*B*b**4*sin(c + d*x)*
*3*cos(c + d*x)**3/(6*d) + 11*B*b**4*sin(c + d*x)*cos(c + d*x)**5/(16*d), Ne(d, 0)), (x*(A + B*cos(c))*(a + b*
cos(c))**4*cos(c), True))

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Giac [A]  time = 1.71884, size = 355, normalized size = 1.09 \begin{align*} \frac{B b^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{1}{16} \,{\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} x + \frac{{\left (4 \, B a b^{3} + A b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{{\left (12 \, B a^{2} b^{2} + 8 \, A a b^{3} + 3 \, B b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 20 \, B a b^{3} + 5 \, A b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (16 \, B a^{4} + 64 \, A a^{3} b + 96 \, B a^{2} b^{2} + 64 \, A a b^{3} + 15 \, B b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (8 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 20 \, B a b^{3} + 5 \, A b^{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+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="giac")

[Out]

1/192*B*b^4*sin(6*d*x + 6*c)/d + 1/16*(8*B*a^4 + 32*A*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 5*B*b^4)*x + 1/80*(4
*B*a*b^3 + A*b^4)*sin(5*d*x + 5*c)/d + 1/64*(12*B*a^2*b^2 + 8*A*a*b^3 + 3*B*b^4)*sin(4*d*x + 4*c)/d + 1/48*(16
*B*a^3*b + 24*A*a^2*b^2 + 20*B*a*b^3 + 5*A*b^4)*sin(3*d*x + 3*c)/d + 1/64*(16*B*a^4 + 64*A*a^3*b + 96*B*a^2*b^
2 + 64*A*a*b^3 + 15*B*b^4)*sin(2*d*x + 2*c)/d + 1/8*(8*A*a^4 + 24*B*a^3*b + 36*A*a^2*b^2 + 20*B*a*b^3 + 5*A*b^
4)*sin(d*x + c)/d