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

Optimal. Leaf size=327 \[ \frac{\sin (c+d x) \left (5 a^3 (3 A+2 C)+30 a^2 b B+6 a b^2 (5 A+4 C)+8 b^3 B\right )}{15 d}+\frac{b \sin (c+d x) \cos ^3(c+d x) \left (6 a^2 C+42 a b B+30 A b^2+25 b^2 C\right )}{120 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (12 a^2 b B+a^3 C+3 a b^2 (5 A+4 C)+4 b^3 B\right )}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left (6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right )}{16 d}+\frac{1}{16} x \left (6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right )+\frac{(a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{10 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d} \]

[Out]

((8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*x)/16 + ((30*a^2*b*B + 8*b^3*B + 5*a^3*(3*A +
2*C) + 6*a*b^2*(5*A + 4*C))*Sin[c + d*x])/(15*d) + ((8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5
*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((12*a^2*b*B + 4*b^3*B + a^3*C + 3*a*b^2*(5*A + 4*C))*Cos[c + d*x]^2*
Sin[c + d*x])/(15*d) + (b*(30*A*b^2 + 42*a*b*B + 6*a^2*C + 25*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + ((
2*b*B + a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(10*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x
])^3*Sin[c + d*x])/(6*d)

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Rubi [A]  time = 0.606592, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3049, 3033, 3023, 2734} \[ \frac{\sin (c+d x) \left (5 a^3 (3 A+2 C)+30 a^2 b B+6 a b^2 (5 A+4 C)+8 b^3 B\right )}{15 d}+\frac{b \sin (c+d x) \cos ^3(c+d x) \left (6 a^2 C+42 a b B+30 A b^2+25 b^2 C\right )}{120 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (12 a^2 b B+a^3 C+3 a b^2 (5 A+4 C)+4 b^3 B\right )}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left (6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right )}{16 d}+\frac{1}{16} x \left (6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right )+\frac{(a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{10 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*x)/16 + ((30*a^2*b*B + 8*b^3*B + 5*a^3*(3*A +
2*C) + 6*a*b^2*(5*A + 4*C))*Sin[c + d*x])/(15*d) + ((8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5
*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((12*a^2*b*B + 4*b^3*B + a^3*C + 3*a*b^2*(5*A + 4*C))*Cos[c + d*x]^2*
Sin[c + d*x])/(15*d) + (b*(30*A*b^2 + 42*a*b*B + 6*a^2*C + 25*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + ((
2*b*B + a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(10*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x
])^3*Sin[c + d*x])/(6*d)

Rule 3049

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

Rule 3033

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b
*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*
(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e +
f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

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 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))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac{1}{6} \int \cos (c+d x) (a+b \cos (c+d x))^2 \left (2 a (3 A+C)+(6 A b+6 a B+5 b C) \cos (c+d x)+3 (2 b B+a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{(2 b B+a C) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac{1}{30} \int \cos (c+d x) (a+b \cos (c+d x)) \left (2 a (15 a A+6 b B+8 a C)+\left (30 a^2 B+24 b^2 B+a b (60 A+47 C)\right ) \cos (c+d x)+\left (30 A b^2+42 a b B+6 a^2 C+25 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{b \left (30 A b^2+42 a b B+6 a^2 C+25 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac{(2 b B+a C) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac{1}{120} \int \cos (c+d x) \left (8 a^2 (15 a A+6 b B+8 a C)+15 \left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \cos (c+d x)+24 \left (12 a^2 b B+4 b^3 B+a^3 C+3 a b^2 (5 A+4 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{\left (12 a^2 b B+4 b^3 B+a^3 C+3 a b^2 (5 A+4 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac{b \left (30 A b^2+42 a b B+6 a^2 C+25 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac{(2 b B+a C) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac{1}{360} \int \cos (c+d x) \left (24 \left (30 a^2 b B+8 b^3 B+5 a^3 (3 A+2 C)+6 a b^2 (5 A+4 C)\right )+45 \left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{1}{16} \left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) x+\frac{\left (30 a^2 b B+8 b^3 B+5 a^3 (3 A+2 C)+6 a b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac{\left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{\left (12 a^2 b B+4 b^3 B+a^3 C+3 a b^2 (5 A+4 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac{b \left (30 A b^2+42 a b B+6 a^2 C+25 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac{(2 b B+a C) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac{C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}\\ \end{align*}

Mathematica [A]  time = 1.10767, size = 368, normalized size = 1.13 \[ \frac{120 \sin (c+d x) \left (a^3 (8 A+6 C)+18 a^2 b B+3 a b^2 (6 A+5 C)+5 b^3 B\right )+15 \sin (2 (c+d x)) \left (48 a^2 b (A+C)+16 a^3 B+48 a b^2 B+b^3 (16 A+15 C)\right )+1440 a^2 A b c+1440 a^2 A b d x+240 a^2 b B \sin (3 (c+d x))+90 a^2 b C \sin (4 (c+d x))+1080 a^2 b c C+1080 a^2 b C d x+480 a^3 B c+480 a^3 B d x+80 a^3 C \sin (3 (c+d x))+240 a A b^2 \sin (3 (c+d x))+90 a b^2 B \sin (4 (c+d x))+1080 a b^2 B c+1080 a b^2 B d x+300 a b^2 C \sin (3 (c+d x))+36 a b^2 C \sin (5 (c+d x))+30 A b^3 \sin (4 (c+d x))+360 A b^3 c+360 A b^3 d x+100 b^3 B \sin (3 (c+d x))+12 b^3 B \sin (5 (c+d x))+45 b^3 C \sin (4 (c+d x))+5 b^3 C \sin (6 (c+d x))+300 b^3 c C+300 b^3 C d x}{960 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1440*a^2*A*b*c + 360*A*b^3*c + 480*a^3*B*c + 1080*a*b^2*B*c + 1080*a^2*b*c*C + 300*b^3*c*C + 1440*a^2*A*b*d*x
+ 360*A*b^3*d*x + 480*a^3*B*d*x + 1080*a*b^2*B*d*x + 1080*a^2*b*C*d*x + 300*b^3*C*d*x + 120*(18*a^2*b*B + 5*b
^3*B + 3*a*b^2*(6*A + 5*C) + a^3*(8*A + 6*C))*Sin[c + d*x] + 15*(16*a^3*B + 48*a*b^2*B + 48*a^2*b*(A + C) + b^
3*(16*A + 15*C))*Sin[2*(c + d*x)] + 240*a*A*b^2*Sin[3*(c + d*x)] + 240*a^2*b*B*Sin[3*(c + d*x)] + 100*b^3*B*Si
n[3*(c + d*x)] + 80*a^3*C*Sin[3*(c + d*x)] + 300*a*b^2*C*Sin[3*(c + d*x)] + 30*A*b^3*Sin[4*(c + d*x)] + 90*a*b
^2*B*Sin[4*(c + d*x)] + 90*a^2*b*C*Sin[4*(c + d*x)] + 45*b^3*C*Sin[4*(c + d*x)] + 12*b^3*B*Sin[5*(c + d*x)] +
36*a*b^2*C*Sin[5*(c + d*x)] + 5*b^3*C*Sin[6*(c + d*x)])/(960*d)

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Maple [A]  time = 0.023, size = 370, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( A{b}^{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{{b}^{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 ) }+C{b}^{3} \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 ) +aA{b}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,a{b}^{2}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\,Ca{b}^{2}\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 ) }+3\,A{a}^{2}b \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{a}^{2}bB \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,{a}^{2}bC \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}\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}} \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))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)

[Out]

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

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Maxima [A]  time = 1.02471, size = 486, normalized size = 1.49 \begin{align*} \frac{240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 320 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b - 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} b + 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^{2} b - 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{2} + 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 b^{2} + 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 b^{2} + 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 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 )} B b^{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 b^{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+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.92528, size = 610, normalized size = 1.87 \begin{align*} \frac{15 \,{\left (8 \, B a^{3} + 6 \,{\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} +{\left (6 \, A + 5 \, C\right )} b^{3}\right )} d x +{\left (40 \, C b^{3} \cos \left (d x + c\right )^{5} + 48 \,{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + 80 \,{\left (3 \, A + 2 \, C\right )} a^{3} + 480 \, B a^{2} b + 96 \,{\left (5 \, A + 4 \, C\right )} a b^{2} + 128 \, B b^{3} + 10 \,{\left (18 \, C a^{2} b + 18 \, B a b^{2} +{\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{3} + 16 \,{\left (5 \, C a^{3} + 15 \, B a^{2} b + 3 \,{\left (5 \, A + 4 \, C\right )} a b^{2} + 4 \, B b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (8 \, B a^{3} + 6 \,{\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} +{\left (6 \, A + 5 \, C\right )} b^{3}\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))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/240*(15*(8*B*a^3 + 6*(4*A + 3*C)*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3)*d*x + (40*C*b^3*cos(d*x + c)^5 + 48*(
3*C*a*b^2 + B*b^3)*cos(d*x + c)^4 + 80*(3*A + 2*C)*a^3 + 480*B*a^2*b + 96*(5*A + 4*C)*a*b^2 + 128*B*b^3 + 10*(
18*C*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3)*cos(d*x + c)^3 + 16*(5*C*a^3 + 15*B*a^2*b + 3*(5*A + 4*C)*a*b^2 + 4
*B*b^3)*cos(d*x + c)^2 + 15*(8*B*a^3 + 6*(4*A + 3*C)*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3)*cos(d*x + c))*sin(d
*x + c))/d

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Sympy [A]  time = 6.98618, size = 966, normalized size = 2.95 \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))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)

[Out]

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

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Giac [A]  time = 1.18892, size = 382, normalized size = 1.17 \begin{align*} \frac{C b^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{1}{16} \,{\left (8 \, B a^{3} + 24 \, A a^{2} b + 18 \, C a^{2} b + 18 \, B a b^{2} + 6 \, A b^{3} + 5 \, C b^{3}\right )} x + \frac{{\left (3 \, C a b^{2} + B b^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{{\left (6 \, C a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3} + 3 \, C b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (4 \, C a^{3} + 12 \, B a^{2} b + 12 \, A a b^{2} + 15 \, C a b^{2} + 5 \, B b^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (16 \, B a^{3} + 48 \, A a^{2} b + 48 \, C a^{2} b + 48 \, B a b^{2} + 16 \, A b^{3} + 15 \, C b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (8 \, A a^{3} + 6 \, C a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2} + 15 \, C a b^{2} + 5 \, B b^{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+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/192*C*b^3*sin(6*d*x + 6*c)/d + 1/16*(8*B*a^3 + 24*A*a^2*b + 18*C*a^2*b + 18*B*a*b^2 + 6*A*b^3 + 5*C*b^3)*x +
1/80*(3*C*a*b^2 + B*b^3)*sin(5*d*x + 5*c)/d + 1/64*(6*C*a^2*b + 6*B*a*b^2 + 2*A*b^3 + 3*C*b^3)*sin(4*d*x + 4*
c)/d + 1/48*(4*C*a^3 + 12*B*a^2*b + 12*A*a*b^2 + 15*C*a*b^2 + 5*B*b^3)*sin(3*d*x + 3*c)/d + 1/64*(16*B*a^3 + 4
8*A*a^2*b + 48*C*a^2*b + 48*B*a*b^2 + 16*A*b^3 + 15*C*b^3)*sin(2*d*x + 2*c)/d + 1/8*(8*A*a^3 + 6*C*a^3 + 18*B*
a^2*b + 18*A*a*b^2 + 15*C*a*b^2 + 5*B*b^3)*sin(d*x + c)/d