\(\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [320]

Optimal result

Integrand size = 33, antiderivative size = 166 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} a^3 (20 A+15 B+13 C) x+\frac {a^3 (20 A+15 B+13 C) \sin (c+d x)}{5 d}+\frac {3 a^3 (20 A+15 B+13 C) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}-\frac {a^3 (20 A+15 B+13 C) \sin ^3(c+d x)}{60 d} \]

[Out]

1/8*a^3*(20*A+15*B+13*C)*x+1/5*a^3*(20*A+15*B+13*C)*sin(d*x+c)/d+3/40*a^3*(20*A+15*B+13*C)*cos(d*x+c)*sin(d*x+
c)/d+1/20*(5*B-C)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+1/5*C*(a+a*cos(d*x+c))^4*sin(d*x+c)/a/d-1/60*a^3*(20*A+15*B+
13*C)*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3102, 2830, 2724, 2717, 2715, 8, 2713} \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {a^3 (20 A+15 B+13 C) \sin ^3(c+d x)}{60 d}+\frac {a^3 (20 A+15 B+13 C) \sin (c+d x)}{5 d}+\frac {3 a^3 (20 A+15 B+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac {1}{8} a^3 x (20 A+15 B+13 C)+\frac {(5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d} \]

[In]

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

[Out]

(a^3*(20*A + 15*B + 13*C)*x)/8 + (a^3*(20*A + 15*B + 13*C)*Sin[c + d*x])/(5*d) + (3*a^3*(20*A + 15*B + 13*C)*C
os[c + d*x]*Sin[c + d*x])/(40*d) + ((5*B - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c +
d*x])^4*Sin[c + d*x])/(5*a*d) - (a^3*(20*A + 15*B + 13*C)*Sin[c + d*x]^3)/(60*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

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]

Rule 2715

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] && Integ
erQ[2*n]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2724

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 2830

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*S
in[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 3102

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]

Rubi steps \begin{align*} \text {integral}& = \frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac {\int (a+a \cos (c+d x))^3 (a (5 A+4 C)+a (5 B-C) \cos (c+d x)) \, dx}{5 a} \\ & = \frac {(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac {1}{20} (20 A+15 B+13 C) \int (a+a \cos (c+d x))^3 \, dx \\ & = \frac {(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac {1}{20} (20 A+15 B+13 C) \int \left (a^3+3 a^3 \cos (c+d x)+3 a^3 \cos ^2(c+d x)+a^3 \cos ^3(c+d x)\right ) \, dx \\ & = \frac {1}{20} a^3 (20 A+15 B+13 C) x+\frac {(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac {1}{20} \left (a^3 (20 A+15 B+13 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{20} \left (3 a^3 (20 A+15 B+13 C)\right ) \int \cos (c+d x) \, dx+\frac {1}{20} \left (3 a^3 (20 A+15 B+13 C)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {1}{20} a^3 (20 A+15 B+13 C) x+\frac {3 a^3 (20 A+15 B+13 C) \sin (c+d x)}{20 d}+\frac {3 a^3 (20 A+15 B+13 C) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac {1}{40} \left (3 a^3 (20 A+15 B+13 C)\right ) \int 1 \, dx-\frac {\left (a^3 (20 A+15 B+13 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{20 d} \\ & = \frac {1}{8} a^3 (20 A+15 B+13 C) x+\frac {a^3 (20 A+15 B+13 C) \sin (c+d x)}{5 d}+\frac {3 a^3 (20 A+15 B+13 C) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}-\frac {a^3 (20 A+15 B+13 C) \sin ^3(c+d x)}{60 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^3 \sin (c+d x) \left (30 (20 A+15 B+13 C) \arcsin \left (\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )+\left (8 (55 A+45 B+38 C)+15 (12 A+15 B+13 C) \cos (c+d x)+8 (5 A+15 B+19 C) \cos ^2(c+d x)+30 (B+3 C) \cos ^3(c+d x)+24 C \cos ^4(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{120 d \sqrt {\sin ^2(c+d x)}} \]

[In]

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

[Out]

(a^3*Sin[c + d*x]*(30*(20*A + 15*B + 13*C)*ArcSin[Sqrt[Sin[(c + d*x)/2]^2]] + (8*(55*A + 45*B + 38*C) + 15*(12
*A + 15*B + 13*C)*Cos[c + d*x] + 8*(5*A + 15*B + 19*C)*Cos[c + d*x]^2 + 30*(B + 3*C)*Cos[c + d*x]^3 + 24*C*Cos
[c + d*x]^4)*Sqrt[Sin[c + d*x]^2]))/(120*d*Sqrt[Sin[c + d*x]^2])

Maple [A] (verified)

Time = 5.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.64

method	result	size
parallelrisch	\(\frac {a^{3} \left (3 \left (3 A +4 B +4 C \right ) \sin \left (2 d x +2 c \right )+\left (A +3 B +\frac {17 C}{4}\right ) \sin \left (3 d x +3 c \right )+\frac {3 \left (B +3 C \right ) \sin \left (4 d x +4 c \right )}{8}+\frac {3 \sin \left (5 d x +5 c \right ) C}{20}+3 \left (15 A +13 B +\frac {23 C}{2}\right ) \sin \left (d x +c \right )+30 \left (A +\frac {3 B}{4}+\frac {13 C}{20}\right ) x d \right )}{12 d}\)	\(107\)
parts	\(a^{3} A x +\frac {\left (3 A \,a^{3}+B \,a^{3}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (B \,a^{3}+3 C \,a^{3}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (A \,a^{3}+3 B \,a^{3}+3 C \,a^{3}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (3 A \,a^{3}+3 B \,a^{3}+C \,a^{3}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}\)	\(197\)
risch	\(\frac {5 a^{3} A x}{2}+\frac {15 a^{3} B x}{8}+\frac {13 a^{3} C x}{8}+\frac {15 a^{3} A \sin \left (d x +c \right )}{4 d}+\frac {13 a^{3} B \sin \left (d x +c \right )}{4 d}+\frac {23 a^{3} C \sin \left (d x +c \right )}{8 d}+\frac {\sin \left (5 d x +5 c \right ) C \,a^{3}}{80 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{3}}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) C \,a^{3}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{3}}{12 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{4 d}+\frac {17 \sin \left (3 d x +3 c \right ) C \,a^{3}}{48 d}+\frac {3 \sin \left (2 d x +2 c \right ) A \,a^{3}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{d}+\frac {\sin \left (2 d x +2 c \right ) C \,a^{3}}{d}\)	\(228\)
derivativedivides	\(\frac {\frac {A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 A \,a^{3} \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+A \,a^{3} \left (d x +c \right )+B \sin \left (d x +c \right ) a^{3}+C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(295\)
default	\(\frac {\frac {A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 A \,a^{3} \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+A \,a^{3} \left (d x +c \right )+B \sin \left (d x +c \right ) a^{3}+C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(295\)
norman	\(\frac {\frac {a^{3} \left (20 A +15 B +13 C \right ) x}{8}+\frac {32 a^{3} \left (20 A +15 B +13 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {7 a^{3} \left (20 A +15 B +13 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{3} \left (20 A +15 B +13 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {5 a^{3} \left (20 A +15 B +13 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {5 a^{3} \left (20 A +15 B +13 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {5 a^{3} \left (20 A +15 B +13 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {5 a^{3} \left (20 A +15 B +13 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} \left (20 A +15 B +13 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} \left (44 A +49 B +51 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (212 A +183 B +133 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\)	\(312\)

[In]

int((a+cos(d*x+c)*a)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/12*a^3*(3*(3*A+4*B+4*C)*sin(2*d*x+2*c)+(A+3*B+17/4*C)*sin(3*d*x+3*c)+3/8*(B+3*C)*sin(4*d*x+4*c)+3/20*sin(5*d
*x+5*c)*C+3*(15*A+13*B+23/2*C)*sin(d*x+c)+30*(A+3/4*B+13/20*C)*x*d)/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.73 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (20 \, A + 15 \, B + 13 \, C\right )} a^{3} d x + {\left (24 \, C a^{3} \cos \left (d x + c\right )^{4} + 30 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 15 \, B + 19 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (12 \, A + 15 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (55 \, A + 45 \, B + 38 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{120 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/120*(15*(20*A + 15*B + 13*C)*a^3*d*x + (24*C*a^3*cos(d*x + c)^4 + 30*(B + 3*C)*a^3*cos(d*x + c)^3 + 8*(5*A +
15*B + 19*C)*a^3*cos(d*x + c)^2 + 15*(12*A + 15*B + 13*C)*a^3*cos(d*x + c) + 8*(55*A + 45*B + 38*C)*a^3)*sin(
d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (150) = 300\).

Time = 0.31 (sec) , antiderivative size = 658, normalized size of antiderivative = 3.96 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {3 A a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + A a^{3} x + \frac {2 A a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {3 A a^{3} \sin {\left (c + d x \right )}}{d} + \frac {3 B a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 B a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 B a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 B a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {5 B a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 B a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 B a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {B a^{3} \sin {\left (c + d x \right )}}{d} + \frac {9 C a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {C a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {9 C a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {C a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {8 C a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {9 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 C a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {15 C a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 C a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + a\right )^{3} \left (A + B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((a+a*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)

[Out]

Piecewise((3*A*a**3*x*sin(c + d*x)**2/2 + 3*A*a**3*x*cos(c + d*x)**2/2 + A*a**3*x + 2*A*a**3*sin(c + d*x)**3/(
3*d) + A*a**3*sin(c + d*x)*cos(c + d*x)**2/d + 3*A*a**3*sin(c + d*x)*cos(c + d*x)/(2*d) + 3*A*a**3*sin(c + d*x
)/d + 3*B*a**3*x*sin(c + d*x)**4/8 + 3*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*B*a**3*x*sin(c + d*x)**2
/2 + 3*B*a**3*x*cos(c + d*x)**4/8 + 3*B*a**3*x*cos(c + d*x)**2/2 + 3*B*a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d)
+ 2*B*a**3*sin(c + d*x)**3/d + 5*B*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 3*B*a**3*sin(c + d*x)*cos(c + d*
x)**2/d + 3*B*a**3*sin(c + d*x)*cos(c + d*x)/(2*d) + B*a**3*sin(c + d*x)/d + 9*C*a**3*x*sin(c + d*x)**4/8 + 9*
C*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + C*a**3*x*sin(c + d*x)**2/2 + 9*C*a**3*x*cos(c + d*x)**4/8 + C*a**
3*x*cos(c + d*x)**2/2 + 8*C*a**3*sin(c + d*x)**5/(15*d) + 4*C*a**3*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*C
*a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 2*C*a**3*sin(c + d*x)**3/d + C*a**3*sin(c + d*x)*cos(c + d*x)**4/d
+ 15*C*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 3*C*a**3*sin(c + d*x)*cos(c + d*x)**2/d + C*a**3*sin(c + d*x)
*cos(c + d*x)/(2*d), Ne(d, 0)), (x*(a*cos(c) + a)**3*(A + B*cos(c) + C*cos(c)**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.70 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 480 \, {\left (d x + c\right )} A a^{3} + 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} + 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 1440 \, A a^{3} \sin \left (d x + c\right ) - 480 \, B a^{3} \sin \left (d x + c\right )}{480 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/480*(160*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 - 360*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^3 - 480*(d*x +
c)*A*a^3 + 480*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3 - 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x +
2*c))*B*a^3 - 360*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^3 - 32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d
*x + c))*C*a^3 + 480*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 - 45*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*
d*x + 2*c))*C*a^3 - 120*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^3 - 1440*A*a^3*sin(d*x + c) - 480*B*a^3*sin(d*x +
c))/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.98 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{3} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {1}{8} \, {\left (20 \, A a^{3} + 15 \, B a^{3} + 13 \, C a^{3}\right )} x + \frac {{\left (B a^{3} + 3 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (4 \, A a^{3} + 12 \, B a^{3} + 17 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (3 \, A a^{3} + 4 \, B a^{3} + 4 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (30 \, A a^{3} + 26 \, B a^{3} + 23 \, C a^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/80*C*a^3*sin(5*d*x + 5*c)/d + 1/8*(20*A*a^3 + 15*B*a^3 + 13*C*a^3)*x + 1/32*(B*a^3 + 3*C*a^3)*sin(4*d*x + 4*
c)/d + 1/48*(4*A*a^3 + 12*B*a^3 + 17*C*a^3)*sin(3*d*x + 3*c)/d + 1/4*(3*A*a^3 + 4*B*a^3 + 4*C*a^3)*sin(2*d*x +
2*c)/d + 1/8*(30*A*a^3 + 26*B*a^3 + 23*C*a^3)*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 2.92 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.94 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (5\,A\,a^3+\frac {15\,B\,a^3}{4}+\frac {13\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {70\,A\,a^3}{3}+\frac {35\,B\,a^3}{2}+\frac {91\,C\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {128\,A\,a^3}{3}+32\,B\,a^3+\frac {416\,C\,a^3}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {106\,A\,a^3}{3}+\frac {61\,B\,a^3}{2}+\frac {133\,C\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A\,a^3+\frac {49\,B\,a^3}{4}+\frac {51\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (20\,A+15\,B+13\,C\right )}{4\,\left (5\,A\,a^3+\frac {15\,B\,a^3}{4}+\frac {13\,C\,a^3}{4}\right )}\right )\,\left (20\,A+15\,B+13\,C\right )}{4\,d}-\frac {a^3\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (20\,A+15\,B+13\,C\right )}{4\,d} \]

[In]

int((a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)

[Out]

(tan(c/2 + (d*x)/2)^9*(5*A*a^3 + (15*B*a^3)/4 + (13*C*a^3)/4) + tan(c/2 + (d*x)/2)^7*((70*A*a^3)/3 + (35*B*a^3
)/2 + (91*C*a^3)/6) + tan(c/2 + (d*x)/2)^3*((106*A*a^3)/3 + (61*B*a^3)/2 + (133*C*a^3)/6) + tan(c/2 + (d*x)/2)
^5*((128*A*a^3)/3 + 32*B*a^3 + (416*C*a^3)/15) + tan(c/2 + (d*x)/2)*(11*A*a^3 + (49*B*a^3)/4 + (51*C*a^3)/4))/
(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(
c/2 + (d*x)/2)^10 + 1)) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(20*A + 15*B + 13*C))/(4*(5*A*a^3 + (15*B*a^3)/4 +
(13*C*a^3)/4)))*(20*A + 15*B + 13*C))/(4*d) - (a^3*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(20*A + 15*B + 13*C))
/(4*d)