\(\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx\) [19]

Optimal result

Integrand size = 31, antiderivative size = 201 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {1}{16} a^3 (26 A+23 B) x+\frac {a^3 (19 A+17 B) \sin (c+d x)}{5 d}+\frac {a^3 (26 A+23 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (22 A+21 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 A+4 B) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (19 A+17 B) \sin ^3(c+d x)}{15 d} \]

[Out]

1/16*a^3*(26*A+23*B)*x+1/5*a^3*(19*A+17*B)*sin(d*x+c)/d+1/16*a^3*(26*A+23*B)*cos(d*x+c)*sin(d*x+c)/d+1/40*a^3*
(22*A+21*B)*cos(d*x+c)^3*sin(d*x+c)/d+1/6*a*B*cos(d*x+c)^3*(a+a*cos(d*x+c))^2*sin(d*x+c)/d+1/15*(3*A+4*B)*cos(
d*x+c)^3*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d-1/15*a^3*(19*A+17*B)*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3055, 3047, 3102, 2827, 2715, 8, 2713} \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=-\frac {a^3 (19 A+17 B) \sin ^3(c+d x)}{15 d}+\frac {a^3 (19 A+17 B) \sin (c+d x)}{5 d}+\frac {a^3 (22 A+21 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {(3 A+4 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{15 d}+\frac {a^3 (26 A+23 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^3 x (26 A+23 B)+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d} \]

[In]

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

[Out]

(a^3*(26*A + 23*B)*x)/16 + (a^3*(19*A + 17*B)*Sin[c + d*x])/(5*d) + (a^3*(26*A + 23*B)*Cos[c + d*x]*Sin[c + d*
x])/(16*d) + (a^3*(22*A + 21*B)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*B*Cos[c + d*x]^3*(a + a*Cos[c + d*x])
^2*Sin[c + d*x])/(6*d) + ((3*A + 4*B)*Cos[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(19*
A + 17*B)*Sin[c + d*x]^3)/(15*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 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3047

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 3055

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

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 {a B \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^2(c+d x) (a+a \cos (c+d x))^2 (3 a (2 A+B)+2 a (3 A+4 B) \cos (c+d x)) \, dx \\ & = \frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 A+4 B) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{30} \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^2 (16 A+13 B)+3 a^2 (22 A+21 B) \cos (c+d x)\right ) \, dx \\ & = \frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 A+4 B) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{30} \int \cos ^2(c+d x) \left (3 a^3 (16 A+13 B)+\left (3 a^3 (16 A+13 B)+3 a^3 (22 A+21 B)\right ) \cos (c+d x)+3 a^3 (22 A+21 B) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {a^3 (22 A+21 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 A+4 B) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{120} \int \cos ^2(c+d x) \left (15 a^3 (26 A+23 B)+24 a^3 (19 A+17 B) \cos (c+d x)\right ) \, dx \\ & = \frac {a^3 (22 A+21 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 A+4 B) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{5} \left (a^3 (19 A+17 B)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{8} \left (a^3 (26 A+23 B)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {a^3 (26 A+23 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (22 A+21 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 A+4 B) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{16} \left (a^3 (26 A+23 B)\right ) \int 1 \, dx-\frac {\left (a^3 (19 A+17 B)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {1}{16} a^3 (26 A+23 B) x+\frac {a^3 (19 A+17 B) \sin (c+d x)}{5 d}+\frac {a^3 (26 A+23 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (22 A+21 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 A+4 B) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (19 A+17 B) \sin ^3(c+d x)}{15 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.67 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {a^3 (1380 B c+1560 A d x+1380 B d x+120 (23 A+21 B) \sin (c+d x)+15 (64 A+63 B) \sin (2 (c+d x))+340 A \sin (3 (c+d x))+380 B \sin (3 (c+d x))+90 A \sin (4 (c+d x))+135 B \sin (4 (c+d x))+12 A \sin (5 (c+d x))+36 B \sin (5 (c+d x))+5 B \sin (6 (c+d x)))}{960 d} \]

[In]

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

[Out]

(a^3*(1380*B*c + 1560*A*d*x + 1380*B*d*x + 120*(23*A + 21*B)*Sin[c + d*x] + 15*(64*A + 63*B)*Sin[2*(c + d*x)]
+ 340*A*Sin[3*(c + d*x)] + 380*B*Sin[3*(c + d*x)] + 90*A*Sin[4*(c + d*x)] + 135*B*Sin[4*(c + d*x)] + 12*A*Sin[
5*(c + d*x)] + 36*B*Sin[5*(c + d*x)] + 5*B*Sin[6*(c + d*x)]))/(960*d)

Maple [A] (verified)

Time = 4.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.56

method	result	size
parallelrisch	\(\frac {3 \left (\left (\frac {32 A}{3}+\frac {21 B}{2}\right ) \sin \left (2 d x +2 c \right )+\frac {2 \left (17 A +19 B \right ) \sin \left (3 d x +3 c \right )}{9}+\left (\frac {3 B}{2}+A \right ) \sin \left (4 d x +4 c \right )+\frac {2 \left (\frac {A}{3}+B \right ) \sin \left (5 d x +5 c \right )}{5}+\frac {B \sin \left (6 d x +6 c \right )}{18}+4 \left (\frac {23 A}{3}+7 B \right ) \sin \left (d x +c \right )+\frac {52 \left (\frac {23 B}{26}+A \right ) x d}{3}\right ) a^{3}}{32 d}\)	\(112\)
risch	\(\frac {13 a^{3} A x}{8}+\frac {23 a^{3} B x}{16}+\frac {23 a^{3} A \sin \left (d x +c \right )}{8 d}+\frac {21 a^{3} B \sin \left (d x +c \right )}{8 d}+\frac {B \,a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {\sin \left (5 d x +5 c \right ) A \,a^{3}}{80 d}+\frac {3 \sin \left (5 d x +5 c \right ) B \,a^{3}}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{3}}{32 d}+\frac {9 \sin \left (4 d x +4 c \right ) B \,a^{3}}{64 d}+\frac {17 \sin \left (3 d x +3 c \right ) A \,a^{3}}{48 d}+\frac {19 \sin \left (3 d x +3 c \right ) B \,a^{3}}{48 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3}}{d}+\frac {63 \sin \left (2 d x +2 c \right ) B \,a^{3}}{64 d}\)	\(207\)
parts	\(\frac {\left (A \,a^{3}+3 B \,a^{3}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (3 A \,a^{3}+B \,a^{3}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (3 A \,a^{3}+3 B \,a^{3}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {B \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(209\)
derivativedivides	\(\frac {\frac {A \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+3 A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 B \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(266\)
default	\(\frac {\frac {A \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+3 A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 B \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(266\)
norman	\(\frac {\frac {a^{3} \left (26 A +23 B \right ) x}{16}+\frac {33 a^{3} \left (26 A +23 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {17 a^{3} \left (26 A +23 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a^{3} \left (26 A +23 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {3 a^{3} \left (26 A +23 B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {15 a^{3} \left (26 A +23 B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {5 a^{3} \left (26 A +23 B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {15 a^{3} \left (26 A +23 B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 a^{3} \left (26 A +23 B \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} \left (26 A +23 B \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 a^{3} \left (34 A +35 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{3} \left (838 A +633 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a^{3} \left (998 A +969 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\)	\(329\)

[In]

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

[Out]

3/32*((32/3*A+21/2*B)*sin(2*d*x+2*c)+2/9*(17*A+19*B)*sin(3*d*x+3*c)+(3/2*B+A)*sin(4*d*x+4*c)+2/5*(1/3*A+B)*sin
(5*d*x+5*c)+1/18*B*sin(6*d*x+6*c)+4*(23/3*A+7*B)*sin(d*x+c)+52/3*(23/26*B+A)*x*d)*a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.65 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {15 \, {\left (26 \, A + 23 \, B\right )} a^{3} d x + {\left (40 \, B a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} + 10 \, {\left (18 \, A + 23 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (19 \, A + 17 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (26 \, A + 23 \, B\right )} a^{3} \cos \left (d x + c\right ) + 32 \, {\left (19 \, A + 17 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{240 \, d} \]

[In]

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

[Out]

1/240*(15*(26*A + 23*B)*a^3*d*x + (40*B*a^3*cos(d*x + c)^5 + 48*(A + 3*B)*a^3*cos(d*x + c)^4 + 10*(18*A + 23*B
)*a^3*cos(d*x + c)^3 + 16*(19*A + 17*B)*a^3*cos(d*x + c)^2 + 15*(26*A + 23*B)*a^3*cos(d*x + c) + 32*(19*A + 17
*B)*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. 695 vs. \(2 (184) = 368\).

Time = 0.41 (sec) , antiderivative size = 695, normalized size of antiderivative = 3.46 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\begin {cases} \frac {9 A a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 A a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {A a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {9 A a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {A a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {8 A a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {9 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 A a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {A a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {15 A a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 A a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {A a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {5 B a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 B a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 B a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {15 B a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {9 B a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {5 B a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {9 B a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {5 B a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {8 B a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {5 B a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {4 B a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 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 )}}{3 d} + \frac {11 B a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {3 B a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {15 B a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {B a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{3} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.30 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3}}{960 \, d} \]

[In]

integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^3*(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^3 - 960*(sin(d*x + c)^3 - 3*sin(d*x + c
))*A*a^3 + 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 + 240*(2*d*x + 2*c + sin(2*d*x + 2
*c))*A*a^3 + 192*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^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))*B*a^3 - 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3 +
90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.83 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {B a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {1}{16} \, {\left (26 \, A a^{3} + 23 \, B a^{3}\right )} x + \frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {3 \, {\left (2 \, A a^{3} + 3 \, B a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (17 \, A a^{3} + 19 \, B a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (64 \, A a^{3} + 63 \, B a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (23 \, A a^{3} + 21 \, B a^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \]

[In]

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

[Out]

1/192*B*a^3*sin(6*d*x + 6*c)/d + 1/16*(26*A*a^3 + 23*B*a^3)*x + 1/80*(A*a^3 + 3*B*a^3)*sin(5*d*x + 5*c)/d + 3/
64*(2*A*a^3 + 3*B*a^3)*sin(4*d*x + 4*c)/d + 1/48*(17*A*a^3 + 19*B*a^3)*sin(3*d*x + 3*c)/d + 1/64*(64*A*a^3 + 6
3*B*a^3)*sin(2*d*x + 2*c)/d + 1/8*(23*A*a^3 + 21*B*a^3)*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 1.63 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.57 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {\left (\frac {13\,A\,a^3}{4}+\frac {23\,B\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {221\,A\,a^3}{12}+\frac {391\,B\,a^3}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {429\,A\,a^3}{10}+\frac {759\,B\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {499\,A\,a^3}{10}+\frac {969\,B\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {419\,A\,a^3}{12}+\frac {211\,B\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,A\,a^3}{4}+\frac {105\,B\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^3\,\left (26\,A+23\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (26\,A+23\,B\right )}{8\,\left (\frac {13\,A\,a^3}{4}+\frac {23\,B\,a^3}{8}\right )}\right )\,\left (26\,A+23\,B\right )}{8\,d} \]

[In]

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

[Out]

(tan(c/2 + (d*x)/2)*((51*A*a^3)/4 + (105*B*a^3)/8) + tan(c/2 + (d*x)/2)^11*((13*A*a^3)/4 + (23*B*a^3)/8) + tan
(c/2 + (d*x)/2)^3*((419*A*a^3)/12 + (211*B*a^3)/8) + tan(c/2 + (d*x)/2)^9*((221*A*a^3)/12 + (391*B*a^3)/24) +
tan(c/2 + (d*x)/2)^7*((429*A*a^3)/10 + (759*B*a^3)/20) + tan(c/2 + (d*x)/2)^5*((499*A*a^3)/10 + (969*B*a^3)/20
))/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 +
6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) - (a^3*(26*A + 23*B)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2
))/(8*d) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(26*A + 23*B))/(8*((13*A*a^3)/4 + (23*B*a^3)/8)))*(26*A + 23*B))/
(8*d)