\(\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \, dx\) [18]

Optimal result

Integrand size = 33, antiderivative size = 237 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{16} a^3 (26 A+21 C) x+\frac {a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac {a^3 (26 A+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}-\frac {a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d} \]

[Out]

1/16*a^3*(26*A+21*C)*x+1/35*a^3*(133*A+108*C)*sin(d*x+c)/d+1/16*a^3*(26*A+21*C)*cos(d*x+c)*sin(d*x+c)/d+1/280*
a^3*(154*A+129*C)*cos(d*x+c)^3*sin(d*x+c)/d+1/7*C*cos(d*x+c)^3*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+1/14*C*cos(d*x+
c)^3*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/a/d+1/5*(A+C)*cos(d*x+c)^3*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d-1/105*a^3*
(133*A+108*C)*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3125, 3055, 3047, 3102, 2827, 2715, 8, 2713} \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d}+\frac {a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac {a^3 (154 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {(A+C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}+\frac {a^3 (26 A+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^3 x (26 A+21 C)+\frac {C \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{14 a d}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]

[In]

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

[Out]

(a^3*(26*A + 21*C)*x)/16 + (a^3*(133*A + 108*C)*Sin[c + d*x])/(35*d) + (a^3*(26*A + 21*C)*Cos[c + d*x]*Sin[c +
d*x])/(16*d) + (a^3*(154*A + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d
*x])^3*Sin[c + d*x])/(7*d) + (C*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(14*a*d) + ((A + C)*Co
s[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(5*d) - (a^3*(133*A + 108*C)*Sin[c + d*x]^3)/(105*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]

Rule 3125

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Si
mp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^
(-1)] && NeQ[m + n + 2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (a (7 A+3 C)+3 a C \cos (c+d x)) \, dx}{7 a} \\ & = \frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (3 a^2 (14 A+9 C)+42 a^2 (A+C) \cos (c+d x)\right ) \, dx}{42 a} \\ & = \frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^3 (112 A+87 C)+3 a^3 (154 A+129 C) \cos (c+d x)\right ) \, dx}{210 a} \\ & = \frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {\int \cos ^2(c+d x) \left (3 a^4 (112 A+87 C)+\left (3 a^4 (112 A+87 C)+3 a^4 (154 A+129 C)\right ) \cos (c+d x)+3 a^4 (154 A+129 C) \cos ^2(c+d x)\right ) \, dx}{210 a} \\ & = \frac {a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {\int \cos ^2(c+d x) \left (105 a^4 (26 A+21 C)+24 a^4 (133 A+108 C) \cos (c+d x)\right ) \, dx}{840 a} \\ & = \frac {a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {1}{8} \left (a^3 (26 A+21 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{35} \left (a^3 (133 A+108 C)\right ) \int \cos ^3(c+d x) \, dx \\ & = \frac {a^3 (26 A+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {1}{16} \left (a^3 (26 A+21 C)\right ) \int 1 \, dx-\frac {\left (a^3 (133 A+108 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d} \\ & = \frac {1}{16} a^3 (26 A+21 C) x+\frac {a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac {a^3 (26 A+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}-\frac {a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.61 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^3 (5460 c C+10920 A d x+8820 C d x+105 (184 A+155 C) \sin (c+d x)+105 (64 A+61 C) \sin (2 (c+d x))+2380 A \sin (3 (c+d x))+2835 C \sin (3 (c+d x))+630 A \sin (4 (c+d x))+1155 C \sin (4 (c+d x))+84 A \sin (5 (c+d x))+399 C \sin (5 (c+d x))+105 C \sin (6 (c+d x))+15 C \sin (7 (c+d x)))}{6720 d} \]

[In]

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

[Out]

(a^3*(5460*c*C + 10920*A*d*x + 8820*C*d*x + 105*(184*A + 155*C)*Sin[c + d*x] + 105*(64*A + 61*C)*Sin[2*(c + d*
x)] + 2380*A*Sin[3*(c + d*x)] + 2835*C*Sin[3*(c + d*x)] + 630*A*Sin[4*(c + d*x)] + 1155*C*Sin[4*(c + d*x)] + 8
4*A*Sin[5*(c + d*x)] + 399*C*Sin[5*(c + d*x)] + 105*C*Sin[6*(c + d*x)] + 15*C*Sin[7*(c + d*x)]))/(6720*d)

Maple [A] (verified)

Time = 9.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.52

method	result	size
parallelrisch	\(\frac {3 a^{3} \left (\left (\frac {32 A}{3}+\frac {61 C}{6}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {34 A}{9}+\frac {9 C}{2}\right ) \sin \left (3 d x +3 c \right )+\left (A +\frac {11 C}{6}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {2 A}{15}+\frac {19 C}{30}\right ) \sin \left (5 d x +5 c \right )+\frac {\sin \left (6 d x +6 c \right ) C}{6}+\frac {\sin \left (7 d x +7 c \right ) C}{42}+\left (\frac {92 A}{3}+\frac {155 C}{6}\right ) \sin \left (d x +c \right )+\frac {52 \left (A +\frac {21 C}{26}\right ) x d}{3}\right )}{32 d}\)	\(123\)
risch	\(\frac {13 a^{3} A x}{8}+\frac {21 a^{3} C x}{16}+\frac {23 a^{3} A \sin \left (d x +c \right )}{8 d}+\frac {155 a^{3} C \sin \left (d x +c \right )}{64 d}+\frac {C \,a^{3} \sin \left (7 d x +7 c \right )}{448 d}+\frac {C \,a^{3} \sin \left (6 d x +6 c \right )}{64 d}+\frac {\sin \left (5 d x +5 c \right ) A \,a^{3}}{80 d}+\frac {19 \sin \left (5 d x +5 c \right ) C \,a^{3}}{320 d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{3}}{32 d}+\frac {11 \sin \left (4 d x +4 c \right ) C \,a^{3}}{64 d}+\frac {17 \sin \left (3 d x +3 c \right ) A \,a^{3}}{48 d}+\frac {27 \sin \left (3 d x +3 c \right ) C \,a^{3}}{64 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3}}{d}+\frac {61 \sin \left (2 d x +2 c \right ) C \,a^{3}}{64 d}\)	\(225\)
parts	\(\frac {\left (A \,a^{3}+3 C \,a^{3}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (3 A \,a^{3}+C \,a^{3}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{3} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}+\frac {3 C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(245\)
derivativedivides	\(\frac {\frac {A \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {C \,a^{3} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+3 A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {3 C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(286\)
default	\(\frac {\frac {A \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {C \,a^{3} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+3 A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {3 C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(286\)
norman	\(\frac {\frac {a^{3} \left (26 A +21 C \right ) x}{16}+\frac {283 a^{3} \left (26 A +21 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}+\frac {5 a^{3} \left (26 A +21 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{3} \left (26 A +21 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {7 a^{3} \left (26 A +21 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{3} \left (26 A +21 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{3} \left (26 A +21 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{3} \left (26 A +21 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{3} \left (26 A +21 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {7 a^{3} \left (26 A +21 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{3} \left (26 A +21 C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{3} \left (102 A +107 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {16 a^{3} \left (203 A +163 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {a^{3} \left (286 A +183 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{3} \left (10178 A +9033 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\)	\(379\)

[In]

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

[Out]

3/32*a^3*((32/3*A+61/6*C)*sin(2*d*x+2*c)+(34/9*A+9/2*C)*sin(3*d*x+3*c)+(A+11/6*C)*sin(4*d*x+4*c)+(2/15*A+19/30
*C)*sin(5*d*x+5*c)+1/6*sin(6*d*x+6*c)*C+1/42*sin(7*d*x+7*c)*C+(92/3*A+155/6*C)*sin(d*x+c)+52/3*(A+21/26*C)*x*d
)/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.62 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (26 \, A + 21 \, C\right )} a^{3} d x + {\left (240 \, C a^{3} \cos \left (d x + c\right )^{6} + 840 \, C a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (7 \, A + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 210 \, {\left (6 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (133 \, A + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 105 \, {\left (26 \, A + 21 \, C\right )} a^{3} \cos \left (d x + c\right ) + 32 \, {\left (133 \, A + 108 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{1680 \, d} \]

[In]

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

[Out]

1/1680*(105*(26*A + 21*C)*a^3*d*x + (240*C*a^3*cos(d*x + c)^6 + 840*C*a^3*cos(d*x + c)^5 + 48*(7*A + 27*C)*a^3
*cos(d*x + c)^4 + 210*(6*A + 7*C)*a^3*cos(d*x + c)^3 + 16*(133*A + 108*C)*a^3*cos(d*x + c)^2 + 105*(26*A + 21*
C)*a^3*cos(d*x + c) + 32*(133*A + 108*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. 750 vs. \(2 (218) = 436\).

Time = 0.55 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.16 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, 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 {15 C a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {45 C a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 C a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {45 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {15 C a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 C a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {16 C a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 C a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {15 C a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {8 C a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {2 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {4 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} + \frac {33 C a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {3 C a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\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+C*cos(d*x+c)**2),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) + 15*C*a**3*x*sin(c + d*x)**6/16 + 45*C*a**3*x
*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*C*a**3*x*sin(c + d*x)**4/8 + 45*C*a**3*x*sin(c + d*x)**2*cos(c + d*x)*
*4/16 + 3*C*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 15*C*a**3*x*cos(c + d*x)**6/16 + 3*C*a**3*x*cos(c + d*x
)**4/8 + 16*C*a**3*sin(c + d*x)**7/(35*d) + 8*C*a**3*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 15*C*a**3*sin(c +
d*x)**5*cos(c + d*x)/(16*d) + 8*C*a**3*sin(c + d*x)**5/(5*d) + 2*C*a**3*sin(c + d*x)**3*cos(c + d*x)**4/d + 5
*C*a**3*sin(c + d*x)**3*cos(c + d*x)**3/(2*d) + 4*C*a**3*sin(c + d*x)**3*cos(c + d*x)**2/d + 3*C*a**3*sin(c +
d*x)**3*cos(c + d*x)/(8*d) + C*a**3*sin(c + d*x)*cos(c + d*x)**6/d + 33*C*a**3*sin(c + d*x)*cos(c + d*x)**5/(1
6*d) + 3*C*a**3*sin(c + d*x)*cos(c + d*x)**4/d + 5*C*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d), Ne(d, 0)), (x*(A
+ C*cos(c)**2)*(a*cos(c) + a)**3*cos(c)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.20 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {448 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} - 6720 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 630 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{3} + 1344 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} - 105 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3}}{6720 \, d} \]

[In]

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

[Out]

1/6720*(448*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^3 - 6720*(sin(d*x + c)^3 - 3*sin(d*x
+ c))*A*a^3 + 630*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 + 1680*(2*d*x + 2*c + sin(2*d*
x + 2*c))*A*a^3 - 192*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*C*a^3 + 134
4*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^3 - 105*(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*a^3 + 210*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))
*C*a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.78 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{3} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {C a^{3} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} + \frac {1}{16} \, {\left (26 \, A a^{3} + 21 \, C a^{3}\right )} x + \frac {{\left (4 \, A a^{3} + 19 \, C a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (6 \, A a^{3} + 11 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (68 \, A a^{3} + 81 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (64 \, A a^{3} + 61 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (184 \, A a^{3} + 155 \, C a^{3}\right )} \sin \left (d x + c\right )}{64 \, d} \]

[In]

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

[Out]

1/448*C*a^3*sin(7*d*x + 7*c)/d + 1/64*C*a^3*sin(6*d*x + 6*c)/d + 1/16*(26*A*a^3 + 21*C*a^3)*x + 1/320*(4*A*a^3
+ 19*C*a^3)*sin(5*d*x + 5*c)/d + 1/64*(6*A*a^3 + 11*C*a^3)*sin(4*d*x + 4*c)/d + 1/192*(68*A*a^3 + 81*C*a^3)*s
in(3*d*x + 3*c)/d + 1/64*(64*A*a^3 + 61*C*a^3)*sin(2*d*x + 2*c)/d + 1/64*(184*A*a^3 + 155*C*a^3)*sin(d*x + c)/
d

Mupad [B] (verification not implemented)

Time = 2.45 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.49 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {13\,A\,a^3}{4}+\frac {21\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {65\,A\,a^3}{3}+\frac {35\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3679\,A\,a^3}{60}+\frac {1981\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {464\,A\,a^3}{5}+\frac {2608\,C\,a^3}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {5089\,A\,a^3}{60}+\frac {3011\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {143\,A\,a^3}{3}+\frac {61\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,A\,a^3}{4}+\frac {107\,C\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^3\,\left (26\,A+21\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (26\,A+21\,C\right )}{8\,\left (\frac {13\,A\,a^3}{4}+\frac {21\,C\,a^3}{8}\right )}\right )\,\left (26\,A+21\,C\right )}{8\,d} \]

[In]

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

[Out]

(tan(c/2 + (d*x)/2)*((51*A*a^3)/4 + (107*C*a^3)/8) + tan(c/2 + (d*x)/2)^13*((13*A*a^3)/4 + (21*C*a^3)/8) + tan
(c/2 + (d*x)/2)^11*((65*A*a^3)/3 + (35*C*a^3)/2) + tan(c/2 + (d*x)/2)^3*((143*A*a^3)/3 + (61*C*a^3)/2) + tan(c
/2 + (d*x)/2)^7*((464*A*a^3)/5 + (2608*C*a^3)/35) + tan(c/2 + (d*x)/2)^9*((3679*A*a^3)/60 + (1981*C*a^3)/40) +
tan(c/2 + (d*x)/2)^5*((5089*A*a^3)/60 + (3011*C*a^3)/40))/(d*(7*tan(c/2 + (d*x)/2)^2 + 21*tan(c/2 + (d*x)/2)^
4 + 35*tan(c/2 + (d*x)/2)^6 + 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7*tan(c/2 + (d*x)/2)^12 + t
an(c/2 + (d*x)/2)^14 + 1)) - (a^3*(26*A + 21*C)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(8*d) + (a^3*atan((a^3*t
an(c/2 + (d*x)/2)*(26*A + 21*C))/(8*((13*A*a^3)/4 + (21*C*a^3)/8)))*(26*A + 21*C))/(8*d)