\(\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [328]

Optimal result

Integrand size = 41, antiderivative size = 304 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{128} a^4 (392 A+352 B+323 C) x+\frac {a^4 (252 A+227 B+208 C) \sin (c+d x)}{35 d}+\frac {a^4 (392 A+352 B+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^4 (2408 A+2208 B+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}-\frac {a^4 (252 A+227 B+208 C) \sin ^3(c+d x)}{105 d} \]

[Out]

1/128*a^4*(392*A+352*B+323*C)*x+1/35*a^4*(252*A+227*B+208*C)*sin(d*x+c)/d+1/128*a^4*(392*A+352*B+323*C)*cos(d*
x+c)*sin(d*x+c)/d+1/2240*a^4*(2408*A+2208*B+2007*C)*cos(d*x+c)^3*sin(d*x+c)/d+1/14*a*(2*B+C)*cos(d*x+c)^3*(a+a
*cos(d*x+c))^3*sin(d*x+c)/d+1/8*C*cos(d*x+c)^3*(a+a*cos(d*x+c))^4*sin(d*x+c)/d+1/336*(56*A+80*B+61*C)*cos(d*x+
c)^3*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d+7/120*(8*A+8*B+7*C)*cos(d*x+c)^3*(a^4+a^4*cos(d*x+c))*sin(d*x+c)/d-1/
105*a^4*(252*A+227*B+208*C)*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {3124, 3055, 3047, 3102, 2827, 2715, 8, 2713} \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {a^4 (252 A+227 B+208 C) \sin ^3(c+d x)}{105 d}+\frac {a^4 (252 A+227 B+208 C) \sin (c+d x)}{35 d}+\frac {a^4 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac {7 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{120 d}+\frac {a^4 (392 A+352 B+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {1}{128} a^4 x (392 A+352 B+323 C)+\frac {(56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{336 d}+\frac {a (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d} \]

[In]

Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(a^4*(392*A + 352*B + 323*C)*x)/128 + (a^4*(252*A + 227*B + 208*C)*Sin[c + d*x])/(35*d) + (a^4*(392*A + 352*B
+ 323*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^4*(2408*A + 2208*B + 2007*C)*Cos[c + d*x]^3*Sin[c + d*x])/(22
40*d) + (a*(2*B + C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(14*d) + (C*Cos[c + d*x]^3*(a + a*Cos
[c + d*x])^4*Sin[c + d*x])/(8*d) + ((56*A + 80*B + 61*C)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x
])/(336*d) + (7*(8*A + 8*B + 7*C)*Cos[c + d*x]^3*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(120*d) - (a^4*(252*A
+ 227*B + 208*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 3124

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/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f
*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*
B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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))^4 \sin (c+d x)}{8 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (a (8 A+3 C)+4 a (2 B+C) \cos (c+d x)) \, dx}{8 a} \\ & = \frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (a^2 (56 A+24 B+33 C)+a^2 (56 A+80 B+61 C) \cos (c+d x)\right ) \, dx}{56 a} \\ & = \frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (3 a^3 (168 A+128 B+127 C)+98 a^3 (8 A+8 B+7 C) \cos (c+d x)\right ) \, dx}{336 a} \\ & = \frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^4 (1624 A+1424 B+1321 C)+3 a^4 (2408 A+2208 B+2007 C) \cos (c+d x)\right ) \, dx}{1680 a} \\ & = \frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {\int \cos ^2(c+d x) \left (3 a^5 (1624 A+1424 B+1321 C)+\left (3 a^5 (1624 A+1424 B+1321 C)+3 a^5 (2408 A+2208 B+2007 C)\right ) \cos (c+d x)+3 a^5 (2408 A+2208 B+2007 C) \cos ^2(c+d x)\right ) \, dx}{1680 a} \\ & = \frac {a^4 (2408 A+2208 B+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {\int \cos ^2(c+d x) \left (105 a^5 (392 A+352 B+323 C)+192 a^5 (252 A+227 B+208 C) \cos (c+d x)\right ) \, dx}{6720 a} \\ & = \frac {a^4 (2408 A+2208 B+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {1}{35} \left (a^4 (252 A+227 B+208 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{64} \left (a^4 (392 A+352 B+323 C)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {a^4 (392 A+352 B+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^4 (2408 A+2208 B+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {1}{128} \left (a^4 (392 A+352 B+323 C)\right ) \int 1 \, dx-\frac {\left (a^4 (252 A+227 B+208 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d} \\ & = \frac {1}{128} a^4 (392 A+352 B+323 C) x+\frac {a^4 (252 A+227 B+208 C) \sin (c+d x)}{35 d}+\frac {a^4 (392 A+352 B+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^4 (2408 A+2208 B+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}-\frac {a^4 (252 A+227 B+208 C) \sin ^3(c+d x)}{105 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.78 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^4 (295680 B c+164640 c C+329280 A d x+295680 B d x+271320 C d x+1680 (352 A+323 B+300 C) \sin (c+d x)+1680 (127 A+124 B+120 C) \sin (2 (c+d x))+80640 A \sin (3 (c+d x))+87920 B \sin (3 (c+d x))+91840 C \sin (3 (c+d x))+25200 A \sin (4 (c+d x))+33600 B \sin (4 (c+d x))+39480 C \sin (4 (c+d x))+5376 A \sin (5 (c+d x))+10416 B \sin (5 (c+d x))+14784 C \sin (5 (c+d x))+560 A \sin (6 (c+d x))+2240 B \sin (6 (c+d x))+4480 C \sin (6 (c+d x))+240 B \sin (7 (c+d x))+960 C \sin (7 (c+d x))+105 C \sin (8 (c+d x)))}{107520 d} \]

[In]

Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(a^4*(295680*B*c + 164640*c*C + 329280*A*d*x + 295680*B*d*x + 271320*C*d*x + 1680*(352*A + 323*B + 300*C)*Sin[
c + d*x] + 1680*(127*A + 124*B + 120*C)*Sin[2*(c + d*x)] + 80640*A*Sin[3*(c + d*x)] + 87920*B*Sin[3*(c + d*x)]
+ 91840*C*Sin[3*(c + d*x)] + 25200*A*Sin[4*(c + d*x)] + 33600*B*Sin[4*(c + d*x)] + 39480*C*Sin[4*(c + d*x)] +
5376*A*Sin[5*(c + d*x)] + 10416*B*Sin[5*(c + d*x)] + 14784*C*Sin[5*(c + d*x)] + 560*A*Sin[6*(c + d*x)] + 2240
*B*Sin[6*(c + d*x)] + 4480*C*Sin[6*(c + d*x)] + 240*B*Sin[7*(c + d*x)] + 960*C*Sin[7*(c + d*x)] + 105*C*Sin[8*
(c + d*x)]))/(107520*d)

Maple [A] (verified)

Time = 12.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.54

method	result	size
parallelrisch	\(\frac {3 \left (\frac {\left (\frac {127 A}{24}+\frac {31 B}{6}+5 C \right ) \sin \left (2 d x +2 c \right )}{2}+\left (A +\frac {157 B}{144}+\frac {41 C}{36}\right ) \sin \left (3 d x +3 c \right )+\frac {\left (\frac {5 A}{4}+\frac {5 B}{3}+\frac {47 C}{24}\right ) \sin \left (4 d x +4 c \right )}{4}+\frac {\left (A +\frac {31 B}{16}+\frac {11 C}{4}\right ) \sin \left (5 d x +5 c \right )}{15}+\frac {\left (\frac {A}{8}+\frac {B}{2}+C \right ) \sin \left (6 d x +6 c \right )}{18}+\frac {\left (\frac {B}{4}+C \right ) \sin \left (7 d x +7 c \right )}{84}+\frac {C \sin \left (8 d x +8 c \right )}{768}+\left (\frac {22 A}{3}+\frac {323 B}{48}+\frac {25 C}{4}\right ) \sin \left (d x +c \right )+\frac {49 x \left (A +\frac {44 B}{49}+\frac {323 C}{392}\right ) d}{12}\right ) a^{4}}{4 d}\)	\(165\)
parts	\(\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {\left (a^{4} A +4 B \,a^{4}+6 C \,a^{4}\right ) \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+C \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{4} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\)	\(349\)
risch	\(\frac {49 a^{4} x A}{16}+\frac {11 a^{4} B x}{4}+\frac {323 a^{4} C x}{128}+\frac {11 \sin \left (d x +c \right ) a^{4} A}{2 d}+\frac {323 \sin \left (d x +c \right ) B \,a^{4}}{64 d}+\frac {75 \sin \left (d x +c \right ) C \,a^{4}}{16 d}+\frac {C \,a^{4} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {\sin \left (7 d x +7 c \right ) B \,a^{4}}{448 d}+\frac {\sin \left (7 d x +7 c \right ) C \,a^{4}}{112 d}+\frac {\sin \left (6 d x +6 c \right ) a^{4} A}{192 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{4}}{48 d}+\frac {\sin \left (6 d x +6 c \right ) C \,a^{4}}{24 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} A}{20 d}+\frac {31 \sin \left (5 d x +5 c \right ) B \,a^{4}}{320 d}+\frac {11 \sin \left (5 d x +5 c \right ) C \,a^{4}}{80 d}+\frac {15 \sin \left (4 d x +4 c \right ) a^{4} A}{64 d}+\frac {5 \sin \left (4 d x +4 c \right ) B \,a^{4}}{16 d}+\frac {47 \sin \left (4 d x +4 c \right ) C \,a^{4}}{128 d}+\frac {3 \sin \left (3 d x +3 c \right ) a^{4} A}{4 d}+\frac {157 \sin \left (3 d x +3 c \right ) B \,a^{4}}{192 d}+\frac {41 \sin \left (3 d x +3 c \right ) C \,a^{4}}{48 d}+\frac {127 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+\frac {31 \sin \left (2 d x +2 c \right ) B \,a^{4}}{16 d}+\frac {15 \sin \left (2 d x +2 c \right ) C \,a^{4}}{8 d}\)	\(392\)
norman	\(\frac {\frac {a^{4} \left (392 A +352 B +323 C \right ) x}{128}+\frac {5053 a^{4} \left (392 A +352 B +323 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6720 d}+\frac {383 a^{4} \left (392 A +352 B +323 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 d}+\frac {23 a^{4} \left (392 A +352 B +323 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {a^{4} \left (392 A +352 B +323 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {7 a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {7 a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {7 a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {7 a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {a^{4} \left (1656 A +1696 B +1725 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {a^{4} \left (21704 A +18528 B +15099 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {a^{4} \left (236936 A +211296 B +206019 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 d}+\frac {a^{4} \left (2277016 A +2090016 B +1872009 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6720 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\)	\(480\)
derivativedivides	\(\frac {a^{4} A \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 )+\frac {B \,a^{4} \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}+C \,a^{4} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {4 a^{4} A \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}+4 B \,a^{4} \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 )+\frac {4 C \,a^{4} \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}+6 a^{4} A \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 {6 B \,a^{4} \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}+6 C \,a^{4} \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 )+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \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 {4 C \,a^{4} \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^{4} A \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^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,a^{4} \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}\)	\(577\)
default	\(\frac {a^{4} A \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 )+\frac {B \,a^{4} \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}+C \,a^{4} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {4 a^{4} A \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}+4 B \,a^{4} \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 )+\frac {4 C \,a^{4} \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}+6 a^{4} A \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 {6 B \,a^{4} \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}+6 C \,a^{4} \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 )+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \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 {4 C \,a^{4} \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^{4} A \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^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,a^{4} \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}\)	\(577\)

[In]

int(cos(d*x+c)^2*(a+cos(d*x+c)*a)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

3/4*(1/2*(127/24*A+31/6*B+5*C)*sin(2*d*x+2*c)+(A+157/144*B+41/36*C)*sin(3*d*x+3*c)+1/4*(5/4*A+5/3*B+47/24*C)*s
in(4*d*x+4*c)+1/15*(A+31/16*B+11/4*C)*sin(5*d*x+5*c)+1/18*(1/8*A+1/2*B+C)*sin(6*d*x+6*c)+1/84*(1/4*B+C)*sin(7*
d*x+7*c)+1/768*C*sin(8*d*x+8*c)+(22/3*A+323/48*B+25/4*C)*sin(d*x+c)+49/12*x*(A+44/49*B+323/392*C)*d)*a^4/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.63 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (392 \, A + 352 \, B + 323 \, C\right )} a^{4} d x + {\left (1680 \, C a^{4} \cos \left (d x + c\right )^{7} + 1920 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (8 \, A + 32 \, B + 55 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 1536 \, {\left (7 \, A + 12 \, B + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (328 \, A + 352 \, B + 323 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 128 \, {\left (252 \, A + 227 \, B + 208 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (392 \, A + 352 \, B + 323 \, C\right )} a^{4} \cos \left (d x + c\right ) + 256 \, {\left (252 \, A + 227 \, B + 208 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{13440 \, d} \]

[In]

integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/13440*(105*(392*A + 352*B + 323*C)*a^4*d*x + (1680*C*a^4*cos(d*x + c)^7 + 1920*(B + 4*C)*a^4*cos(d*x + c)^6
+ 280*(8*A + 32*B + 55*C)*a^4*cos(d*x + c)^5 + 1536*(7*A + 12*B + 13*C)*a^4*cos(d*x + c)^4 + 70*(328*A + 352*B
+ 323*C)*a^4*cos(d*x + c)^3 + 128*(252*A + 227*B + 208*C)*a^4*cos(d*x + c)^2 + 105*(392*A + 352*B + 323*C)*a^
4*cos(d*x + c) + 256*(252*A + 227*B + 208*C)*a^4)*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1640 vs. \(2 (289) = 578\).

Time = 0.86 (sec) , antiderivative size = 1640, normalized size of antiderivative = 5.39 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)**2*(a+a*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)

[Out]

Piecewise((5*A*a**4*x*sin(c + d*x)**6/16 + 15*A*a**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 9*A*a**4*x*sin(c +
d*x)**4/4 + 15*A*a**4*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 9*A*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2/2 + A
*a**4*x*sin(c + d*x)**2/2 + 5*A*a**4*x*cos(c + d*x)**6/16 + 9*A*a**4*x*cos(c + d*x)**4/4 + A*a**4*x*cos(c + d*
x)**2/2 + 5*A*a**4*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 32*A*a**4*sin(c + d*x)**5/(15*d) + 5*A*a**4*sin(c + d
*x)**3*cos(c + d*x)**3/(6*d) + 16*A*a**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*A*a**4*sin(c + d*x)**3*cos(
c + d*x)/(4*d) + 8*A*a**4*sin(c + d*x)**3/(3*d) + 11*A*a**4*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 4*A*a**4*sin
(c + d*x)*cos(c + d*x)**4/d + 15*A*a**4*sin(c + d*x)*cos(c + d*x)**3/(4*d) + 4*A*a**4*sin(c + d*x)*cos(c + d*x
)**2/d + A*a**4*sin(c + d*x)*cos(c + d*x)/(2*d) + 5*B*a**4*x*sin(c + d*x)**6/4 + 15*B*a**4*x*sin(c + d*x)**4*c
os(c + d*x)**2/4 + 3*B*a**4*x*sin(c + d*x)**4/2 + 15*B*a**4*x*sin(c + d*x)**2*cos(c + d*x)**4/4 + 3*B*a**4*x*s
in(c + d*x)**2*cos(c + d*x)**2 + 5*B*a**4*x*cos(c + d*x)**6/4 + 3*B*a**4*x*cos(c + d*x)**4/2 + 16*B*a**4*sin(c
+ d*x)**7/(35*d) + 8*B*a**4*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 5*B*a**4*sin(c + d*x)**5*cos(c + d*x)/(4*
d) + 16*B*a**4*sin(c + d*x)**5/(5*d) + 2*B*a**4*sin(c + d*x)**3*cos(c + d*x)**4/d + 10*B*a**4*sin(c + d*x)**3*
cos(c + d*x)**3/(3*d) + 8*B*a**4*sin(c + d*x)**3*cos(c + d*x)**2/d + 3*B*a**4*sin(c + d*x)**3*cos(c + d*x)/(2*
d) + 2*B*a**4*sin(c + d*x)**3/(3*d) + B*a**4*sin(c + d*x)*cos(c + d*x)**6/d + 11*B*a**4*sin(c + d*x)*cos(c + d
*x)**5/(4*d) + 6*B*a**4*sin(c + d*x)*cos(c + d*x)**4/d + 5*B*a**4*sin(c + d*x)*cos(c + d*x)**3/(2*d) + B*a**4*
sin(c + d*x)*cos(c + d*x)**2/d + 35*C*a**4*x*sin(c + d*x)**8/128 + 35*C*a**4*x*sin(c + d*x)**6*cos(c + d*x)**2
/32 + 15*C*a**4*x*sin(c + d*x)**6/8 + 105*C*a**4*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 45*C*a**4*x*sin(c + d*
x)**4*cos(c + d*x)**2/8 + 3*C*a**4*x*sin(c + d*x)**4/8 + 35*C*a**4*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 45*C
*a**4*x*sin(c + d*x)**2*cos(c + d*x)**4/8 + 3*C*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 35*C*a**4*x*cos(c +
d*x)**8/128 + 15*C*a**4*x*cos(c + d*x)**6/8 + 3*C*a**4*x*cos(c + d*x)**4/8 + 35*C*a**4*sin(c + d*x)**7*cos(c
+ d*x)/(128*d) + 64*C*a**4*sin(c + d*x)**7/(35*d) + 385*C*a**4*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 32*C*
a**4*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 15*C*a**4*sin(c + d*x)**5*cos(c + d*x)/(8*d) + 32*C*a**4*sin(c +
d*x)**5/(15*d) + 511*C*a**4*sin(c + d*x)**3*cos(c + d*x)**5/(384*d) + 8*C*a**4*sin(c + d*x)**3*cos(c + d*x)**4
/d + 5*C*a**4*sin(c + d*x)**3*cos(c + d*x)**3/d + 16*C*a**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 3*C*a**4*s
in(c + d*x)**3*cos(c + d*x)/(8*d) + 93*C*a**4*sin(c + d*x)*cos(c + d*x)**7/(128*d) + 4*C*a**4*sin(c + d*x)*cos
(c + d*x)**6/d + 33*C*a**4*sin(c + d*x)*cos(c + d*x)**5/(8*d) + 4*C*a**4*sin(c + d*x)*cos(c + d*x)**4/d + 5*C*
a**4*sin(c + d*x)*cos(c + d*x)**3/(8*d), Ne(d, 0)), (x*(a*cos(c) + a)**4*(A + B*cos(c) + C*cos(c)**2)*cos(c)**
2, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (286) = 572\).

Time = 0.22 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.90 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {28672 \, {\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^{4} - 560 \, {\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 )} A a^{4} - 143360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 20160 \, {\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^{4} + 26880 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 3072 \, {\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 )} B a^{4} + 43008 \, {\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^{4} - 2240 \, {\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^{4} - 35840 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 13440 \, {\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^{4} - 12288 \, {\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^{4} + 28672 \, {\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^{4} - 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 3360 \, {\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^{4} + 3360 \, {\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^{4}}{107520 \, d} \]

[In]

integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

1/107520*(28672*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 - 560*(4*sin(2*d*x + 2*c)^3 - 6
0*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*A*a^4 - 143360*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^
4 + 20160*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 26880*(2*d*x + 2*c + sin(2*d*x + 2*c
))*A*a^4 - 3072*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*B*a^4 + 43008*(3*
sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^4 - 2240*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*s
in(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*B*a^4 - 35840*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^4 + 13440*(12*d*x +
12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 - 12288*(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^4 + 28672*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^4 - 35*
(128*sin(2*d*x + 2*c)^3 - 840*d*x - 840*c - 3*sin(8*d*x + 8*c) - 168*sin(4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*
C*a^4 - 3360*(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^4 + 3360*(1
2*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4)/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{4} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {1}{128} \, {\left (392 \, A a^{4} + 352 \, B a^{4} + 323 \, C a^{4}\right )} x + \frac {{\left (B a^{4} + 4 \, C a^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {{\left (A a^{4} + 4 \, B a^{4} + 8 \, C a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (16 \, A a^{4} + 31 \, B a^{4} + 44 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (30 \, A a^{4} + 40 \, B a^{4} + 47 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (144 \, A a^{4} + 157 \, B a^{4} + 164 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (127 \, A a^{4} + 124 \, B a^{4} + 120 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (352 \, A a^{4} + 323 \, B a^{4} + 300 \, C a^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]

[In]

integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/1024*C*a^4*sin(8*d*x + 8*c)/d + 1/128*(392*A*a^4 + 352*B*a^4 + 323*C*a^4)*x + 1/448*(B*a^4 + 4*C*a^4)*sin(7*
d*x + 7*c)/d + 1/192*(A*a^4 + 4*B*a^4 + 8*C*a^4)*sin(6*d*x + 6*c)/d + 1/320*(16*A*a^4 + 31*B*a^4 + 44*C*a^4)*s
in(5*d*x + 5*c)/d + 1/128*(30*A*a^4 + 40*B*a^4 + 47*C*a^4)*sin(4*d*x + 4*c)/d + 1/192*(144*A*a^4 + 157*B*a^4 +
164*C*a^4)*sin(3*d*x + 3*c)/d + 1/64*(127*A*a^4 + 124*B*a^4 + 120*C*a^4)*sin(2*d*x + 2*c)/d + 1/64*(352*A*a^4
+ 323*B*a^4 + 300*C*a^4)*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 3.61 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.49 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {49\,A\,a^4}{8}+\frac {11\,B\,a^4}{2}+\frac {323\,C\,a^4}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+\left (\frac {1127\,A\,a^4}{24}+\frac {253\,B\,a^4}{6}+\frac {7429\,C\,a^4}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {18767\,A\,a^4}{120}+\frac {4213\,B\,a^4}{30}+\frac {123709\,C\,a^4}{960}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {35371\,A\,a^4}{120}+\frac {55583\,B\,a^4}{210}+\frac {1632119\,C\,a^4}{6720}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {40661\,A\,a^4}{120}+\frac {21771\,B\,a^4}{70}+\frac {624003\,C\,a^4}{2240}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {29617\,A\,a^4}{120}+\frac {2201\,B\,a^4}{10}+\frac {68673\,C\,a^4}{320}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {2713\,A\,a^4}{24}+\frac {193\,B\,a^4}{2}+\frac {5033\,C\,a^4}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+\frac {53\,B\,a^4}{2}+\frac {1725\,C\,a^4}{64}\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 )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A+352\,B+323\,C\right )}{64\,\left (\frac {49\,A\,a^4}{8}+\frac {11\,B\,a^4}{2}+\frac {323\,C\,a^4}{64}\right )}\right )\,\left (392\,A+352\,B+323\,C\right )}{64\,d}-\frac {a^4\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (392\,A+352\,B+323\,C\right )}{64\,d} \]

[In]

int(cos(c + d*x)^2*(a + a*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)

[Out]

(tan(c/2 + (d*x)/2)^15*((49*A*a^4)/8 + (11*B*a^4)/2 + (323*C*a^4)/64) + tan(c/2 + (d*x)/2)^3*((2713*A*a^4)/24
+ (193*B*a^4)/2 + (5033*C*a^4)/64) + tan(c/2 + (d*x)/2)^13*((1127*A*a^4)/24 + (253*B*a^4)/6 + (7429*C*a^4)/192
) + tan(c/2 + (d*x)/2)^5*((29617*A*a^4)/120 + (2201*B*a^4)/10 + (68673*C*a^4)/320) + tan(c/2 + (d*x)/2)^11*((1
8767*A*a^4)/120 + (4213*B*a^4)/30 + (123709*C*a^4)/960) + tan(c/2 + (d*x)/2)^7*((40661*A*a^4)/120 + (21771*B*a
^4)/70 + (624003*C*a^4)/2240) + tan(c/2 + (d*x)/2)^9*((35371*A*a^4)/120 + (55583*B*a^4)/210 + (1632119*C*a^4)/
6720) + tan(c/2 + (d*x)/2)*((207*A*a^4)/8 + (53*B*a^4)/2 + (1725*C*a^4)/64))/(d*(8*tan(c/2 + (d*x)/2)^2 + 28*t
an(c/2 + (d*x)/2)^4 + 56*tan(c/2 + (d*x)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 + 56*tan(c/2 + (d*x)/2)^10 + 28*tan(c/
2 + (d*x)/2)^12 + 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1)) + (a^4*atan((a^4*tan(c/2 + (d*x)/2)*(3
92*A + 352*B + 323*C))/(64*((49*A*a^4)/8 + (11*B*a^4)/2 + (323*C*a^4)/64)))*(392*A + 352*B + 323*C))/(64*d) -
(a^4*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(392*A + 352*B + 323*C))/(64*d)