3.28 \(\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=279 \[ -\frac{4 a^4 (63 A+52 C) \sin ^3(c+d x)}{105 d}+\frac{4 a^4 (63 A+52 C) \sin (c+d x)}{35 d}+\frac{a^4 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac{(56 A+61 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{336 d}+\frac{7 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{120 d}+\frac{a^4 (392 A+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a^4 x (392 A+323 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d} \]

[Out]

(a^4*(392*A + 323*C)*x)/128 + (4*a^4*(63*A + 52*C)*Sin[c + d*x])/(35*d) + (a^4*(392*A + 323*C)*Cos[c + d*x]*Si
n[c + d*x])/(128*d) + (a^4*(2408*A + 2007*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2240*d) + (a*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
+ 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 + 7*C)*Cos[c + d*x]^3*(a^4
+ a^4*Cos[c + d*x])*Sin[c + d*x])/(120*d) - (4*a^4*(63*A + 52*C)*Sin[c + d*x]^3)/(105*d)

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Rubi [A]  time = 0.792916, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {3046, 2976, 2968, 3023, 2748, 2635, 8, 2633} \[ -\frac{4 a^4 (63 A+52 C) \sin ^3(c+d x)}{105 d}+\frac{4 a^4 (63 A+52 C) \sin (c+d x)}{35 d}+\frac{a^4 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac{(56 A+61 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{336 d}+\frac{7 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{120 d}+\frac{a^4 (392 A+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a^4 x (392 A+323 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(392*A + 323*C)*x)/128 + (4*a^4*(63*A + 52*C)*Sin[c + d*x])/(35*d) + (a^4*(392*A + 323*C)*Cos[c + d*x]*Si
n[c + d*x])/(128*d) + (a^4*(2408*A + 2007*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2240*d) + (a*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
+ 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 + 7*C)*Cos[c + d*x]^3*(a^4
+ a^4*Cos[c + d*x])*Sin[c + d*x])/(120*d) - (4*a^4*(63*A + 52*C)*Sin[c + d*x]^3)/(105*d)

Rule 3046

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*Simp
[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]

Rule 2976

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)))*S
in[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 2968

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 3023

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

Rule 2748

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 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 2633

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]

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\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 C \cos (c+d x)) \, dx}{8 a}\\ &=\frac{a 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+33 C)+a^2 (56 A+61 C) \cos (c+d x)\right ) \, dx}{56 a}\\ &=\frac{a 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+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+127 C)+98 a^3 (8 A+7 C) \cos (c+d x)\right ) \, dx}{336 a}\\ &=\frac{a 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+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+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+1321 C)+3 a^4 (2408 A+2007 C) \cos (c+d x)\right ) \, dx}{1680 a}\\ &=\frac{a 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+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+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+1321 C)+\left (3 a^5 (1624 A+1321 C)+3 a^5 (2408 A+2007 C)\right ) \cos (c+d x)+3 a^5 (2408 A+2007 C) \cos ^2(c+d x)\right ) \, dx}{1680 a}\\ &=\frac{a^4 (2408 A+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac{a 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+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+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+323 C)+768 a^5 (63 A+52 C) \cos (c+d x)\right ) \, dx}{6720 a}\\ &=\frac{a^4 (2408 A+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac{a 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+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+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 (4 a^4 (63 A+52 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{64} \left (a^4 (392 A+323 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{a^4 (392 A+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{a^4 (2408 A+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac{a 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+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+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+323 C)\right ) \int 1 \, dx-\frac{\left (4 a^4 (63 A+52 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac{1}{128} a^4 (392 A+323 C) x+\frac{4 a^4 (63 A+52 C) \sin (c+d x)}{35 d}+\frac{a^4 (392 A+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{a^4 (2408 A+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac{a 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+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+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}-\frac{4 a^4 (63 A+52 C) \sin ^3(c+d x)}{105 d}\\ \end{align*}

Mathematica [A]  time = 0.928706, size = 167, normalized size = 0.6 \[ \frac{a^4 (6720 (88 A+75 C) \sin (c+d x)+1680 (127 A+120 C) \sin (2 (c+d x))+80640 A \sin (3 (c+d x))+25200 A \sin (4 (c+d x))+5376 A \sin (5 (c+d x))+560 A \sin (6 (c+d x))+329280 A d x+91840 C \sin (3 (c+d x))+39480 C \sin (4 (c+d x))+14784 C \sin (5 (c+d x))+4480 C \sin (6 (c+d x))+960 C \sin (7 (c+d x))+105 C \sin (8 (c+d x))+164640 c C+271320 C d x)}{107520 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(164640*c*C + 329280*A*d*x + 271320*C*d*x + 6720*(88*A + 75*C)*Sin[c + d*x] + 1680*(127*A + 120*C)*Sin[2*
(c + d*x)] + 80640*A*Sin[3*(c + d*x)] + 91840*C*Sin[3*(c + d*x)] + 25200*A*Sin[4*(c + d*x)] + 39480*C*Sin[4*(c
+ d*x)] + 5376*A*Sin[5*(c + d*x)] + 14784*C*Sin[5*(c + d*x)] + 560*A*Sin[6*(c + d*x)] + 4480*C*Sin[6*(c + d*x
)] + 960*C*Sin[7*(c + d*x)] + 105*C*Sin[8*(c + d*x)]))/(107520*d)

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Maple [A]  time = 0.057, size = 393, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(A*a^4*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+a^4*C*(1/8*(cos(d*
x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos(d*x+c))*sin(d*x+c)+35/128*d*x+35/128*c)+4/5*A*a^4*(8/3+co
s(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4/7*a^4*C*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*
x+c)+6*A*a^4*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+6*a^4*C*(1/6*(cos(d*x+c)^5+5/4*cos(d
*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+4/3*A*a^4*(2+cos(d*x+c)^2)*sin(d*x+c)+4/5*a^4*C*(8/3+cos(
d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+A*a^4*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+a^4*C*(1/4*(cos(d*x+c)^3
+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c))

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Maxima [A]  time = 1.06925, size = 531, normalized size = 1.9 \begin{align*} \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} - 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^4*(A+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 - 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*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*
x + 2*c))*C*a^4)/d

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Fricas [A]  time = 1.53497, size = 448, normalized size = 1.61 \begin{align*} \frac{105 \,{\left (392 \, A + 323 \, C\right )} a^{4} d x +{\left (1680 \, C a^{4} \cos \left (d x + c\right )^{7} + 7680 \, C a^{4} \cos \left (d x + c\right )^{6} + 280 \,{\left (8 \, A + 55 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 1536 \,{\left (7 \, A + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \,{\left (328 \, A + 323 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 512 \,{\left (63 \, A + 52 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \,{\left (392 \, A + 323 \, C\right )} a^{4} \cos \left (d x + c\right ) + 1024 \,{\left (63 \, A + 52 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{13440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13440*(105*(392*A + 323*C)*a^4*d*x + (1680*C*a^4*cos(d*x + c)^7 + 7680*C*a^4*cos(d*x + c)^6 + 280*(8*A + 55*
C)*a^4*cos(d*x + c)^5 + 1536*(7*A + 13*C)*a^4*cos(d*x + c)^4 + 70*(328*A + 323*C)*a^4*cos(d*x + c)^3 + 512*(63
*A + 52*C)*a^4*cos(d*x + c)^2 + 105*(392*A + 323*C)*a^4*cos(d*x + c) + 1024*(63*A + 52*C)*a^4)*sin(d*x + c))/d

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Sympy [A]  time = 20.364, size = 1149, normalized size = 4.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*(a+a*cos(d*x+c))**4*(A+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) + 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) + 3
2*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*sin(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 + C*cos(c)**2)*(a*cos(c) + a)**4*co
s(c)**2, True))

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Giac [A]  time = 1.24415, size = 285, normalized size = 1.02 \begin{align*} \frac{C a^{4} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac{C a^{4} \sin \left (7 \, d x + 7 \, c\right )}{112 \, d} + \frac{1}{128} \,{\left (392 \, A a^{4} + 323 \, C a^{4}\right )} x + \frac{{\left (A a^{4} + 8 \, C a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{{\left (4 \, A a^{4} + 11 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{{\left (30 \, A a^{4} + 47 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{{\left (36 \, A a^{4} + 41 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (127 \, A a^{4} + 120 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (88 \, A a^{4} + 75 \, C a^{4}\right )} \sin \left (d x + c\right )}{16 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1024*C*a^4*sin(8*d*x + 8*c)/d + 1/112*C*a^4*sin(7*d*x + 7*c)/d + 1/128*(392*A*a^4 + 323*C*a^4)*x + 1/192*(A*
a^4 + 8*C*a^4)*sin(6*d*x + 6*c)/d + 1/80*(4*A*a^4 + 11*C*a^4)*sin(5*d*x + 5*c)/d + 1/128*(30*A*a^4 + 47*C*a^4)
*sin(4*d*x + 4*c)/d + 1/48*(36*A*a^4 + 41*C*a^4)*sin(3*d*x + 3*c)/d + 1/64*(127*A*a^4 + 120*C*a^4)*sin(2*d*x +
2*c)/d + 1/16*(88*A*a^4 + 75*C*a^4)*sin(d*x + c)/d