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

Optimal. Leaf size=241 \[ -\frac{a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d}+\frac{a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac{a^4 (301 A+276 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(7 A+10 B) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{42 d}+\frac{7 (A+B) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{15 d}+\frac{a^4 (49 A+44 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^4 x (49 A+44 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]

[Out]

(a^4*(49*A + 44*B)*x)/16 + (a^4*(252*A + 227*B)*Sin[c + d*x])/(35*d) + (a^4*(49*A + 44*B)*Cos[c + d*x]*Sin[c +
d*x])/(16*d) + (a^4*(301*A + 276*B)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (a*B*Cos[c + d*x]^3*(a + a*Cos[c +
d*x])^3*Sin[c + d*x])/(7*d) + ((7*A + 10*B)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(42*d) +
(7*(A + B)*Cos[c + d*x]^3*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^4*(252*A + 227*B)*Sin[c + d*x]^3)
/(105*d)

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Rubi [A]  time = 0.592742, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {2976, 2968, 3023, 2748, 2635, 8, 2633} \[ -\frac{a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d}+\frac{a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac{a^4 (301 A+276 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(7 A+10 B) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{42 d}+\frac{7 (A+B) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{15 d}+\frac{a^4 (49 A+44 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^4 x (49 A+44 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(49*A + 44*B)*x)/16 + (a^4*(252*A + 227*B)*Sin[c + d*x])/(35*d) + (a^4*(49*A + 44*B)*Cos[c + d*x]*Sin[c +
d*x])/(16*d) + (a^4*(301*A + 276*B)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (a*B*Cos[c + d*x]^3*(a + a*Cos[c +
d*x])^3*Sin[c + d*x])/(7*d) + ((7*A + 10*B)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(42*d) +
(7*(A + B)*Cos[c + d*x]^3*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^4*(252*A + 227*B)*Sin[c + d*x]^3)
/(105*d)

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 (A+B \cos (c+d x)) \, dx &=\frac{a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{1}{7} \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (a (7 A+3 B)+a (7 A+10 B) \cos (c+d x)) \, dx\\ &=\frac{a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{1}{42} \int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (3 a^2 (21 A+16 B)+98 a^2 (A+B) \cos (c+d x)\right ) \, dx\\ &=\frac{a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{210} \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^3 (203 A+178 B)+3 a^3 (301 A+276 B) \cos (c+d x)\right ) \, dx\\ &=\frac{a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{210} \int \cos ^2(c+d x) \left (3 a^4 (203 A+178 B)+\left (3 a^4 (203 A+178 B)+3 a^4 (301 A+276 B)\right ) \cos (c+d x)+3 a^4 (301 A+276 B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{840} \int \cos ^2(c+d x) \left (105 a^4 (49 A+44 B)+24 a^4 (252 A+227 B) \cos (c+d x)\right ) \, dx\\ &=\frac{a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{8} \left (a^4 (49 A+44 B)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{35} \left (a^4 (252 A+227 B)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac{a^4 (49 A+44 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{16} \left (a^4 (49 A+44 B)\right ) \int 1 \, dx-\frac{\left (a^4 (252 A+227 B)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac{1}{16} a^4 (49 A+44 B) x+\frac{a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac{a^4 (49 A+44 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac{a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d}\\ \end{align*}

Mathematica [A]  time = 0.793152, size = 156, normalized size = 0.65 \[ \frac{a^4 (105 (352 A+323 B) \sin (c+d x)+105 (127 A+124 B) \sin (2 (c+d x))+5040 A \sin (3 (c+d x))+1575 A \sin (4 (c+d x))+336 A \sin (5 (c+d x))+35 A \sin (6 (c+d x))+20580 A d x+5495 B \sin (3 (c+d x))+2100 B \sin (4 (c+d x))+651 B \sin (5 (c+d x))+140 B \sin (6 (c+d x))+15 B \sin (7 (c+d x))+18480 B c+18480 B d x)}{6720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(18480*B*c + 20580*A*d*x + 18480*B*d*x + 105*(352*A + 323*B)*Sin[c + d*x] + 105*(127*A + 124*B)*Sin[2*(c
+ d*x)] + 5040*A*Sin[3*(c + d*x)] + 5495*B*Sin[3*(c + d*x)] + 1575*A*Sin[4*(c + d*x)] + 2100*B*Sin[4*(c + d*x)
] + 336*A*Sin[5*(c + d*x)] + 651*B*Sin[5*(c + d*x)] + 35*A*Sin[6*(c + d*x)] + 140*B*Sin[6*(c + d*x)] + 15*B*Si
n[7*(c + d*x)]))/(6720*d)

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Maple [A]  time = 0.067, size = 358, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( A{a}^{4} \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +{\frac{{a}^{4}B\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) }+{\frac{4\,A{a}^{4}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+4\,{a}^{4}B \left ( 1/6\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +6\,A{a}^{4} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{6\,{a}^{4}B\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+{\frac{4\,A{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+4\,{a}^{4}B \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +A{a}^{4} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{\frac{{a}^{4}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(a+cos(d*x+c)*a)^4*(A+B*cos(d*x+c)),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)+1/7*a^4*B*(16/5+co
s(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c)+4/5*A*a^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x
+c)+4*a^4*B*(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)+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)+6/5*a^4*B*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4
/3*A*a^4*(2+cos(d*x+c)^2)*sin(d*x+c)+4*a^4*B*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+A*a^
4*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+1/3*a^4*B*(2+cos(d*x+c)^2)*sin(d*x+c))

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Maxima [A]  time = 1.01811, size = 481, normalized size = 2. \begin{align*} \frac{1792 \,{\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} - 35 \,{\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} - 8960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 1260 \,{\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} + 1680 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 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 )} B a^{4} + 2688 \,{\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} - 140 \,{\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} - 2240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 840 \,{\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}}{6720 \, 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+B*cos(d*x+c)),x, algorithm="maxima")

[Out]

1/6720*(1792*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 - 35*(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))*A*a^4 - 8960*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 + 12
60*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 1680*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4
- 192*(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 + 2688*(3*sin(d*x +
c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^4 - 140*(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^4 - 2240*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^4 + 840*(12*d*x + 12*c + sin(4*
d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4)/d

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Fricas [A]  time = 1.4319, size = 396, normalized size = 1.64 \begin{align*} \frac{105 \,{\left (49 \, A + 44 \, B\right )} a^{4} d x +{\left (240 \, B a^{4} \cos \left (d x + c\right )^{6} + 280 \,{\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{5} + 192 \,{\left (7 \, A + 12 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \,{\left (41 \, A + 44 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \,{\left (252 \, A + 227 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \,{\left (49 \, A + 44 \, B\right )} a^{4} \cos \left (d x + c\right ) + 32 \,{\left (252 \, A + 227 \, B\right )} a^{4}\right )} \sin \left (d x + c\right )}{1680 \, 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+B*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/1680*(105*(49*A + 44*B)*a^4*d*x + (240*B*a^4*cos(d*x + c)^6 + 280*(A + 4*B)*a^4*cos(d*x + c)^5 + 192*(7*A +
12*B)*a^4*cos(d*x + c)^4 + 70*(41*A + 44*B)*a^4*cos(d*x + c)^3 + 16*(252*A + 227*B)*a^4*cos(d*x + c)^2 + 105*(
49*A + 44*B)*a^4*cos(d*x + c) + 32*(252*A + 227*B)*a^4)*sin(d*x + c))/d

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Sympy [A]  time = 11.627, size = 960, normalized size = 3.98 \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+B*cos(d*x+c)),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, Ne(d, 0)), (x*(A + B*cos(c))*(a*cos(c) + a)**4*cos(c)**2, True))

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Giac [A]  time = 1.31712, size = 261, normalized size = 1.08 \begin{align*} \frac{B a^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{1}{16} \,{\left (49 \, A a^{4} + 44 \, B a^{4}\right )} x + \frac{{\left (A a^{4} + 4 \, B a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{{\left (16 \, A a^{4} + 31 \, B a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{5 \,{\left (3 \, A a^{4} + 4 \, B a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (144 \, A a^{4} + 157 \, B a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{{\left (127 \, A a^{4} + 124 \, B a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (352 \, A a^{4} + 323 \, B a^{4}\right )} \sin \left (d x + c\right )}{64 \, 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+B*cos(d*x+c)),x, algorithm="giac")

[Out]

1/448*B*a^4*sin(7*d*x + 7*c)/d + 1/16*(49*A*a^4 + 44*B*a^4)*x + 1/192*(A*a^4 + 4*B*a^4)*sin(6*d*x + 6*c)/d + 1
/320*(16*A*a^4 + 31*B*a^4)*sin(5*d*x + 5*c)/d + 5/64*(3*A*a^4 + 4*B*a^4)*sin(4*d*x + 4*c)/d + 1/192*(144*A*a^4
+ 157*B*a^4)*sin(3*d*x + 3*c)/d + 1/64*(127*A*a^4 + 124*B*a^4)*sin(2*d*x + 2*c)/d + 1/64*(352*A*a^4 + 323*B*a
^4)*sin(d*x + c)/d