3.606 \(\int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=209 \[ -\frac{3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{7 a^3 \cos ^9(c+d x)}{9 d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac{41 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac{41 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{41 a^3 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{41 a^3 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{41 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{41 a^3 x}{1024} \]

[Out]

(41*a^3*x)/1024 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (7*a^3*Cos[c + d*x]^9)/(9*d) - (3*a^3*Cos[c + d*x]^11)/(11*d)
+ (41*a^3*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (41*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + (41*a^3*Cos[c
+ d*x]^5*Sin[c + d*x])/(1920*d) - (41*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (41*a^3*Cos[c + d*x]^7*Sin[c
+ d*x]^3)/(120*d) - (a^3*Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*d)

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Rubi [A]  time = 0.412502, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2873, 2565, 14, 2568, 2635, 8, 270} \[ -\frac{3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{7 a^3 \cos ^9(c+d x)}{9 d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac{41 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac{41 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{41 a^3 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{41 a^3 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{41 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{41 a^3 x}{1024} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]

[Out]

(41*a^3*x)/1024 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (7*a^3*Cos[c + d*x]^9)/(9*d) - (3*a^3*Cos[c + d*x]^11)/(11*d)
+ (41*a^3*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (41*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + (41*a^3*Cos[c
+ d*x]^5*Sin[c + d*x])/(1920*d) - (41*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (41*a^3*Cos[c + d*x]^7*Sin[c
+ d*x]^3)/(120*d) - (a^3*Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*d)

Rule 2873

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cos ^6(c+d x) \sin ^3(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^4(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^5(c+d x)+a^3 \cos ^6(c+d x) \sin ^6(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+a^3 \int \cos ^6(c+d x) \sin ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx\\ &=-\frac{3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{1}{12} \left (5 a^3\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+\frac{1}{10} \left (9 a^3\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{9 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{1}{80} \left (9 a^3\right ) \int \cos ^6(c+d x) \, dx+\frac{1}{8} a^3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{7 a^3 \cos ^9(c+d x)}{9 d}-\frac{3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{3 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac{41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{1}{64} a^3 \int \cos ^6(c+d x) \, dx+\frac{1}{32} \left (3 a^3\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{7 a^3 \cos ^9(c+d x)}{9 d}-\frac{3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{3 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{41 a^3 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{1}{384} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{128} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{7 a^3 \cos ^9(c+d x)}{9 d}-\frac{3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{9 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{41 a^3 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac{41 a^3 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{1}{512} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{256} \left (9 a^3\right ) \int 1 \, dx\\ &=\frac{9 a^3 x}{256}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{7 a^3 \cos ^9(c+d x)}{9 d}-\frac{3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{41 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac{41 a^3 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac{41 a^3 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{\left (5 a^3\right ) \int 1 \, dx}{1024}\\ &=\frac{41 a^3 x}{1024}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{7 a^3 \cos ^9(c+d x)}{9 d}-\frac{3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{41 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac{41 a^3 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac{41 a^3 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}\\ \end{align*}

Mathematica [A]  time = 1.51411, size = 136, normalized size = 0.65 \[ \frac{a^3 (166320 \sin (2 (c+d x))-384615 \sin (4 (c+d x))-83160 \sin (6 (c+d x))+51975 \sin (8 (c+d x))+16632 \sin (10 (c+d x))-1155 \sin (12 (c+d x))-1496880 \cos (c+d x)-572880 \cos (3 (c+d x))+83160 \cos (5 (c+d x))+106920 \cos (7 (c+d x))+3080 \cos (9 (c+d x))-7560 \cos (11 (c+d x))+1247400 c+1136520 d x)}{28385280 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(1247400*c + 1136520*d*x - 1496880*Cos[c + d*x] - 572880*Cos[3*(c + d*x)] + 83160*Cos[5*(c + d*x)] + 1069
20*Cos[7*(c + d*x)] + 3080*Cos[9*(c + d*x)] - 7560*Cos[11*(c + d*x)] + 166320*Sin[2*(c + d*x)] - 384615*Sin[4*
(c + d*x)] - 83160*Sin[6*(c + d*x)] + 51975*Sin[8*(c + d*x)] + 16632*Sin[10*(c + d*x)] - 1155*Sin[12*(c + d*x)
]))/(28385280*d)

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Maple [A]  time = 0.041, size = 272, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{12}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{24}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{64}}+{\frac{\sin \left ( dx+c \right ) }{384} \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}{1024}}+{\frac{5\,c}{1024}} \right ) +3\,{a}^{3} \left ( -1/11\, \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{99}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{693}} \right ) +3\,{a}^{3} \left ( -1/10\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80}}+{\frac{\sin \left ( dx+c \right ) }{160} \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 ) }+{\frac{3\,dx}{256}}+{\frac{3\,c}{256}} \right ) +{a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)^3*(a+a*sin(d*x+c))^3,x)

[Out]

1/d*(a^3*(-1/12*sin(d*x+c)^5*cos(d*x+c)^7-1/24*sin(d*x+c)^3*cos(d*x+c)^7-1/64*sin(d*x+c)*cos(d*x+c)^7+1/384*(c
os(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/1024*d*x+5/1024*c)+3*a^3*(-1/11*sin(d*x+c)^4*cos(d*
x+c)^7-4/99*sin(d*x+c)^2*cos(d*x+c)^7-8/693*cos(d*x+c)^7)+3*a^3*(-1/10*sin(d*x+c)^3*cos(d*x+c)^7-3/80*sin(d*x+
c)*cos(d*x+c)^7+1/160*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+3/256*d*x+3/256*c)+a^3*(-1/9*
sin(d*x+c)^2*cos(d*x+c)^7-2/63*cos(d*x+c)^7))

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Maxima [A]  time = 1.06464, size = 221, normalized size = 1.06 \begin{align*} -\frac{122880 \,{\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 450560 \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 8316 \,{\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 1155 \,{\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{28385280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/28385280*(122880*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^3 - 450560*(7*cos(d*x + c)
^9 - 9*cos(d*x + c)^7)*a^3 - 8316*(32*sin(2*d*x + 2*c)^5 + 120*d*x + 120*c + 5*sin(8*d*x + 8*c) - 40*sin(4*d*x
+ 4*c))*a^3 - 1155*(4*sin(4*d*x + 4*c)^3 + 120*d*x + 120*c + 9*sin(8*d*x + 8*c) - 48*sin(4*d*x + 4*c))*a^3)/d

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Fricas [A]  time = 1.3848, size = 390, normalized size = 1.87 \begin{align*} -\frac{967680 \, a^{3} \cos \left (d x + c\right )^{11} - 2759680 \, a^{3} \cos \left (d x + c\right )^{9} + 2027520 \, a^{3} \cos \left (d x + c\right )^{7} - 142065 \, a^{3} d x + 231 \,{\left (1280 \, a^{3} \cos \left (d x + c\right )^{11} - 7808 \, a^{3} \cos \left (d x + c\right )^{9} + 8496 \, a^{3} \cos \left (d x + c\right )^{7} - 328 \, a^{3} \cos \left (d x + c\right )^{5} - 410 \, a^{3} \cos \left (d x + c\right )^{3} - 615 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/3548160*(967680*a^3*cos(d*x + c)^11 - 2759680*a^3*cos(d*x + c)^9 + 2027520*a^3*cos(d*x + c)^7 - 142065*a^3*
d*x + 231*(1280*a^3*cos(d*x + c)^11 - 7808*a^3*cos(d*x + c)^9 + 8496*a^3*cos(d*x + c)^7 - 328*a^3*cos(d*x + c)
^5 - 410*a^3*cos(d*x + c)^3 - 615*a^3*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 86.8045, size = 699, normalized size = 3.34 \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)**6*sin(d*x+c)**3*(a+a*sin(d*x+c))**3,x)

[Out]

Piecewise((5*a**3*x*sin(c + d*x)**12/1024 + 15*a**3*x*sin(c + d*x)**10*cos(c + d*x)**2/512 + 9*a**3*x*sin(c +
d*x)**10/256 + 75*a**3*x*sin(c + d*x)**8*cos(c + d*x)**4/1024 + 45*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256
+ 25*a**3*x*sin(c + d*x)**6*cos(c + d*x)**6/256 + 45*a**3*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 75*a**3*x*si
n(c + d*x)**4*cos(c + d*x)**8/1024 + 45*a**3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 15*a**3*x*sin(c + d*x)**2
*cos(c + d*x)**10/512 + 45*a**3*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 5*a**3*x*cos(c + d*x)**12/1024 + 9*a**
3*x*cos(c + d*x)**10/256 + 5*a**3*sin(c + d*x)**11*cos(c + d*x)/(1024*d) + 85*a**3*sin(c + d*x)**9*cos(c + d*x
)**3/(3072*d) + 9*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 33*a**3*sin(c + d*x)**7*cos(c + d*x)**5/(512*d)
+ 21*a**3*sin(c + d*x)**7*cos(c + d*x)**3/(128*d) - 33*a**3*sin(c + d*x)**5*cos(c + d*x)**7/(512*d) + 3*a**3*s
in(c + d*x)**5*cos(c + d*x)**5/(10*d) - 3*a**3*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 85*a**3*sin(c + d*x)**3
*cos(c + d*x)**9/(3072*d) - 21*a**3*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - 4*a**3*sin(c + d*x)**2*cos(c + d
*x)**9/(21*d) - a**3*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 5*a**3*sin(c + d*x)*cos(c + d*x)**11/(1024*d) - 9
*a**3*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 8*a**3*cos(c + d*x)**11/(231*d) - 2*a**3*cos(c + d*x)**9/(63*d),
Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)**3*cos(c)**6, True))

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Giac [A]  time = 1.31328, size = 281, normalized size = 1.34 \begin{align*} \frac{41}{1024} \, a^{3} x - \frac{3 \, a^{3} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac{a^{3} \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac{27 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac{3 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac{31 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac{27 \, a^{3} \cos \left (d x + c\right )}{512 \, d} - \frac{a^{3} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} + \frac{3 \, a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{15 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac{3 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac{111 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac{3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

41/1024*a^3*x - 3/11264*a^3*cos(11*d*x + 11*c)/d + 1/9216*a^3*cos(9*d*x + 9*c)/d + 27/7168*a^3*cos(7*d*x + 7*c
)/d + 3/1024*a^3*cos(5*d*x + 5*c)/d - 31/1536*a^3*cos(3*d*x + 3*c)/d - 27/512*a^3*cos(d*x + c)/d - 1/24576*a^3
*sin(12*d*x + 12*c)/d + 3/5120*a^3*sin(10*d*x + 10*c)/d + 15/8192*a^3*sin(8*d*x + 8*c)/d - 3/1024*a^3*sin(6*d*
x + 6*c)/d - 111/8192*a^3*sin(4*d*x + 4*c)/d + 3/512*a^3*sin(2*d*x + 2*c)/d