3.739 \(\int \frac{\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=161 \[ -\frac{3 \cos ^7(c+d x)}{7 a^3 d}+\frac{7 \cos ^5(c+d x)}{5 a^3 d}-\frac{4 \cos ^3(c+d x)}{3 a^3 d}+\frac{\sin ^5(c+d x) \cos ^3(c+d x)}{8 a^3 d}+\frac{29 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a^3 d}+\frac{29 \sin (c+d x) \cos ^3(c+d x)}{64 a^3 d}-\frac{29 \sin (c+d x) \cos (c+d x)}{128 a^3 d}-\frac{29 x}{128 a^3} \]

[Out]

(-29*x)/(128*a^3) - (4*Cos[c + d*x]^3)/(3*a^3*d) + (7*Cos[c + d*x]^5)/(5*a^3*d) - (3*Cos[c + d*x]^7)/(7*a^3*d)
- (29*Cos[c + d*x]*Sin[c + d*x])/(128*a^3*d) + (29*Cos[c + d*x]^3*Sin[c + d*x])/(64*a^3*d) + (29*Cos[c + d*x]
^3*Sin[c + d*x]^3)/(48*a^3*d) + (Cos[c + d*x]^3*Sin[c + d*x]^5)/(8*a^3*d)

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Rubi [A]  time = 0.477072, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2875, 2873, 2565, 14, 2568, 2635, 8, 270} \[ -\frac{3 \cos ^7(c+d x)}{7 a^3 d}+\frac{7 \cos ^5(c+d x)}{5 a^3 d}-\frac{4 \cos ^3(c+d x)}{3 a^3 d}+\frac{\sin ^5(c+d x) \cos ^3(c+d x)}{8 a^3 d}+\frac{29 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a^3 d}+\frac{29 \sin (c+d x) \cos ^3(c+d x)}{64 a^3 d}-\frac{29 \sin (c+d x) \cos (c+d x)}{128 a^3 d}-\frac{29 x}{128 a^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]

[Out]

(-29*x)/(128*a^3) - (4*Cos[c + d*x]^3)/(3*a^3*d) + (7*Cos[c + d*x]^5)/(5*a^3*d) - (3*Cos[c + d*x]^7)/(7*a^3*d)
- (29*Cos[c + d*x]*Sin[c + d*x])/(128*a^3*d) + (29*Cos[c + d*x]^3*Sin[c + d*x])/(64*a^3*d) + (29*Cos[c + d*x]
^3*Sin[c + d*x]^3)/(48*a^3*d) + (Cos[c + d*x]^3*Sin[c + d*x]^5)/(8*a^3*d)

Rule 2875

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[((g*Cos[e + f*x])^(2*m + 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] && ILtQ[m, 0]

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 \frac{\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \cos ^2(c+d x) \sin ^3(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac{\int \left (a^3 \cos ^2(c+d x) \sin ^3(c+d x)-3 a^3 \cos ^2(c+d x) \sin ^4(c+d x)+3 a^3 \cos ^2(c+d x) \sin ^5(c+d x)-a^3 \cos ^2(c+d x) \sin ^6(c+d x)\right ) \, dx}{a^6}\\ &=\frac{\int \cos ^2(c+d x) \sin ^3(c+d x) \, dx}{a^3}-\frac{\int \cos ^2(c+d x) \sin ^6(c+d x) \, dx}{a^3}-\frac{3 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{a^3}+\frac{3 \int \cos ^2(c+d x) \sin ^5(c+d x) \, dx}{a^3}\\ &=\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{2 a^3 d}+\frac{\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac{5 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{8 a^3}-\frac{3 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{2 a^3}-\frac{\operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=\frac{3 \cos ^3(c+d x) \sin (c+d x)}{8 a^3 d}+\frac{29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac{\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac{5 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{16 a^3}-\frac{3 \int \cos ^2(c+d x) \, dx}{8 a^3}-\frac{\operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=-\frac{4 \cos ^3(c+d x)}{3 a^3 d}+\frac{7 \cos ^5(c+d x)}{5 a^3 d}-\frac{3 \cos ^7(c+d x)}{7 a^3 d}-\frac{3 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac{29 \cos ^3(c+d x) \sin (c+d x)}{64 a^3 d}+\frac{29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac{\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac{5 \int \cos ^2(c+d x) \, dx}{64 a^3}-\frac{3 \int 1 \, dx}{16 a^3}\\ &=-\frac{3 x}{16 a^3}-\frac{4 \cos ^3(c+d x)}{3 a^3 d}+\frac{7 \cos ^5(c+d x)}{5 a^3 d}-\frac{3 \cos ^7(c+d x)}{7 a^3 d}-\frac{29 \cos (c+d x) \sin (c+d x)}{128 a^3 d}+\frac{29 \cos ^3(c+d x) \sin (c+d x)}{64 a^3 d}+\frac{29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac{\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac{5 \int 1 \, dx}{128 a^3}\\ &=-\frac{29 x}{128 a^3}-\frac{4 \cos ^3(c+d x)}{3 a^3 d}+\frac{7 \cos ^5(c+d x)}{5 a^3 d}-\frac{3 \cos ^7(c+d x)}{7 a^3 d}-\frac{29 \cos (c+d x) \sin (c+d x)}{128 a^3 d}+\frac{29 \cos ^3(c+d x) \sin (c+d x)}{64 a^3 d}+\frac{29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac{\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}\\ \end{align*}

Mathematica [B]  time = 4.18506, size = 482, normalized size = 2.99 \[ \frac{-48720 d x \sin \left (\frac{c}{2}\right )+38640 \sin \left (\frac{c}{2}+d x\right )-38640 \sin \left (\frac{3 c}{2}+d x\right )+6720 \sin \left (\frac{3 c}{2}+2 d x\right )+6720 \sin \left (\frac{5 c}{2}+2 d x\right )+3920 \sin \left (\frac{5 c}{2}+3 d x\right )-3920 \sin \left (\frac{7 c}{2}+3 d x\right )+5880 \sin \left (\frac{7 c}{2}+4 d x\right )+5880 \sin \left (\frac{9 c}{2}+4 d x\right )-4368 \sin \left (\frac{9 c}{2}+5 d x\right )+4368 \sin \left (\frac{11 c}{2}+5 d x\right )-2240 \sin \left (\frac{11 c}{2}+6 d x\right )-2240 \sin \left (\frac{13 c}{2}+6 d x\right )+720 \sin \left (\frac{13 c}{2}+7 d x\right )-720 \sin \left (\frac{15 c}{2}+7 d x\right )+105 \sin \left (\frac{15 c}{2}+8 d x\right )+105 \sin \left (\frac{17 c}{2}+8 d x\right )+84 \cos \left (\frac{c}{2}\right ) (12870 c-580 d x-7)-38640 \cos \left (\frac{c}{2}+d x\right )-38640 \cos \left (\frac{3 c}{2}+d x\right )+6720 \cos \left (\frac{3 c}{2}+2 d x\right )-6720 \cos \left (\frac{5 c}{2}+2 d x\right )-3920 \cos \left (\frac{5 c}{2}+3 d x\right )-3920 \cos \left (\frac{7 c}{2}+3 d x\right )+5880 \cos \left (\frac{7 c}{2}+4 d x\right )-5880 \cos \left (\frac{9 c}{2}+4 d x\right )+4368 \cos \left (\frac{9 c}{2}+5 d x\right )+4368 \cos \left (\frac{11 c}{2}+5 d x\right )-2240 \cos \left (\frac{11 c}{2}+6 d x\right )+2240 \cos \left (\frac{13 c}{2}+6 d x\right )-720 \cos \left (\frac{13 c}{2}+7 d x\right )-720 \cos \left (\frac{15 c}{2}+7 d x\right )+105 \cos \left (\frac{15 c}{2}+8 d x\right )-105 \cos \left (\frac{17 c}{2}+8 d x\right )+1081080 c \sin \left (\frac{c}{2}\right )-998928 \sin \left (\frac{c}{2}\right )}{215040 a^3 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]

[Out]

(84*(-7 + 12870*c - 580*d*x)*Cos[c/2] - 38640*Cos[c/2 + d*x] - 38640*Cos[(3*c)/2 + d*x] + 6720*Cos[(3*c)/2 + 2
*d*x] - 6720*Cos[(5*c)/2 + 2*d*x] - 3920*Cos[(5*c)/2 + 3*d*x] - 3920*Cos[(7*c)/2 + 3*d*x] + 5880*Cos[(7*c)/2 +
4*d*x] - 5880*Cos[(9*c)/2 + 4*d*x] + 4368*Cos[(9*c)/2 + 5*d*x] + 4368*Cos[(11*c)/2 + 5*d*x] - 2240*Cos[(11*c)
/2 + 6*d*x] + 2240*Cos[(13*c)/2 + 6*d*x] - 720*Cos[(13*c)/2 + 7*d*x] - 720*Cos[(15*c)/2 + 7*d*x] + 105*Cos[(15
*c)/2 + 8*d*x] - 105*Cos[(17*c)/2 + 8*d*x] - 998928*Sin[c/2] + 1081080*c*Sin[c/2] - 48720*d*x*Sin[c/2] + 38640
*Sin[c/2 + d*x] - 38640*Sin[(3*c)/2 + d*x] + 6720*Sin[(3*c)/2 + 2*d*x] + 6720*Sin[(5*c)/2 + 2*d*x] + 3920*Sin[
(5*c)/2 + 3*d*x] - 3920*Sin[(7*c)/2 + 3*d*x] + 5880*Sin[(7*c)/2 + 4*d*x] + 5880*Sin[(9*c)/2 + 4*d*x] - 4368*Si
n[(9*c)/2 + 5*d*x] + 4368*Sin[(11*c)/2 + 5*d*x] - 2240*Sin[(11*c)/2 + 6*d*x] - 2240*Sin[(13*c)/2 + 6*d*x] + 72
0*Sin[(13*c)/2 + 7*d*x] - 720*Sin[(15*c)/2 + 7*d*x] + 105*Sin[(15*c)/2 + 8*d*x] + 105*Sin[(17*c)/2 + 8*d*x])/(
215040*a^3*d*(Cos[c/2] + Sin[c/2]))

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Maple [B]  time = 0.106, size = 517, normalized size = 3.2 \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)^8*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x)

[Out]

-76/105/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8+29/64/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)-608/105/d/a
^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^2+667/192/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)
^3-244/15/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^4-1465/192/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(
1/2*d*x+1/2*c)^5+32/15/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^6-5117/192/d/a^3/(1+tan(1/2*d*x+1/2
*c)^2)^8*tan(1/2*d*x+1/2*c)^7-76/3/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^8+5117/192/d/a^3/(1+tan
(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^9-128/3/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^10+1465/19
2/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^11-4/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)
^12-667/192/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^13-29/64/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(
1/2*d*x+1/2*c)^15-29/64/d/a^3*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.59037, size = 674, normalized size = 4.19 \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)^8*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/6720*((3045*sin(d*x + c)/(cos(d*x + c) + 1) - 38912*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 23345*sin(d*x + c)
^3/(cos(d*x + c) + 1)^3 - 109312*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 51275*sin(d*x + c)^5/(cos(d*x + c) + 1)
^5 + 14336*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 179095*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 170240*sin(d*x +
c)^8/(cos(d*x + c) + 1)^8 + 179095*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 286720*sin(d*x + c)^10/(cos(d*x + c)
+ 1)^10 + 51275*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 26880*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 23345*s
in(d*x + c)^13/(cos(d*x + c) + 1)^13 - 3045*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 4864)/(a^3 + 8*a^3*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 + 28*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 56*a^3*sin(d*x + c)^6/(cos(d*x + c
) + 1)^6 + 70*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 56*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 28*a^3*
sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 8*a^3*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + a^3*sin(d*x + c)^16/(cos
(d*x + c) + 1)^16) - 3045*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d

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Fricas [A]  time = 1.12619, size = 263, normalized size = 1.63 \begin{align*} -\frac{5760 \, \cos \left (d x + c\right )^{7} - 18816 \, \cos \left (d x + c\right )^{5} + 17920 \, \cos \left (d x + c\right )^{3} + 3045 \, d x - 35 \,{\left (48 \, \cos \left (d x + c\right )^{7} - 328 \, \cos \left (d x + c\right )^{5} + 454 \, \cos \left (d x + c\right )^{3} - 87 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/13440*(5760*cos(d*x + c)^7 - 18816*cos(d*x + c)^5 + 17920*cos(d*x + c)^3 + 3045*d*x - 35*(48*cos(d*x + c)^7
- 328*cos(d*x + c)^5 + 454*cos(d*x + c)^3 - 87*cos(d*x + c))*sin(d*x + c))/(a^3*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**8*sin(d*x+c)**3/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.31098, size = 294, normalized size = 1.83 \begin{align*} -\frac{\frac{3045 \,{\left (d x + c\right )}}{a^{3}} + \frac{2 \,{\left (3045 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 23345 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 26880 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} - 51275 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 286720 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 179095 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 170240 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 179095 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 14336 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 51275 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 109312 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 23345 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 38912 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3045 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4864\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{8} a^{3}}}{13440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/13440*(3045*(d*x + c)/a^3 + 2*(3045*tan(1/2*d*x + 1/2*c)^15 + 23345*tan(1/2*d*x + 1/2*c)^13 + 26880*tan(1/2
*d*x + 1/2*c)^12 - 51275*tan(1/2*d*x + 1/2*c)^11 + 286720*tan(1/2*d*x + 1/2*c)^10 - 179095*tan(1/2*d*x + 1/2*c
)^9 + 170240*tan(1/2*d*x + 1/2*c)^8 + 179095*tan(1/2*d*x + 1/2*c)^7 - 14336*tan(1/2*d*x + 1/2*c)^6 + 51275*tan
(1/2*d*x + 1/2*c)^5 + 109312*tan(1/2*d*x + 1/2*c)^4 - 23345*tan(1/2*d*x + 1/2*c)^3 + 38912*tan(1/2*d*x + 1/2*c
)^2 - 3045*tan(1/2*d*x + 1/2*c) + 4864)/((tan(1/2*d*x + 1/2*c)^2 + 1)^8*a^3))/d