3.705 \(\int \frac{\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=209 \[ -\frac{\cos ^{11}(c+d x)}{11 a d}+\frac{2 \cos ^9(c+d x)}{9 a d}-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin ^5(c+d x) \cos ^7(c+d x)}{12 a d}+\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{24 a d}+\frac{\sin (c+d x) \cos ^7(c+d x)}{64 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{384 a d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{1536 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{1024 a d}-\frac{5 x}{1024 a} \]

[Out]

(-5*x)/(1024*a) - Cos[c + d*x]^7/(7*a*d) + (2*Cos[c + d*x]^9)/(9*a*d) - Cos[c + d*x]^11/(11*a*d) - (5*Cos[c +
d*x]*Sin[c + d*x])/(1024*a*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(1536*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(38
4*a*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(64*a*d) + (Cos[c + d*x]^7*Sin[c + d*x]^3)/(24*a*d) + (Cos[c + d*x]^7*S
in[c + d*x]^5)/(12*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.281858, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2839, 2565, 270, 2568, 2635, 8} \[ -\frac{\cos ^{11}(c+d x)}{11 a d}+\frac{2 \cos ^9(c+d x)}{9 a d}-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin ^5(c+d x) \cos ^7(c+d x)}{12 a d}+\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{24 a d}+\frac{\sin (c+d x) \cos ^7(c+d x)}{64 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{384 a d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{1536 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{1024 a d}-\frac{5 x}{1024 a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^8*Sin[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

(-5*x)/(1024*a) - Cos[c + d*x]^7/(7*a*d) + (2*Cos[c + d*x]^9)/(9*a*d) - Cos[c + d*x]^11/(11*a*d) - (5*Cos[c +
d*x]*Sin[c + d*x])/(1024*a*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(1536*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(38
4*a*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(64*a*d) + (Cos[c + d*x]^7*Sin[c + d*x]^3)/(24*a*d) + (Cos[c + d*x]^7*S
in[c + d*x]^5)/(12*a*d)

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 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 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]

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]

Rubi steps

\begin{align*} \int \frac{\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \cos ^6(c+d x) \sin ^5(c+d x) \, dx}{a}-\frac{\int \cos ^6(c+d x) \sin ^6(c+d x) \, dx}{a}\\ &=\frac{\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac{5 \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx}{12 a}-\frac{\operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac{\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac{\int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{8 a}-\frac{\operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac{\cos ^7(c+d x)}{7 a d}+\frac{2 \cos ^9(c+d x)}{9 a d}-\frac{\cos ^{11}(c+d x)}{11 a d}+\frac{\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac{\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac{\int \cos ^6(c+d x) \, dx}{64 a}\\ &=-\frac{\cos ^7(c+d x)}{7 a d}+\frac{2 \cos ^9(c+d x)}{9 a d}-\frac{\cos ^{11}(c+d x)}{11 a d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac{\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac{\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac{5 \int \cos ^4(c+d x) \, dx}{384 a}\\ &=-\frac{\cos ^7(c+d x)}{7 a d}+\frac{2 \cos ^9(c+d x)}{9 a d}-\frac{\cos ^{11}(c+d x)}{11 a d}-\frac{5 \cos ^3(c+d x) \sin (c+d x)}{1536 a d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac{\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac{\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac{5 \int \cos ^2(c+d x) \, dx}{512 a}\\ &=-\frac{\cos ^7(c+d x)}{7 a d}+\frac{2 \cos ^9(c+d x)}{9 a d}-\frac{\cos ^{11}(c+d x)}{11 a d}-\frac{5 \cos (c+d x) \sin (c+d x)}{1024 a d}-\frac{5 \cos ^3(c+d x) \sin (c+d x)}{1536 a d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac{\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac{\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac{5 \int 1 \, dx}{1024 a}\\ &=-\frac{5 x}{1024 a}-\frac{\cos ^7(c+d x)}{7 a d}+\frac{2 \cos ^9(c+d x)}{9 a d}-\frac{\cos ^{11}(c+d x)}{11 a d}-\frac{5 \cos (c+d x) \sin (c+d x)}{1024 a d}-\frac{5 \cos ^3(c+d x) \sin (c+d x)}{1536 a d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac{\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac{\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}\\ \end{align*}

Mathematica [B]  time = 14.4436, size = 518, normalized size = 2.48 \[ -\frac{55440 d x \sin \left (\frac{c}{2}\right )-55440 \sin \left (\frac{c}{2}+d x\right )+55440 \sin \left (\frac{3 c}{2}+d x\right )-18480 \sin \left (\frac{5 c}{2}+3 d x\right )+18480 \sin \left (\frac{7 c}{2}+3 d x\right )-10395 \sin \left (\frac{7 c}{2}+4 d x\right )-10395 \sin \left (\frac{9 c}{2}+4 d x\right )+5544 \sin \left (\frac{9 c}{2}+5 d x\right )-5544 \sin \left (\frac{11 c}{2}+5 d x\right )+3960 \sin \left (\frac{13 c}{2}+7 d x\right )-3960 \sin \left (\frac{15 c}{2}+7 d x\right )+2079 \sin \left (\frac{15 c}{2}+8 d x\right )+2079 \sin \left (\frac{17 c}{2}+8 d x\right )-616 \sin \left (\frac{17 c}{2}+9 d x\right )+616 \sin \left (\frac{19 c}{2}+9 d x\right )-504 \sin \left (\frac{21 c}{2}+11 d x\right )+504 \sin \left (\frac{23 c}{2}+11 d x\right )-231 \sin \left (\frac{23 c}{2}+12 d x\right )-231 \sin \left (\frac{25 c}{2}+12 d x\right )+55440 d x \cos \left (\frac{c}{2}\right )+55440 \cos \left (\frac{c}{2}+d x\right )+55440 \cos \left (\frac{3 c}{2}+d x\right )+18480 \cos \left (\frac{5 c}{2}+3 d x\right )+18480 \cos \left (\frac{7 c}{2}+3 d x\right )-10395 \cos \left (\frac{7 c}{2}+4 d x\right )+10395 \cos \left (\frac{9 c}{2}+4 d x\right )-5544 \cos \left (\frac{9 c}{2}+5 d x\right )-5544 \cos \left (\frac{11 c}{2}+5 d x\right )-3960 \cos \left (\frac{13 c}{2}+7 d x\right )-3960 \cos \left (\frac{15 c}{2}+7 d x\right )+2079 \cos \left (\frac{15 c}{2}+8 d x\right )-2079 \cos \left (\frac{17 c}{2}+8 d x\right )+616 \cos \left (\frac{17 c}{2}+9 d x\right )+616 \cos \left (\frac{19 c}{2}+9 d x\right )+504 \cos \left (\frac{21 c}{2}+11 d x\right )+504 \cos \left (\frac{23 c}{2}+11 d x\right )-231 \cos \left (\frac{23 c}{2}+12 d x\right )+231 \cos \left (\frac{25 c}{2}+12 d x\right )+99792 \sin \left (\frac{c}{2}\right )}{11354112 a d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

-(55440*d*x*Cos[c/2] + 55440*Cos[c/2 + d*x] + 55440*Cos[(3*c)/2 + d*x] + 18480*Cos[(5*c)/2 + 3*d*x] + 18480*Co
s[(7*c)/2 + 3*d*x] - 10395*Cos[(7*c)/2 + 4*d*x] + 10395*Cos[(9*c)/2 + 4*d*x] - 5544*Cos[(9*c)/2 + 5*d*x] - 554
4*Cos[(11*c)/2 + 5*d*x] - 3960*Cos[(13*c)/2 + 7*d*x] - 3960*Cos[(15*c)/2 + 7*d*x] + 2079*Cos[(15*c)/2 + 8*d*x]
- 2079*Cos[(17*c)/2 + 8*d*x] + 616*Cos[(17*c)/2 + 9*d*x] + 616*Cos[(19*c)/2 + 9*d*x] + 504*Cos[(21*c)/2 + 11*
d*x] + 504*Cos[(23*c)/2 + 11*d*x] - 231*Cos[(23*c)/2 + 12*d*x] + 231*Cos[(25*c)/2 + 12*d*x] + 99792*Sin[c/2] +
55440*d*x*Sin[c/2] - 55440*Sin[c/2 + d*x] + 55440*Sin[(3*c)/2 + d*x] - 18480*Sin[(5*c)/2 + 3*d*x] + 18480*Sin
[(7*c)/2 + 3*d*x] - 10395*Sin[(7*c)/2 + 4*d*x] - 10395*Sin[(9*c)/2 + 4*d*x] + 5544*Sin[(9*c)/2 + 5*d*x] - 5544
*Sin[(11*c)/2 + 5*d*x] + 3960*Sin[(13*c)/2 + 7*d*x] - 3960*Sin[(15*c)/2 + 7*d*x] + 2079*Sin[(15*c)/2 + 8*d*x]
+ 2079*Sin[(17*c)/2 + 8*d*x] - 616*Sin[(17*c)/2 + 9*d*x] + 616*Sin[(19*c)/2 + 9*d*x] - 504*Sin[(21*c)/2 + 11*d
*x] + 504*Sin[(23*c)/2 + 11*d*x] - 231*Sin[(23*c)/2 + 12*d*x] - 231*Sin[(25*c)/2 + 12*d*x])/(11354112*a*d*(Cos
[c/2] + Sin[c/2]))

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Maple [B]  time = 0.128, size = 755, normalized size = 3.6 \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)^5/(a+a*sin(d*x+c)),x)

[Out]

-16/693/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12+5/512/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)-64/231/d/a/(1
+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^2+175/1536/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^3-3
2/21/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^4+311/512/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+
1/2*c)^5+352/63/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^6-8361/512/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*
tan(1/2*d*x+1/2*c)^7-192/7/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^8+42259/768/d/a/(1+tan(1/2*d*x+1
/2*c)^2)^12*tan(1/2*d*x+1/2*c)^9+96/7/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^10-25295/256/d/a/(1+t
an(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^11-32/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^12+25295
/256/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^13-32/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*
c)^14-42259/768/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^15+16/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1
/2*d*x+1/2*c)^16+8361/512/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^17-32/3/d/a/(1+tan(1/2*d*x+1/2*c)
^2)^12*tan(1/2*d*x+1/2*c)^18-311/512/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^19-175/1536/d/a/(1+tan
(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^21-5/512/d/a/(1+tan(1/2*d*x+1/2*c)^2)^12*tan(1/2*d*x+1/2*c)^23-5/512/
a/d*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.63672, size = 952, normalized size = 4.56 \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)^5/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/354816*((3465*sin(d*x + c)/(cos(d*x + c) + 1) - 98304*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 40425*sin(d*x +
c)^3/(cos(d*x + c) + 1)^3 - 540672*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 215523*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 + 1982464*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 5794173*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 9732096*si
n(d*x + c)^8/(cos(d*x + c) + 1)^8 + 19523658*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 4866048*sin(d*x + c)^10/(co
s(d*x + c) + 1)^10 - 35058870*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 3784704*sin(d*x + c)^12/(cos(d*x + c) +
1)^12 + 35058870*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 11354112*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 1952
3658*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 + 5677056*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 + 5794173*sin(d*x +
c)^17/(cos(d*x + c) + 1)^17 - 3784704*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 - 215523*sin(d*x + c)^19/(cos(d*x
+ c) + 1)^19 - 40425*sin(d*x + c)^21/(cos(d*x + c) + 1)^21 - 3465*sin(d*x + c)^23/(cos(d*x + c) + 1)^23 - 819
2)/(a + 12*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 66*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 220*a*sin(d*x +
c)^6/(cos(d*x + c) + 1)^6 + 495*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 792*a*sin(d*x + c)^10/(cos(d*x + c) +
1)^10 + 924*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 792*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 495*a*sin(
d*x + c)^16/(cos(d*x + c) + 1)^16 + 220*a*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 + 66*a*sin(d*x + c)^20/(cos(d*
x + c) + 1)^20 + 12*a*sin(d*x + c)^22/(cos(d*x + c) + 1)^22 + a*sin(d*x + c)^24/(cos(d*x + c) + 1)^24) - 3465*
arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

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Fricas [A]  time = 1.20768, size = 324, normalized size = 1.55 \begin{align*} -\frac{64512 \, \cos \left (d x + c\right )^{11} - 157696 \, \cos \left (d x + c\right )^{9} + 101376 \, \cos \left (d x + c\right )^{7} + 3465 \, d x - 231 \,{\left (256 \, \cos \left (d x + c\right )^{11} - 640 \, \cos \left (d x + c\right )^{9} + 432 \, \cos \left (d x + c\right )^{7} - 8 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{709632 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/709632*(64512*cos(d*x + c)^11 - 157696*cos(d*x + c)^9 + 101376*cos(d*x + c)^7 + 3465*d*x - 231*(256*cos(d*x
+ c)^11 - 640*cos(d*x + c)^9 + 432*cos(d*x + c)^7 - 8*cos(d*x + c)^5 - 10*cos(d*x + c)^3 - 15*cos(d*x + c))*s
in(d*x + c))/(a*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)**5/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.29103, size = 417, normalized size = 2. \begin{align*} -\frac{\frac{3465 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (3465 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{23} + 40425 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{21} + 215523 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{19} + 3784704 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{18} - 5794173 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{17} - 5677056 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{16} + 19523658 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 11354112 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} - 35058870 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 3784704 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 35058870 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 4866048 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 19523658 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 9732096 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 5794173 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1982464 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 215523 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 540672 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 40425 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 98304 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3465 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8192\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{12} a}}{709632 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/709632*(3465*(d*x + c)/a + 2*(3465*tan(1/2*d*x + 1/2*c)^23 + 40425*tan(1/2*d*x + 1/2*c)^21 + 215523*tan(1/2
*d*x + 1/2*c)^19 + 3784704*tan(1/2*d*x + 1/2*c)^18 - 5794173*tan(1/2*d*x + 1/2*c)^17 - 5677056*tan(1/2*d*x + 1
/2*c)^16 + 19523658*tan(1/2*d*x + 1/2*c)^15 + 11354112*tan(1/2*d*x + 1/2*c)^14 - 35058870*tan(1/2*d*x + 1/2*c)
^13 + 3784704*tan(1/2*d*x + 1/2*c)^12 + 35058870*tan(1/2*d*x + 1/2*c)^11 - 4866048*tan(1/2*d*x + 1/2*c)^10 - 1
9523658*tan(1/2*d*x + 1/2*c)^9 + 9732096*tan(1/2*d*x + 1/2*c)^8 + 5794173*tan(1/2*d*x + 1/2*c)^7 - 1982464*tan
(1/2*d*x + 1/2*c)^6 - 215523*tan(1/2*d*x + 1/2*c)^5 + 540672*tan(1/2*d*x + 1/2*c)^4 - 40425*tan(1/2*d*x + 1/2*
c)^3 + 98304*tan(1/2*d*x + 1/2*c)^2 - 3465*tan(1/2*d*x + 1/2*c) + 8192)/((tan(1/2*d*x + 1/2*c)^2 + 1)^12*a))/d