3.737 \(\int \frac{\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=218 \[ \frac{2 \cot ^9(c+d x)}{9 a^2 d}+\frac{4 \cot ^7(c+d x)}{7 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{9 \tanh ^{-1}(\cos (c+d x))}{256 a^2 d}-\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{3 \cot (c+d x) \csc ^7(c+d x)}{80 a^2 d}+\frac{9 \cot (c+d x) \csc ^5(c+d x)}{160 a^2 d}-\frac{3 \cot (c+d x) \csc ^3(c+d x)}{128 a^2 d}-\frac{9 \cot (c+d x) \csc (c+d x)}{256 a^2 d} \]

[Out]

(-9*ArcTanh[Cos[c + d*x]])/(256*a^2*d) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (4*Cot[c + d*x]^7)/(7*a^2*d) + (2*Cot[
c + d*x]^9)/(9*a^2*d) - (9*Cot[c + d*x]*Csc[c + d*x])/(256*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*a^2*d
) + (9*Cot[c + d*x]*Csc[c + d*x]^5)/(160*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*a^2*d) + (3*Cot[c + d*x]*
Csc[c + d*x]^7)/(80*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*a^2*d)

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Rubi [A]  time = 0.487213, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2875, 2873, 2611, 3768, 3770, 2607, 270} \[ \frac{2 \cot ^9(c+d x)}{9 a^2 d}+\frac{4 \cot ^7(c+d x)}{7 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{9 \tanh ^{-1}(\cos (c+d x))}{256 a^2 d}-\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{3 \cot (c+d x) \csc ^7(c+d x)}{80 a^2 d}+\frac{9 \cot (c+d x) \csc ^5(c+d x)}{160 a^2 d}-\frac{3 \cot (c+d x) \csc ^3(c+d x)}{128 a^2 d}-\frac{9 \cot (c+d x) \csc (c+d x)}{256 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^8*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]

[Out]

(-9*ArcTanh[Cos[c + d*x]])/(256*a^2*d) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (4*Cot[c + d*x]^7)/(7*a^2*d) + (2*Cot[
c + d*x]^9)/(9*a^2*d) - (9*Cot[c + d*x]*Csc[c + d*x])/(256*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*a^2*d
) + (9*Cot[c + d*x]*Csc[c + d*x]^5)/(160*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*a^2*d) + (3*Cot[c + d*x]*
Csc[c + d*x]^7)/(80*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*a^2*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 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

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{\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \cot ^4(c+d x) \csc ^7(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \cot ^4(c+d x) \csc ^5(c+d x)-2 a^2 \cot ^4(c+d x) \csc ^6(c+d x)+a^2 \cot ^4(c+d x) \csc ^7(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \cot ^4(c+d x) \csc ^5(c+d x) \, dx}{a^2}+\frac{\int \cot ^4(c+d x) \csc ^7(c+d x) \, dx}{a^2}-\frac{2 \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx}{a^2}\\ &=-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}-\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d}-\frac{3 \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx}{10 a^2}-\frac{3 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{8 a^2}-\frac{2 \operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^2 d}\\ &=\frac{\cot (c+d x) \csc ^5(c+d x)}{16 a^2 d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{3 \cot (c+d x) \csc ^7(c+d x)}{80 a^2 d}-\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d}+\frac{3 \int \csc ^7(c+d x) \, dx}{80 a^2}+\frac{\int \csc ^5(c+d x) \, dx}{16 a^2}-\frac{2 \operatorname{Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}\\ &=\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{4 \cot ^7(c+d x)}{7 a^2 d}+\frac{2 \cot ^9(c+d x)}{9 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac{9 \cot (c+d x) \csc ^5(c+d x)}{160 a^2 d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{3 \cot (c+d x) \csc ^7(c+d x)}{80 a^2 d}-\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d}+\frac{\int \csc ^5(c+d x) \, dx}{32 a^2}+\frac{3 \int \csc ^3(c+d x) \, dx}{64 a^2}\\ &=\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{4 \cot ^7(c+d x)}{7 a^2 d}+\frac{2 \cot ^9(c+d x)}{9 a^2 d}-\frac{3 \cot (c+d x) \csc (c+d x)}{128 a^2 d}-\frac{3 \cot (c+d x) \csc ^3(c+d x)}{128 a^2 d}+\frac{9 \cot (c+d x) \csc ^5(c+d x)}{160 a^2 d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{3 \cot (c+d x) \csc ^7(c+d x)}{80 a^2 d}-\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d}+\frac{3 \int \csc (c+d x) \, dx}{128 a^2}+\frac{3 \int \csc ^3(c+d x) \, dx}{128 a^2}\\ &=-\frac{3 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{4 \cot ^7(c+d x)}{7 a^2 d}+\frac{2 \cot ^9(c+d x)}{9 a^2 d}-\frac{9 \cot (c+d x) \csc (c+d x)}{256 a^2 d}-\frac{3 \cot (c+d x) \csc ^3(c+d x)}{128 a^2 d}+\frac{9 \cot (c+d x) \csc ^5(c+d x)}{160 a^2 d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{3 \cot (c+d x) \csc ^7(c+d x)}{80 a^2 d}-\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d}+\frac{3 \int \csc (c+d x) \, dx}{256 a^2}\\ &=-\frac{9 \tanh ^{-1}(\cos (c+d x))}{256 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{4 \cot ^7(c+d x)}{7 a^2 d}+\frac{2 \cot ^9(c+d x)}{9 a^2 d}-\frac{9 \cot (c+d x) \csc (c+d x)}{256 a^2 d}-\frac{3 \cot (c+d x) \csc ^3(c+d x)}{128 a^2 d}+\frac{9 \cot (c+d x) \csc ^5(c+d x)}{160 a^2 d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{3 \cot (c+d x) \csc ^7(c+d x)}{80 a^2 d}-\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d}\\ \end{align*}

Mathematica [A]  time = 1.60567, size = 353, normalized size = 1.62 \[ \frac{\csc ^{10}(c+d x) \left (1720320 \sin (2 (c+d x))+1228800 \sin (4 (c+d x))+184320 \sin (6 (c+d x))-40960 \sin (8 (c+d x))+4096 \sin (10 (c+d x))-3219300 \cos (c+d x)-1237320 \cos (3 (c+d x))+278712 \cos (5 (c+d x))+54810 \cos (7 (c+d x))-5670 \cos (9 (c+d x))+357210 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+595350 \cos (2 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-340200 \cos (4 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+127575 \cos (6 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-28350 \cos (8 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+2835 \cos (10 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-357210 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-595350 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+340200 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-127575 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+28350 \cos (8 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2835 \cos (10 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{41287680 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]

[Out]

(Csc[c + d*x]^10*(-3219300*Cos[c + d*x] - 1237320*Cos[3*(c + d*x)] + 278712*Cos[5*(c + d*x)] + 54810*Cos[7*(c
+ d*x)] - 5670*Cos[9*(c + d*x)] - 357210*Log[Cos[(c + d*x)/2]] + 595350*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]]
- 340200*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 127575*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 28350*Cos[8
*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 2835*Cos[10*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 357210*Log[Sin[(c + d*x)/2]
] - 595350*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 340200*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 127575*Cos
[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 28350*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 2835*Cos[10*(c + d*x)]*Lo
g[Sin[(c + d*x)/2]] + 1720320*Sin[2*(c + d*x)] + 1228800*Sin[4*(c + d*x)] + 184320*Sin[6*(c + d*x)] - 40960*Si
n[8*(c + d*x)] + 4096*Sin[10*(c + d*x)]))/(41287680*a^2*d)

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Maple [B]  time = 0.214, size = 398, normalized size = 1.8 \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*csc(d*x+c)^11/(a+a*sin(d*x+c))^2,x)

[Out]

1/10240/d/a^2*tan(1/2*d*x+1/2*c)^10-1/2304/d/a^2*tan(1/2*d*x+1/2*c)^9+3/4096/d/a^2*tan(1/2*d*x+1/2*c)^8-1/1792
/d/a^2*tan(1/2*d*x+1/2*c)^7-1/2048/d/a^2*tan(1/2*d*x+1/2*c)^6+1/320/d/a^2*tan(1/2*d*x+1/2*c)^5-3/512/d/a^2*tan
(1/2*d*x+1/2*c)^4+1/192/d/a^2*tan(1/2*d*x+1/2*c)^3+1/1024/d/a^2*tan(1/2*d*x+1/2*c)^2-3/128/d/a^2*tan(1/2*d*x+1
/2*c)-1/10240/d/a^2/tan(1/2*d*x+1/2*c)^10+1/1792/d/a^2/tan(1/2*d*x+1/2*c)^7+3/128/d/a^2/tan(1/2*d*x+1/2*c)-3/4
096/d/a^2/tan(1/2*d*x+1/2*c)^8-1/320/d/a^2/tan(1/2*d*x+1/2*c)^5+3/512/d/a^2/tan(1/2*d*x+1/2*c)^4+1/2304/d/a^2/
tan(1/2*d*x+1/2*c)^9+9/256/d/a^2*ln(tan(1/2*d*x+1/2*c))+1/2048/d/a^2/tan(1/2*d*x+1/2*c)^6-1/192/d/a^2/tan(1/2*
d*x+1/2*c)^3-1/1024/d/a^2/tan(1/2*d*x+1/2*c)^2

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Maxima [B]  time = 1.04572, size = 587, normalized size = 2.69 \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*csc(d*x+c)^11/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/1290240*((30240*sin(d*x + c)/(cos(d*x + c) + 1) - 1260*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 6720*sin(d*x +
c)^3/(cos(d*x + c) + 1)^3 + 7560*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 4032*sin(d*x + c)^5/(cos(d*x + c) + 1)
^5 + 630*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 720*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 945*sin(d*x + c)^8/(c
os(d*x + c) + 1)^8 + 560*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 126*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)/a^2
- 45360*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 - (560*sin(d*x + c)/(cos(d*x + c) + 1) - 945*sin(d*x + c)^2/(
cos(d*x + c) + 1)^2 + 720*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 630*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 4032
*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 7560*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 6720*sin(d*x + c)^7/(cos(d*x
+ c) + 1)^7 - 1260*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 30240*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 126)*(co
s(d*x + c) + 1)^10/(a^2*sin(d*x + c)^10))/d

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Fricas [A]  time = 1.20639, size = 817, normalized size = 3.75 \begin{align*} \frac{5670 \, \cos \left (d x + c\right )^{9} - 26460 \, \cos \left (d x + c\right )^{7} + 16128 \, \cos \left (d x + c\right )^{5} + 26460 \, \cos \left (d x + c\right )^{3} - 2835 \,{\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2835 \,{\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 1024 \,{\left (8 \, \cos \left (d x + c\right )^{9} - 36 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right ) - 5670 \, \cos \left (d x + c\right )}{161280 \,{\left (a^{2} d \cos \left (d x + c\right )^{10} - 5 \, a^{2} d \cos \left (d x + c\right )^{8} + 10 \, a^{2} d \cos \left (d x + c\right )^{6} - 10 \, a^{2} d \cos \left (d x + c\right )^{4} + 5 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^11/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/161280*(5670*cos(d*x + c)^9 - 26460*cos(d*x + c)^7 + 16128*cos(d*x + c)^5 + 26460*cos(d*x + c)^3 - 2835*(cos
(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1)*log(1/2*cos(d*
x + c) + 1/2) + 2835*(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x +
c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2) - 1024*(8*cos(d*x + c)^9 - 36*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*sin(
d*x + c) - 5670*cos(d*x + c))/(a^2*d*cos(d*x + c)^10 - 5*a^2*d*cos(d*x + c)^8 + 10*a^2*d*cos(d*x + c)^6 - 10*a
^2*d*cos(d*x + c)^4 + 5*a^2*d*cos(d*x + c)^2 - a^2*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*csc(d*x+c)**11/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.40215, size = 447, normalized size = 2.05 \begin{align*} \frac{\frac{45360 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac{132858 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 30240 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 1260 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 6720 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 7560 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 4032 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 630 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 720 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 945 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 560 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 126}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10}} + \frac{126 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 560 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 945 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 720 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 630 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 4032 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 7560 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 6720 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1260 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 30240 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{20}}}{1290240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^11/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1290240*(45360*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (132858*tan(1/2*d*x + 1/2*c)^10 - 30240*tan(1/2*d*x + 1/
2*c)^9 + 1260*tan(1/2*d*x + 1/2*c)^8 + 6720*tan(1/2*d*x + 1/2*c)^7 - 7560*tan(1/2*d*x + 1/2*c)^6 + 4032*tan(1/
2*d*x + 1/2*c)^5 - 630*tan(1/2*d*x + 1/2*c)^4 - 720*tan(1/2*d*x + 1/2*c)^3 + 945*tan(1/2*d*x + 1/2*c)^2 - 560*
tan(1/2*d*x + 1/2*c) + 126)/(a^2*tan(1/2*d*x + 1/2*c)^10) + (126*a^18*tan(1/2*d*x + 1/2*c)^10 - 560*a^18*tan(1
/2*d*x + 1/2*c)^9 + 945*a^18*tan(1/2*d*x + 1/2*c)^8 - 720*a^18*tan(1/2*d*x + 1/2*c)^7 - 630*a^18*tan(1/2*d*x +
1/2*c)^6 + 4032*a^18*tan(1/2*d*x + 1/2*c)^5 - 7560*a^18*tan(1/2*d*x + 1/2*c)^4 + 6720*a^18*tan(1/2*d*x + 1/2*
c)^3 + 1260*a^18*tan(1/2*d*x + 1/2*c)^2 - 30240*a^18*tan(1/2*d*x + 1/2*c))/a^20)/d