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

Optimal. Leaf size=152 \[ -\frac{\cot ^9(c+d x)}{9 a d}-\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{128 a d}+\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d} \]

[Out]

(-5*ArcTanh[Cos[c + d*x]])/(128*a*d) - Cot[c + d*x]^7/(7*a*d) - Cot[c + d*x]^9/(9*a*d) - (5*Cot[c + d*x]*Csc[c
+ d*x])/(128*a*d) + (5*Cot[c + d*x]*Csc[c + d*x]^3)/(64*a*d) - (5*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*a*d) + (
Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*a*d)

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Rubi [A]  time = 0.235424, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2839, 2607, 14, 2611, 3768, 3770} \[ -\frac{\cot ^9(c+d x)}{9 a d}-\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{128 a d}+\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*ArcTanh[Cos[c + d*x]])/(128*a*d) - Cot[c + d*x]^7/(7*a*d) - Cot[c + d*x]^9/(9*a*d) - (5*Cot[c + d*x]*Csc[c
+ d*x])/(128*a*d) + (5*Cot[c + d*x]*Csc[c + d*x]^3)/(64*a*d) - (5*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*a*d) + (
Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*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 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 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 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]

Rubi steps

\begin{align*} \int \frac{\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot ^6(c+d x) \csc ^3(c+d x) \, dx}{a}+\frac{\int \cot ^6(c+d x) \csc ^4(c+d x) \, dx}{a}\\ &=\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac{5 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx}{8 a}+\frac{\operatorname{Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac{5 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{16 a}+\frac{\operatorname{Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac{\cot ^7(c+d x)}{7 a d}-\frac{\cot ^9(c+d x)}{9 a d}+\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac{5 \int \csc ^3(c+d x) \, dx}{64 a}\\ &=-\frac{\cot ^7(c+d x)}{7 a d}-\frac{\cot ^9(c+d x)}{9 a d}-\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d}+\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac{5 \int \csc (c+d x) \, dx}{128 a}\\ &=-\frac{5 \tanh ^{-1}(\cos (c+d x))}{128 a d}-\frac{\cot ^7(c+d x)}{7 a d}-\frac{\cot ^9(c+d x)}{9 a d}-\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d}+\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}\\ \end{align*}

Mathematica [B]  time = 1.35241, size = 313, normalized size = 2.06 \[ -\frac{\csc ^9(c+d x) \left (-36540 \sin (2 (c+d x))-20916 \sin (4 (c+d x))-16044 \sin (6 (c+d x))-630 \sin (8 (c+d x))+129024 \cos (c+d x)+75264 \cos (3 (c+d x))+23040 \cos (5 (c+d x))+2304 \cos (7 (c+d x))-256 \cos (9 (c+d x))-39690 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+26460 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-11340 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+2835 \sin (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-315 \sin (9 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+39690 \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-26460 \sin (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+11340 \sin (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-2835 \sin (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+315 \sin (9 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{2064384 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csc[c + d*x]^9*(129024*Cos[c + d*x] + 75264*Cos[3*(c + d*x)] + 23040*Cos[5*(c + d*x)] + 2304*Cos[7*(c + d*x)
] - 256*Cos[9*(c + d*x)] + 39690*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 39690*Log[Sin[(c + d*x)/2]]*Sin[c + d*x]
- 36540*Sin[2*(c + d*x)] - 26460*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 26460*Log[Sin[(c + d*x)/2]]*Sin[3*(
c + d*x)] - 20916*Sin[4*(c + d*x)] + 11340*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 11340*Log[Sin[(c + d*x)/2]
]*Sin[5*(c + d*x)] - 16044*Sin[6*(c + d*x)] - 2835*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] + 2835*Log[Sin[(c +
d*x)/2]]*Sin[7*(c + d*x)] - 630*Sin[8*(c + d*x)] + 315*Log[Cos[(c + d*x)/2]]*Sin[9*(c + d*x)] - 315*Log[Sin[(c
+ d*x)/2]]*Sin[9*(c + d*x)]))/(2064384*a*d)

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Maple [B]  time = 0.179, size = 322, normalized size = 2.1 \begin{align*}{\frac{1}{4608\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{1}{2048\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}}-{\frac{3}{3584\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{1}{384\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}-{\frac{1}{256\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}+{\frac{1}{192\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{1}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{3}{256\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{3}{3584\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}+{\frac{3}{256\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{1}{2048\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-8}}+{\frac{1}{256\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}-{\frac{1}{4608\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-9}}+{\frac{5}{128\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{384\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}-{\frac{1}{192\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{1}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*csc(d*x+c)^10/(a+a*sin(d*x+c)),x)

[Out]

1/4608/d/a*tan(1/2*d*x+1/2*c)^9-1/2048/d/a*tan(1/2*d*x+1/2*c)^8-3/3584/d/a*tan(1/2*d*x+1/2*c)^7+1/384/d/a*tan(
1/2*d*x+1/2*c)^6-1/256/d/a*tan(1/2*d*x+1/2*c)^4+1/192/d/a*tan(1/2*d*x+1/2*c)^3-1/128/d/a*tan(1/2*d*x+1/2*c)^2-
3/256/d/a*tan(1/2*d*x+1/2*c)+3/3584/d/a/tan(1/2*d*x+1/2*c)^7+3/256/d/a/tan(1/2*d*x+1/2*c)+1/2048/d/a/tan(1/2*d
*x+1/2*c)^8+1/256/d/a/tan(1/2*d*x+1/2*c)^4-1/4608/d/a/tan(1/2*d*x+1/2*c)^9+5/128/d/a*ln(tan(1/2*d*x+1/2*c))-1/
384/d/a/tan(1/2*d*x+1/2*c)^6-1/192/d/a/tan(1/2*d*x+1/2*c)^3+1/128/d/a/tan(1/2*d*x+1/2*c)^2

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Maxima [B]  time = 1.02169, size = 479, normalized size = 3.15 \begin{align*} -\frac{\frac{\frac{1512 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{1008 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{672 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{504 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{336 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{108 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{63 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{28 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a} - \frac{5040 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{{\left (\frac{63 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{108 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{336 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{504 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{672 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{1008 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{1512 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 28\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}{a \sin \left (d x + c\right )^{9}}}{129024 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/129024*((1512*sin(d*x + c)/(cos(d*x + c) + 1) + 1008*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 672*sin(d*x + c)
^3/(cos(d*x + c) + 1)^3 + 504*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 336*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 +
108*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 63*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 28*sin(d*x + c)^9/(cos(d*x
+ c) + 1)^9)/a - 5040*log(sin(d*x + c)/(cos(d*x + c) + 1))/a - (63*sin(d*x + c)/(cos(d*x + c) + 1) + 108*sin(d
*x + c)^2/(cos(d*x + c) + 1)^2 - 336*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 504*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 - 672*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1008*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 1512*sin(d*x + c)^
8/(cos(d*x + c) + 1)^8 - 28)*(cos(d*x + c) + 1)^9/(a*sin(d*x + c)^9))/d

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Fricas [A]  time = 1.23934, size = 689, normalized size = 4.53 \begin{align*} \frac{512 \, \cos \left (d x + c\right )^{9} - 2304 \, \cos \left (d x + c\right )^{7} - 315 \,{\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 315 \,{\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 42 \,{\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16128 \,{\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/16128*(512*cos(d*x + c)^9 - 2304*cos(d*x + c)^7 - 315*(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4
- 4*cos(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 315*(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*
cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 42*(15*cos(d*x + c)^7 + 73*
cos(d*x + c)^5 - 55*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c))/((a*d*cos(d*x + c)^8 - 4*a*d*cos(d*x + c)^
6 + 6*a*d*cos(d*x + c)^4 - 4*a*d*cos(d*x + c)^2 + a*d)*sin(d*x + c))

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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)**10/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.3315, size = 369, normalized size = 2.43 \begin{align*} \frac{\frac{5040 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} + \frac{28 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 63 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 108 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 336 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 504 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 672 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1008 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1512 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{9}} - \frac{14258 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1512 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 1008 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 672 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 504 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 336 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 108 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 28}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9}}}{129024 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/129024*(5040*log(abs(tan(1/2*d*x + 1/2*c)))/a + (28*a^8*tan(1/2*d*x + 1/2*c)^9 - 63*a^8*tan(1/2*d*x + 1/2*c)
^8 - 108*a^8*tan(1/2*d*x + 1/2*c)^7 + 336*a^8*tan(1/2*d*x + 1/2*c)^6 - 504*a^8*tan(1/2*d*x + 1/2*c)^4 + 672*a^
8*tan(1/2*d*x + 1/2*c)^3 - 1008*a^8*tan(1/2*d*x + 1/2*c)^2 - 1512*a^8*tan(1/2*d*x + 1/2*c))/a^9 - (14258*tan(1
/2*d*x + 1/2*c)^9 - 1512*tan(1/2*d*x + 1/2*c)^8 - 1008*tan(1/2*d*x + 1/2*c)^7 + 672*tan(1/2*d*x + 1/2*c)^6 - 5
04*tan(1/2*d*x + 1/2*c)^5 + 336*tan(1/2*d*x + 1/2*c)^3 - 108*tan(1/2*d*x + 1/2*c)^2 - 63*tan(1/2*d*x + 1/2*c)
+ 28)/(a*tan(1/2*d*x + 1/2*c)^9))/d