\(\int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [719]

Optimal result

Integrand size = 29, antiderivative size = 152 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 \text {arctanh}(\cos (c+d x))}{128 a d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {\cot ^9(c+d x)}{9 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d}+\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d} \]

[Out]

-5/128*arctanh(cos(d*x+c))/a/d-1/7*cot(d*x+c)^7/a/d-1/9*cot(d*x+c)^9/a/d-5/128*cot(d*x+c)*csc(d*x+c)/a/d+5/64*
cot(d*x+c)*csc(d*x+c)^3/a/d-5/48*cot(d*x+c)^3*csc(d*x+c)^3/a/d+1/8*cot(d*x+c)^5*csc(d*x+c)^3/a/d

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2918, 2687, 14, 2691, 3853, 3855} \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 \text {arctanh}(\cos (c+d x))}{128 a d}-\frac {\cot ^9(c+d x)}{9 a d}-\frac {\cot ^7(c+d x)}{7 a d}+\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d} \]

[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 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 2687

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 2691

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 2918

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 3853

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 3855

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

Rubi steps \begin{align*} \text {integral}& = -\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 {\text {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 {\text {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 \text {arctanh}(\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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(313\) vs. \(2(152)=304\).

Time = 2.38 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.06 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^9(c+d x) \left (129024 \cos (c+d x)+75264 \cos (3 (c+d x))+23040 \cos (5 (c+d x))+2304 \cos (7 (c+d x))-256 \cos (9 (c+d x))+39690 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-39690 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-36540 \sin (2 (c+d x))-26460 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+26460 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-20916 \sin (4 (c+d x))+11340 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-11340 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-16044 \sin (6 (c+d x))-2835 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+2835 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-630 \sin (8 (c+d x))+315 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))-315 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))\right )}{2064384 a d} \]

[In]

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

[Out]

-1/2064384*(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]]*Si
n[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)]))/(a*d)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.49

method	result	size
parallelrisch	\(\frac {-28 \left (\cot ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+28 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+63 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-63 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+108 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-108 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-336 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+336 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-504 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-672 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+672 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1008 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1008 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1512 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+5040 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1512 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{129024 d a}\)	\(226\)
derivativedivides	\(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}+\frac {3}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{512 d a}\)	\(228\)
default	\(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}+\frac {3}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{512 d a}\)	\(228\)
risch	\(\frac {315 \,{\mathrm e}^{17 i \left (d x +c \right )}-80640 i {\mathrm e}^{10 i \left (d x +c \right )}+8022 \,{\mathrm e}^{15 i \left (d x +c \right )}-16128 i {\mathrm e}^{14 i \left (d x +c \right )}+10458 \,{\mathrm e}^{13 i \left (d x +c \right )}-6912 i {\mathrm e}^{4 i \left (d x +c \right )}+18270 \,{\mathrm e}^{11 i \left (d x +c \right )}-48384 i {\mathrm e}^{8 i \left (d x +c \right )}-2304 i {\mathrm e}^{2 i \left (d x +c \right )}-18270 \,{\mathrm e}^{7 i \left (d x +c \right )}-48384 i {\mathrm e}^{6 i \left (d x +c \right )}-10458 \,{\mathrm e}^{5 i \left (d x +c \right )}-26880 i {\mathrm e}^{12 i \left (d x +c \right )}-8022 \,{\mathrm e}^{3 i \left (d x +c \right )}+256 i-315 \,{\mathrm e}^{i \left (d x +c \right )}}{4032 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d a}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d a}\)	\(238\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.64 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {512 \, \cos \left (d x + c\right )^{9} - 2304 \, \cos \left (d x + c\right )^{7} - 315 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 315 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 42 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16128 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )} \]

[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))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (138) = 276\).

Time = 0.22 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.34 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {1512 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1008 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {672 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {504 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {336 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {108 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {63 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {28 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a} - \frac {5040 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {{\left (\frac {63 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {108 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {336 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {504 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {672 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1008 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {1512 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 28\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{9}}{a \sin \left (d x + c\right )^{9}}}{129024 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.80 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {5040 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {28 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 63 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 108 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 336 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1008 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1512 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}} - \frac {14258 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1512 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1008 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 672 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 108 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 28}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{129024 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 13.43 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.86 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {28\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-28\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-63\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-108\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}-504\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-1512\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+1512\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+504\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+108\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+5040\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{129024\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]

[In]

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

[Out]

(28*sin(c/2 + (d*x)/2)^18 - 28*cos(c/2 + (d*x)/2)^18 - 63*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^17 + 63*cos(c/
2 + (d*x)/2)^17*sin(c/2 + (d*x)/2) - 108*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^16 + 336*cos(c/2 + (d*x)/2)^3
*sin(c/2 + (d*x)/2)^15 - 504*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^13 + 672*cos(c/2 + (d*x)/2)^6*sin(c/2 + (
d*x)/2)^12 - 1008*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^11 - 1512*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^10
+ 1512*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^8 + 1008*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^7 - 672*cos
(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^6 + 504*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^5 - 336*cos(c/2 + (d*x)
/2)^15*sin(c/2 + (d*x)/2)^3 + 108*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d*x)/2)^2 + 5040*log(sin(c/2 + (d*x)/2)/cos
(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^9)/(129024*a*d*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2
)^9)