\(\int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [737]

Optimal result

Integrand size = 29, antiderivative size = 218 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {9 \text {arctanh}(\cos (c+d x))}{256 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {4 \cot ^7(c+d x)}{7 a^2 d}+\frac {2 \cot ^9(c+d x)}{9 a^2 d}-\frac {9 \cot (c+d x) \csc (c+d x)}{256 a^2 d}-\frac {3 \cot (c+d x) \csc ^3(c+d x)}{128 a^2 d}+\frac {9 \cot (c+d x) \csc ^5(c+d x)}{160 a^2 d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac {3 \cot (c+d x) \csc ^7(c+d x)}{80 a^2 d}-\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d} \]

[Out]

-9/256*arctanh(cos(d*x+c))/a^2/d+2/5*cot(d*x+c)^5/a^2/d+4/7*cot(d*x+c)^7/a^2/d+2/9*cot(d*x+c)^9/a^2/d-9/256*co
t(d*x+c)*csc(d*x+c)/a^2/d-3/128*cot(d*x+c)*csc(d*x+c)^3/a^2/d+9/160*cot(d*x+c)*csc(d*x+c)^5/a^2/d-1/8*cot(d*x+
c)^3*csc(d*x+c)^5/a^2/d+3/80*cot(d*x+c)*csc(d*x+c)^7/a^2/d-1/10*cot(d*x+c)^3*csc(d*x+c)^7/a^2/d

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2954, 2952, 2691, 3853, 3855, 2687, 276} \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {9 \text {arctanh}(\cos (c+d x))}{256 a^2 d}+\frac {2 \cot ^9(c+d x)}{9 a^2 d}+\frac {4 \cot ^7(c+d x)}{7 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac {3 \cot (c+d x) \csc ^7(c+d x)}{80 a^2 d}+\frac {9 \cot (c+d x) \csc ^5(c+d x)}{160 a^2 d}-\frac {3 \cot (c+d x) \csc ^3(c+d x)}{128 a^2 d}-\frac {9 \cot (c+d x) \csc (c+d x)}{256 a^2 d} \]

[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 276

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 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 2952

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 2954

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 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 ^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 \text {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 \text {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 \text {arctanh}(\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 \text {arctanh}(\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] (verified)

Time = 3.06 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.62 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\csc ^{10}(c+d x) \left (-3219300 \cos (c+d x)-1237320 \cos (3 (c+d x))+278712 \cos (5 (c+d x))+54810 \cos (7 (c+d x))-5670 \cos (9 (c+d x))-357210 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+595350 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-340200 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+127575 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-28350 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2835 \cos (10 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+357210 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-595350 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+340200 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-127575 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+28350 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2835 \cos (10 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1720320 \sin (2 (c+d x))+1228800 \sin (4 (c+d x))+184320 \sin (6 (c+d x))-40960 \sin (8 (c+d x))+4096 \sin (10 (c+d x))\right )}{41287680 a^2 d} \]

[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)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.60 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.19

method	result	size
risch	\(\frac {2835 \,{\mathrm e}^{19 i \left (d x +c \right )}-27405 \,{\mathrm e}^{17 i \left (d x +c \right )}+860160 i {\mathrm e}^{14 i \left (d x +c \right )}-139356 \,{\mathrm e}^{15 i \left (d x +c \right )}-1290240 i {\mathrm e}^{8 i \left (d x +c \right )}+618660 \,{\mathrm e}^{13 i \left (d x +c \right )}-368640 i {\mathrm e}^{6 i \left (d x +c \right )}+1609650 \,{\mathrm e}^{11 i \left (d x +c \right )}+430080 i {\mathrm e}^{12 i \left (d x +c \right )}+1609650 \,{\mathrm e}^{9 i \left (d x +c \right )}+516096 i {\mathrm e}^{10 i \left (d x +c \right )}+618660 \,{\mathrm e}^{7 i \left (d x +c \right )}-184320 i {\mathrm e}^{4 i \left (d x +c \right )}-139356 \,{\mathrm e}^{5 i \left (d x +c \right )}+40960 i {\mathrm e}^{2 i \left (d x +c \right )}-27405 \,{\mathrm e}^{3 i \left (d x +c \right )}-4096 i+2835 \,{\mathrm e}^{i \left (d x +c \right )}}{40320 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}-\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d \,a^{2}}+\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d \,a^{2}}\)	\(260\)
derivativedivides	\(\frac {\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}-\frac {4 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-6 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {16 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+36 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {24}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {4}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-\frac {16}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {4}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}-\frac {16}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {3}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}}{1024 d \,a^{2}}\)	\(278\)
default	\(\frac {\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}-\frac {4 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-6 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {16 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+36 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {24}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {4}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-\frac {16}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {4}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}-\frac {16}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {3}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}}{1024 d \,a^{2}}\)	\(278\)
parallelrisch	\(\frac {126 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-126 \left (\cot ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-560 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+560 \left (\cot ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+945 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-945 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-720 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+720 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-630 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+630 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4032 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4032 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7560 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7560 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6720 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6720 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1260 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1260 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45360 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-30240 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+30240 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{1290240 d \,a^{2}}\)	\(278\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^11/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/40320*(2835*exp(19*I*(d*x+c))-27405*exp(17*I*(d*x+c))+860160*I*exp(14*I*(d*x+c))-139356*exp(15*I*(d*x+c))-12
90240*I*exp(8*I*(d*x+c))+618660*exp(13*I*(d*x+c))-368640*I*exp(6*I*(d*x+c))+1609650*exp(11*I*(d*x+c))+430080*I
*exp(12*I*(d*x+c))+1609650*exp(9*I*(d*x+c))+516096*I*exp(10*I*(d*x+c))+618660*exp(7*I*(d*x+c))-184320*I*exp(4*
I*(d*x+c))-139356*exp(5*I*(d*x+c))+40960*I*exp(2*I*(d*x+c))-27405*exp(3*I*(d*x+c))-4096*I+2835*exp(I*(d*x+c)))
/d/a^2/(exp(2*I*(d*x+c))-1)^10-9/256/d/a^2*ln(exp(I*(d*x+c))+1)+9/256/d/a^2*ln(exp(I*(d*x+c))-1)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.35 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {5670 \, \cos \left (d x + c\right )^{9} - 26460 \, \cos \left (d x + c\right )^{7} + 16128 \, \cos \left (d x + c\right )^{5} + 26460 \, \cos \left (d x + c\right )^{3} - 2835 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2835 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 1024 \, {\left (8 \, \cos \left (d x + c\right )^{9} - 36 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right ) - 5670 \, \cos \left (d x + c\right )}{161280 \, {\left (a^{2} d \cos \left (d x + c\right )^{10} - 5 \, a^{2} d \cos \left (d x + c\right )^{8} + 10 \, a^{2} d \cos \left (d x + c\right )^{6} - 10 \, a^{2} d \cos \left (d x + c\right )^{4} + 5 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**11/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (198) = 396\).

Time = 0.22 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.00 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {30240 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1260 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6720 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {7560 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4032 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {630 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {720 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {945 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {560 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {126 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}}{a^{2}} - \frac {45360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {{\left (\frac {560 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {945 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {720 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {630 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4032 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {7560 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {6720 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {1260 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {30240 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 126\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{10}}{a^{2} \sin \left (d x + c\right )^{10}}}{1290240 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.52 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {45360 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {132858 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 30240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1260 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4032 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 630 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 126}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}} + \frac {126 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 560 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 945 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 720 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 630 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4032 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 7560 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 6720 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1260 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30240 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{20}}}{1290240 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 14.99 (sec) , antiderivative size = 531, normalized size of antiderivative = 2.44 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {126\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}-126\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}-560\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}+560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+945\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-630\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+4032\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}-7560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+1260\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-30240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+30240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-1260\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+7560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-4032\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+630\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-945\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+45360\,\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 )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{1290240\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^11*(a + a*sin(c + d*x))^2),x)

[Out]

(126*sin(c/2 + (d*x)/2)^20 - 126*cos(c/2 + (d*x)/2)^20 - 560*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^19 + 560*co
s(c/2 + (d*x)/2)^19*sin(c/2 + (d*x)/2) + 945*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^18 - 720*cos(c/2 + (d*x)/
2)^3*sin(c/2 + (d*x)/2)^17 - 630*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^16 + 4032*cos(c/2 + (d*x)/2)^5*sin(c/
2 + (d*x)/2)^15 - 7560*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^14 + 6720*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/
2)^13 + 1260*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^12 - 30240*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^11 + 3
0240*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^9 - 1260*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^8 - 6720*cos(c
/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^7 + 7560*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^6 - 4032*cos(c/2 + (d*x)
/2)^15*sin(c/2 + (d*x)/2)^5 + 630*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d*x)/2)^4 + 720*cos(c/2 + (d*x)/2)^17*sin(c
/2 + (d*x)/2)^3 - 945*cos(c/2 + (d*x)/2)^18*sin(c/2 + (d*x)/2)^2 + 45360*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x
)/2))*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^10)/(1290240*a^2*d*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^10)