\(\int \frac {\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [750]

Optimal result

Integrand size = 27, antiderivative size = 166 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {29 \text {arctanh}(\cos (c+d x))}{128 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {7 \cot ^5(c+d x)}{5 a^3 d}+\frac {3 \cot ^7(c+d x)}{7 a^3 d}+\frac {29 \cot (c+d x) \csc (c+d x)}{128 a^3 d}+\frac {29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d} \]

[Out]

29/128*arctanh(cos(d*x+c))/a^3/d+4/3*cot(d*x+c)^3/a^3/d+7/5*cot(d*x+c)^5/a^3/d+3/7*cot(d*x+c)^7/a^3/d+29/128*c
ot(d*x+c)*csc(d*x+c)/a^3/d+29/192*cot(d*x+c)*csc(d*x+c)^3/a^3/d-23/48*cot(d*x+c)*csc(d*x+c)^5/a^3/d-1/8*cot(d*
x+c)*csc(d*x+c)^7/a^3/d

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2954, 2952, 2687, 14, 2691, 3853, 3855, 276} \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {29 \text {arctanh}(\cos (c+d x))}{128 a^3 d}+\frac {3 \cot ^7(c+d x)}{7 a^3 d}+\frac {7 \cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}+\frac {29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}+\frac {29 \cot (c+d x) \csc (c+d x)}{128 a^3 d} \]

[In]

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

[Out]

(29*ArcTanh[Cos[c + d*x]])/(128*a^3*d) + (4*Cot[c + d*x]^3)/(3*a^3*d) + (7*Cot[c + d*x]^5)/(5*a^3*d) + (3*Cot[
c + d*x]^7)/(7*a^3*d) + (29*Cot[c + d*x]*Csc[c + d*x])/(128*a^3*d) + (29*Cot[c + d*x]*Csc[c + d*x]^3)/(192*a^3
*d) - (23*Cot[c + d*x]*Csc[c + d*x]^5)/(48*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^7)/(8*a^3*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 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 ^2(c+d x) \csc ^7(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (-a^3 \cot ^2(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^5(c+d x)-3 a^3 \cot ^2(c+d x) \csc ^6(c+d x)+a^3 \cot ^2(c+d x) \csc ^7(c+d x)\right ) \, dx}{a^6} \\ & = -\frac {\int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a^3}+\frac {\int \cot ^2(c+d x) \csc ^7(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{a^3}-\frac {3 \int \cot ^2(c+d x) \csc ^6(c+d x) \, dx}{a^3} \\ & = -\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac {\int \csc ^7(c+d x) \, dx}{8 a^3}-\frac {\int \csc ^5(c+d x) \, dx}{2 a^3}-\frac {\text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^2 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = \frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac {5 \int \csc ^5(c+d x) \, dx}{48 a^3}-\frac {3 \int \csc ^3(c+d x) \, dx}{8 a^3}-\frac {\text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (x^2+2 x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = \frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {7 \cot ^5(c+d x)}{5 a^3 d}+\frac {3 \cot ^7(c+d x)}{7 a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac {29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac {5 \int \csc ^3(c+d x) \, dx}{64 a^3}-\frac {3 \int \csc (c+d x) \, dx}{16 a^3} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{16 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {7 \cot ^5(c+d x)}{5 a^3 d}+\frac {3 \cot ^7(c+d x)}{7 a^3 d}+\frac {29 \cot (c+d x) \csc (c+d x)}{128 a^3 d}+\frac {29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac {5 \int \csc (c+d x) \, dx}{128 a^3} \\ & = \frac {29 \text {arctanh}(\cos (c+d x))}{128 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {7 \cot ^5(c+d x)}{5 a^3 d}+\frac {3 \cot ^7(c+d x)}{7 a^3 d}+\frac {29 \cot (c+d x) \csc (c+d x)}{128 a^3 d}+\frac {29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.01 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.91 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (\csc ^4\left (\frac {1}{2} (c+d x)\right ) (1328-210 \csc (c+d x))+15 \csc ^8\left (\frac {1}{2} (c+d x)\right ) (-24+7 \csc (c+d x))+4 \csc ^6\left (\frac {1}{2} (c+d x)\right ) (-276+455 \csc (c+d x))-4 \csc ^2\left (\frac {1}{2} (c+d x)\right ) (-4864+3045 \csc (c+d x))-8 \left (6090 \csc (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\frac {1}{4} (2833+4616 \cos (c+d x)+1907 \cos (2 (c+d x))+304 \cos (3 (c+d x))) \sec ^8\left (\frac {1}{2} (c+d x)\right )-6090 \csc ^3(c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )-420 \csc ^5(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+14560 \csc ^7(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )+3360 \csc ^9(c+d x) \sin ^8\left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin ^7(c+d x)}{13762560 a^3 d (1+\sin (c+d x))^3} \]

[In]

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

[Out]

-1/13762560*((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^6*(Csc[(c + d*x)/2]^4*(1328 - 210*Csc[c + d*x]) + 15*Csc[(c
+ d*x)/2]^8*(-24 + 7*Csc[c + d*x]) + 4*Csc[(c + d*x)/2]^6*(-276 + 455*Csc[c + d*x]) - 4*Csc[(c + d*x)/2]^2*(-
4864 + 3045*Csc[c + d*x]) - 8*(6090*Csc[c + d*x]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + ((2833 + 46
16*Cos[c + d*x] + 1907*Cos[2*(c + d*x)] + 304*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^8)/4 - 6090*Csc[c + d*x]^3*Si
n[(c + d*x)/2]^2 - 420*Csc[c + d*x]^5*Sin[(c + d*x)/2]^4 + 14560*Csc[c + d*x]^7*Sin[(c + d*x)/2]^6 + 3360*Csc[
c + d*x]^9*Sin[(c + d*x)/2]^8))*Sin[c + d*x]^7)/(a^3*d*(1 + Sin[c + d*x])^3)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.36

method	result	size
risch	\(-\frac {3045 \,{\mathrm e}^{15 i \left (d x +c \right )}-26880 i {\mathrm e}^{12 i \left (d x +c \right )}-23345 \,{\mathrm e}^{13 i \left (d x +c \right )}+286720 i {\mathrm e}^{10 i \left (d x +c \right )}-51275 \,{\mathrm e}^{11 i \left (d x +c \right )}-170240 i {\mathrm e}^{8 i \left (d x +c \right )}+179095 \,{\mathrm e}^{9 i \left (d x +c \right )}-14336 i {\mathrm e}^{6 i \left (d x +c \right )}+179095 \,{\mathrm e}^{7 i \left (d x +c \right )}-109312 i {\mathrm e}^{4 i \left (d x +c \right )}-51275 \,{\mathrm e}^{5 i \left (d x +c \right )}+38912 i {\mathrm e}^{2 i \left (d x +c \right )}-23345 \,{\mathrm e}^{3 i \left (d x +c \right )}-4864 i+3045 \,{\mathrm e}^{i \left (d x +c \right )}}{6720 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}-\frac {29 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d \,a^{3}}+\frac {29 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d \,a^{3}}\)	\(226\)
parallelrisch	\(\frac {105 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-105 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-720 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+720 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2240 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2240 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4368 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4368 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5880 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5880 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3920 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3920 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6720 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6720 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+38640 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-38640 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-48720 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{215040 d \,a^{3}}\)	\(226\)
derivativedivides	\(\frac {\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {8 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {26 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+46 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {26}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-58 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {6}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {8}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {46}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {7}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d \,a^{3}}\)	\(228\)
default	\(\frac {\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {8 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {26 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+46 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {26}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-58 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {6}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {8}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {46}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {7}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d \,a^{3}}\)	\(228\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/6720*(3045*exp(15*I*(d*x+c))-26880*I*exp(12*I*(d*x+c))-23345*exp(13*I*(d*x+c))+286720*I*exp(10*I*(d*x+c))-5
1275*exp(11*I*(d*x+c))-170240*I*exp(8*I*(d*x+c))+179095*exp(9*I*(d*x+c))-14336*I*exp(6*I*(d*x+c))+179095*exp(7
*I*(d*x+c))-109312*I*exp(4*I*(d*x+c))-51275*exp(5*I*(d*x+c))+38912*I*exp(2*I*(d*x+c))-23345*exp(3*I*(d*x+c))-4
864*I+3045*exp(I*(d*x+c)))/d/a^3/(exp(2*I*(d*x+c))-1)^8-29/128/d/a^3*ln(exp(I*(d*x+c))-1)+29/128/d/a^3*ln(exp(
I*(d*x+c))+1)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.50 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {6090 \, \cos \left (d x + c\right )^{7} - 22330 \, \cos \left (d x + c\right )^{5} + 13510 \, \cos \left (d x + c\right )^{3} - 3045 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3045 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 256 \, {\left (38 \, \cos \left (d x + c\right )^{7} - 133 \, \cos \left (d x + c\right )^{5} + 140 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) + 6090 \, \cos \left (d x + c\right )}{26880 \, {\left (a^{3} d \cos \left (d x + c\right )^{8} - 4 \, a^{3} d \cos \left (d x + c\right )^{6} + 6 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/26880*(6090*cos(d*x + c)^7 - 22330*cos(d*x + c)^5 + 13510*cos(d*x + c)^3 - 3045*(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) + 3045*(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) - 256*(38*cos(d*x + c)^7 -
133*cos(d*x + c)^5 + 140*cos(d*x + c)^3)*sin(d*x + c) + 6090*cos(d*x + c))/(a^3*d*cos(d*x + c)^8 - 4*a^3*d*cos
(d*x + c)^6 + 6*a^3*d*cos(d*x + c)^4 - 4*a^3*d*cos(d*x + c)^2 + a^3*d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**9/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (150) = 300\).

Time = 0.25 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.13 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {38640 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6720 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3920 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5880 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4368 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {2240 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {720 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {105 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}{a^{3}} - \frac {48720 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {{\left (\frac {720 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2240 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4368 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {5880 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3920 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {6720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {38640 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 105\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{8}}{a^{3} \sin \left (d x + c\right )^{8}}}{215040 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/215040*((38640*sin(d*x + c)/(cos(d*x + c) + 1) - 6720*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 3920*sin(d*x + c
)^3/(cos(d*x + c) + 1)^3 + 5880*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 4368*sin(d*x + c)^5/(cos(d*x + c) + 1)^5
+ 2240*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 720*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 105*sin(d*x + c)^8/(co
s(d*x + c) + 1)^8)/a^3 - 48720*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 + (720*sin(d*x + c)/(cos(d*x + c) + 1)
- 2240*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 4368*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 5880*sin(d*x + c)^4/(
cos(d*x + c) + 1)^4 + 3920*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 6720*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 38
640*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 105)*(cos(d*x + c) + 1)^8/(a^3*sin(d*x + c)^8))/d

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.65 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {48720 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {132414 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 38640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3920 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4368 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}} - \frac {105 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 720 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2240 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4368 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5880 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3920 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6720 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 38640 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{24}}}{215040 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/215040*(48720*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (132414*tan(1/2*d*x + 1/2*c)^8 - 38640*tan(1/2*d*x + 1/2
*c)^7 + 6720*tan(1/2*d*x + 1/2*c)^6 + 3920*tan(1/2*d*x + 1/2*c)^5 - 5880*tan(1/2*d*x + 1/2*c)^4 + 4368*tan(1/2
*d*x + 1/2*c)^3 - 2240*tan(1/2*d*x + 1/2*c)^2 + 720*tan(1/2*d*x + 1/2*c) - 105)/(a^3*tan(1/2*d*x + 1/2*c)^8) -
(105*a^21*tan(1/2*d*x + 1/2*c)^8 - 720*a^21*tan(1/2*d*x + 1/2*c)^7 + 2240*a^21*tan(1/2*d*x + 1/2*c)^6 - 4368*
a^21*tan(1/2*d*x + 1/2*c)^5 + 5880*a^21*tan(1/2*d*x + 1/2*c)^4 - 3920*a^21*tan(1/2*d*x + 1/2*c)^3 - 6720*a^21*
tan(1/2*d*x + 1/2*c)^2 + 38640*a^21*tan(1/2*d*x + 1/2*c))/a^24)/d

Mupad [B] (verification not implemented)

Time = 12.30 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.62 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-105\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+720\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}-720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+4368\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-5880\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+3920\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-38640\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+38640\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3920\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+5880\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4368\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+48720\,\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 )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{215040\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^9*(a + a*sin(c + d*x))^3),x)

[Out]

-(105*cos(c/2 + (d*x)/2)^16 - 105*sin(c/2 + (d*x)/2)^16 + 720*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^15 - 720*c
os(c/2 + (d*x)/2)^15*sin(c/2 + (d*x)/2) - 2240*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^14 + 4368*cos(c/2 + (d*
x)/2)^3*sin(c/2 + (d*x)/2)^13 - 5880*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^12 + 3920*cos(c/2 + (d*x)/2)^5*si
n(c/2 + (d*x)/2)^11 + 6720*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^10 - 38640*cos(c/2 + (d*x)/2)^7*sin(c/2 + (
d*x)/2)^9 + 38640*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^7 - 6720*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^6
- 3920*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^5 + 5880*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^4 - 4368*cos
(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^3 + 2240*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^2 + 48720*log(sin(c/2
+ (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^8)/(215040*a^3*d*cos(c/2 + (d*x)/2)^8*s
in(c/2 + (d*x)/2)^8)