\(\int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [720]

Optimal result

Integrand size = 29, antiderivative size = 176 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{256 a d}+\frac {\cot ^7(c+d x)}{7 a d}+\frac {\cot ^9(c+d x)}{9 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d} \]

[Out]

3/256*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+3/256*cot(d*x+c)*csc(d*x+c)/a/d+1/128*
cot(d*x+c)*csc(d*x+c)^3/a/d-1/32*cot(d*x+c)*csc(d*x+c)^5/a/d+1/16*cot(d*x+c)^3*csc(d*x+c)^5/a/d-1/10*cot(d*x+c
)^5*csc(d*x+c)^5/a/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2918, 2691, 3853, 3855, 2687, 14} \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{256 a d}+\frac {\cot ^9(c+d x)}{9 a d}+\frac {\cot ^7(c+d x)}{7 a d}-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d} \]

[In]

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

[Out]

(3*ArcTanh[Cos[c + d*x]])/(256*a*d) + Cot[c + d*x]^7/(7*a*d) + Cot[c + d*x]^9/(9*a*d) + (3*Cot[c + d*x]*Csc[c
+ d*x])/(256*a*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(128*a*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(32*a*d) + (Cot[c
+ d*x]^3*Csc[c + d*x]^5)/(16*a*d) - (Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*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 ^4(c+d x) \, dx}{a}+\frac {\int \cot ^6(c+d x) \csc ^5(c+d x) \, dx}{a} \\ & = -\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac {\int \cot ^4(c+d x) \csc ^5(c+d x) \, dx}{2 a}-\frac {\text {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a d} \\ & = \frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac {3 \int \cot ^2(c+d x) \csc ^5(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 {\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac {\int \csc ^5(c+d x) \, dx}{32 a} \\ & = \frac {\cot ^7(c+d x)}{7 a d}+\frac {\cot ^9(c+d x)}{9 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac {3 \int \csc ^3(c+d x) \, dx}{128 a} \\ & = \frac {\cot ^7(c+d x)}{7 a d}+\frac {\cot ^9(c+d x)}{9 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac {3 \int \csc (c+d x) \, dx}{256 a} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{256 a d}+\frac {\cot ^7(c+d x)}{7 a d}+\frac {\cot ^9(c+d x)}{9 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(386\) vs. \(2(176)=352\).

Time = 2.53 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.19 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^9(c+d x) \left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (2367540 \cos (c+d x)+1307880 \cos (3 (c+d x))+436968 \cos (5 (c+d x))+18270 \cos (7 (c+d x))-1890 \cos (9 (c+d x))-119070 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+198450 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-113400 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+42525 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-9450 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+945 \cos (10 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+119070 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-198450 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+113400 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-42525 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+9450 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-945 \cos (10 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-537600 \sin (2 (c+d x))-522240 \sin (4 (c+d x))-207360 \sin (6 (c+d x))-25600 \sin (8 (c+d x))+2560 \sin (10 (c+d x))\right )}{165150720 a d (1+\csc (c+d x))} \]

[In]

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

[Out]

-1/165150720*(Csc[c + d*x]^9*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(2367540*Cos[c + d*x] + 1307880*Cos[3*(c
+ d*x)] + 436968*Cos[5*(c + d*x)] + 18270*Cos[7*(c + d*x)] - 1890*Cos[9*(c + d*x)] - 119070*Log[Cos[(c + d*x)/
2]] + 198450*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 113400*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 42525*Co
s[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 9450*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 945*Cos[10*(c + d*x)]*Log
[Cos[(c + d*x)/2]] + 119070*Log[Sin[(c + d*x)/2]] - 198450*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 113400*Cos
[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 42525*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 9450*Cos[8*(c + d*x)]*Log
[Sin[(c + d*x)/2]] - 945*Cos[10*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 537600*Sin[2*(c + d*x)] - 522240*Sin[4*(c +
d*x)] - 207360*Sin[6*(c + d*x)] - 25600*Sin[8*(c + d*x)] + 2560*Sin[10*(c + d*x)]))/(a*d*(1 + Csc[c + d*x]))

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.43

method	result	size
derivativedivides	\(\frac {\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}-\frac {2 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {12}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {1}{10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {6}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}+\frac {16}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{1024 d a}\)	\(252\)
default	\(\frac {\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}-\frac {2 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {12}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {1}{10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {6}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}+\frac {16}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{1024 d a}\)	\(252\)
parallelrisch	\(\frac {126 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-126 \left (\cot ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-280 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+280 \left (\cot ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-315 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1080 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1080 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-630 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+630 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2520 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2520 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6720 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6720 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1260 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1260 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15120 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15120 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{1290240 d a}\)	\(252\)
risch	\(-\frac {945 \,{\mathrm e}^{19 i \left (d x +c \right )}-107520 i {\mathrm e}^{14 i \left (d x +c \right )}-9135 \,{\mathrm e}^{17 i \left (d x +c \right )}-161280 i {\mathrm e}^{16 i \left (d x +c \right )}-218484 \,{\mathrm e}^{15 i \left (d x +c \right )}+414720 i {\mathrm e}^{6 i \left (d x +c \right )}-653940 \,{\mathrm e}^{13 i \left (d x +c \right )}-537600 i {\mathrm e}^{12 i \left (d x +c \right )}-1183770 \,{\mathrm e}^{11 i \left (d x +c \right )}-1183770 \,{\mathrm e}^{9 i \left (d x +c \right )}+322560 i {\mathrm e}^{10 i \left (d x +c \right )}-653940 \,{\mathrm e}^{7 i \left (d x +c \right )}+46080 i {\mathrm e}^{4 i \left (d x +c \right )}-218484 \,{\mathrm e}^{5 i \left (d x +c \right )}+25600 i {\mathrm e}^{2 i \left (d x +c \right )}-9135 \,{\mathrm e}^{3 i \left (d x +c \right )}-2560 i+945 \,{\mathrm e}^{i \left (d x +c \right )}}{40320 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d a}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d a}\)	\(260\)

[In]

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

[Out]

1/1024/d/a*(1/10*tan(1/2*d*x+1/2*c)^10-2/9*tan(1/2*d*x+1/2*c)^9-1/4*tan(1/2*d*x+1/2*c)^8+6/7*tan(1/2*d*x+1/2*c
)^7-1/2*tan(1/2*d*x+1/2*c)^6+2*tan(1/2*d*x+1/2*c)^4-16/3*tan(1/2*d*x+1/2*c)^3+tan(1/2*d*x+1/2*c)^2+12*tan(1/2*
d*x+1/2*c)-12/tan(1/2*d*x+1/2*c)+1/4/tan(1/2*d*x+1/2*c)^8-1/10/tan(1/2*d*x+1/2*c)^10+1/2/tan(1/2*d*x+1/2*c)^6-
1/tan(1/2*d*x+1/2*c)^2-12*ln(tan(1/2*d*x+1/2*c))-6/7/tan(1/2*d*x+1/2*c)^7-2/tan(1/2*d*x+1/2*c)^4+2/9/tan(1/2*d
*x+1/2*c)^9+16/3/tan(1/2*d*x+1/2*c)^3)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.55 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {1890 \, \cos \left (d x + c\right )^{9} - 8820 \, \cos \left (d x + c\right )^{7} - 16128 \, \cos \left (d x + c\right )^{5} + 8820 \, \cos \left (d x + c\right )^{3} - 945 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 945 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2560 \, {\left (2 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right ) - 1890 \, \cos \left (d x + c\right )}{161280 \, {\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \]

[In]

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

[Out]

-1/161280*(1890*cos(d*x + c)^9 - 8820*cos(d*x + c)^7 - 16128*cos(d*x + c)^5 + 8820*cos(d*x + c)^3 - 945*(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) + 945*(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) - 2560*(2*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*sin(d*x + c) - 1890*cos(d*x
+ c))/(a*d*cos(d*x + c)^10 - 5*a*d*cos(d*x + c)^8 + 10*a*d*cos(d*x + c)^6 - 10*a*d*cos(d*x + c)^4 + 5*a*d*cos(
d*x + c)^2 - a*d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**11/(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. 394 vs. \(2 (160) = 320\).

Time = 0.21 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.24 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {15120 \, \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 {2520 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {630 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1080 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {315 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {280 \, \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} - \frac {15120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {280 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {315 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1080 \, \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 {2520 \, \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 {15120 \, \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 \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)),x, algorithm="maxima")

[Out]

1/1290240*((15120*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 + 2520*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 630*sin(d*x + c)^6/(cos(d*x + c) + 1)^6
+ 1080*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 315*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 280*sin(d*x + c)^9/(co
s(d*x + c) + 1)^9 + 126*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)/a - 15120*log(sin(d*x + c)/(cos(d*x + c) + 1))/
a + (280*sin(d*x + c)/(cos(d*x + c) + 1) + 315*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1080*sin(d*x + c)^3/(cos(
d*x + c) + 1)^3 + 630*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 2520*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 6720*si
n(d*x + c)^7/(cos(d*x + c) + 1)^7 - 1260*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 15120*sin(d*x + c)^9/(cos(d*x +
c) + 1)^9 - 126)*(cos(d*x + c) + 1)^10/(a*sin(d*x + c)^10))/d

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.72 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {15120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {126 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 280 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 315 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1080 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 630 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2520 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 6720 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1260 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15120 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{10}} - \frac {44286 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 15120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1260 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 630 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1080 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 126}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{1290240 \, d} \]

[In]

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

[Out]

-1/1290240*(15120*log(abs(tan(1/2*d*x + 1/2*c)))/a - (126*a^9*tan(1/2*d*x + 1/2*c)^10 - 280*a^9*tan(1/2*d*x +
1/2*c)^9 - 315*a^9*tan(1/2*d*x + 1/2*c)^8 + 1080*a^9*tan(1/2*d*x + 1/2*c)^7 - 630*a^9*tan(1/2*d*x + 1/2*c)^6 +
2520*a^9*tan(1/2*d*x + 1/2*c)^4 - 6720*a^9*tan(1/2*d*x + 1/2*c)^3 + 1260*a^9*tan(1/2*d*x + 1/2*c)^2 + 15120*a
^9*tan(1/2*d*x + 1/2*c))/a^10 - (44286*tan(1/2*d*x + 1/2*c)^10 - 15120*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 - 2520*tan(1/2*d*x + 1/2*c)^6 + 630*tan(1/2*d*x + 1/2*c)^4 - 1080*
tan(1/2*d*x + 1/2*c)^3 + 315*tan(1/2*d*x + 1/2*c)^2 + 280*tan(1/2*d*x + 1/2*c) - 126)/(a*tan(1/2*d*x + 1/2*c)^
10))/d

Mupad [B] (verification not implemented)

Time = 15.08 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.74 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {126\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}-126\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+280\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}-280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-1080\,{\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}-2520\,{\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}-15120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+15120\,{\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+2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-630\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+1080\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+15120\,\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\,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))),x)

[Out]

-(126*cos(c/2 + (d*x)/2)^20 - 126*sin(c/2 + (d*x)/2)^20 + 280*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^19 - 280*c
os(c/2 + (d*x)/2)^19*sin(c/2 + (d*x)/2) + 315*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^18 - 1080*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 - 2520*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 - 15120*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^11 + 15120*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 + 2520*cos
(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^6 - 630*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d*x)/2)^4 + 1080*cos(c/2 + (d*x
)/2)^17*sin(c/2 + (d*x)/2)^3 - 315*cos(c/2 + (d*x)/2)^18*sin(c/2 + (d*x)/2)^2 + 15120*log(sin(c/2 + (d*x)/2)/c
os(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^10)/(1290240*a*d*cos(c/2 + (d*x)/2)^10*sin(c/2 + (
d*x)/2)^10)