Integrand size = 29, antiderivative size = 168 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{64 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {3 \cot ^7(c+d x)}{7 a^2 d}-\frac {\cot ^9(c+d x)}{9 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{64 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{32 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{4 a^2 d} \] Output:
3/64*arctanh(cos(d*x+c))/a^2/d-2/5*cot(d*x+c)^5/a^2/d-3/7*cot(d*x+c)^7/a^2 /d-1/9*cot(d*x+c)^9/a^2/d+3/64*cot(d*x+c)*csc(d*x+c)/a^2/d+1/32*cot(d*x+c) *csc(d*x+c)^3/a^2/d-1/8*cot(d*x+c)*csc(d*x+c)^5/a^2/d+1/4*cot(d*x+c)^3*csc (d*x+c)^5/a^2/d
Time = 4.93 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.86 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\csc ^9(c+d x) \left (-451584 \cos (c+d x)-155904 \cos (3 (c+d x))+20736 \cos (5 (c+d x))+14976 \cos (7 (c+d x))-1664 \cos (9 (c+d x))+119070 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-119070 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+212940 \sin (2 (c+d x))-79380 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+79380 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+195300 \sin (4 (c+d x))+34020 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-34020 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+16380 \sin (6 (c+d x))-8505 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+8505 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-1890 \sin (8 (c+d x))+945 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))-945 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))\right )}{5160960 a^2 d} \] Input:
Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]
Output:
(Csc[c + d*x]^9*(-451584*Cos[c + d*x] - 155904*Cos[3*(c + d*x)] + 20736*Co s[5*(c + d*x)] + 14976*Cos[7*(c + d*x)] - 1664*Cos[9*(c + d*x)] + 119070*L og[Cos[(c + d*x)/2]]*Sin[c + d*x] - 119070*Log[Sin[(c + d*x)/2]]*Sin[c + d *x] + 212940*Sin[2*(c + d*x)] - 79380*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x )] + 79380*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 195300*Sin[4*(c + d*x) ] + 34020*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 34020*Log[Sin[(c + d*x) /2]]*Sin[5*(c + d*x)] + 16380*Sin[6*(c + d*x)] - 8505*Log[Cos[(c + d*x)/2] ]*Sin[7*(c + d*x)] + 8505*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 1890*Si n[8*(c + d*x)] + 945*Log[Cos[(c + d*x)/2]]*Sin[9*(c + d*x)] - 945*Log[Sin[ (c + d*x)/2]]*Sin[9*(c + d*x)]))/(5160960*a^2*d)
Time = 0.68 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3354, 3042, 3352, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^{10} (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cot ^4(c+d x) \csc ^6(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x)^4 (a-a \sin (c+d x))^2}{\sin (c+d x)^{10}}dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^2 \cot ^4(c+d x) \csc ^6(c+d x)-2 a^2 \cot ^4(c+d x) \csc ^5(c+d x)+a^2 \cot ^4(c+d x) \csc ^4(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{64 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{64 d}}{a^4}\) |
Input:
Int[(Cot[c + d*x]^8*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]
Output:
((3*a^2*ArcTanh[Cos[c + d*x]])/(64*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (3* a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(9*d) + (3*a^2*Cot[c + d* x]*Csc[c + d*x])/(64*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(32*d) - (a^2* Cot[c + d*x]*Csc[c + d*x]^5)/(8*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/( 4*d))/a^4
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] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(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]
Time = 9.88 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {18}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {16}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-\frac {5}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {8}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}}{512 d \,a^{2}}\) | \(202\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {18}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {16}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-\frac {5}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {8}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}}{512 d \,a^{2}}\) | \(202\) |
| risch | \(-\frac {945 \,{\mathrm e}^{17 i \left (d x +c \right )}+330624 i {\mathrm e}^{8 i \left (d x +c \right )}-8190 \,{\mathrm e}^{15 i \left (d x +c \right )}+120960 i {\mathrm e}^{10 i \left (d x +c \right )}-97650 \,{\mathrm e}^{13 i \left (d x +c \right )}-40320 i {\mathrm e}^{14 i \left (d x +c \right )}-106470 \,{\mathrm e}^{11 i \left (d x +c \right )}+8064 i {\mathrm e}^{6 i \left (d x +c \right )}+147840 i {\mathrm e}^{12 i \left (d x +c \right )}+106470 \,{\mathrm e}^{7 i \left (d x +c \right )}-14976 i {\mathrm e}^{2 i \left (d x +c \right )}+97650 \,{\mathrm e}^{5 i \left (d x +c \right )}+19584 i {\mathrm e}^{4 i \left (d x +c \right )}+8190 \,{\mathrm e}^{3 i \left (d x +c \right )}+1664 i-945 \,{\mathrm e}^{i \left (d x +c \right )}}{10080 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{64 d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{64 d \,a^{2}}\) | \(238\) |
Input:
int(cot(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/512/d/a^2*(1/9*tan(1/2*d*x+1/2*c)^9-1/2*tan(1/2*d*x+1/2*c)^8+5/7*tan(1/2 *d*x+1/2*c)^7-8/5*tan(1/2*d*x+1/2*c)^5+4*tan(1/2*d*x+1/2*c)^4-16/3*tan(1/2 *d*x+1/2*c)^3+18*tan(1/2*d*x+1/2*c)-18/tan(1/2*d*x+1/2*c)-24*ln(tan(1/2*d* x+1/2*c))+16/3/tan(1/2*d*x+1/2*c)^3-4/tan(1/2*d*x+1/2*c)^4-1/9/tan(1/2*d*x +1/2*c)^9-5/7/tan(1/2*d*x+1/2*c)^7+1/2/tan(1/2*d*x+1/2*c)^8+8/5/tan(1/2*d* x+1/2*c)^5)
Time = 0.09 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.60 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3328 \, \cos \left (d x + c\right )^{9} - 14976 \, \cos \left (d x + c\right )^{7} + 16128 \, \cos \left (d x + c\right )^{5} - 945 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 945 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 630 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, {\left (a^{2} d \cos \left (d x + c\right )^{8} - 4 \, a^{2} d \cos \left (d x + c\right )^{6} + 6 \, a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/40320*(3328*cos(d*x + c)^9 - 14976*cos(d*x + c)^7 + 16128*cos(d*x + c)^ 5 - 945*(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) + 945*(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*co s(d*x + c) + 1/2)*sin(d*x + c) + 630*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*cos(d*x + c)^3 + 3*cos(d*x + c))*sin(d*x + c))/((a^2*d*cos(d*x + c)^ 8 - 4*a^2*d*cos(d*x + c)^6 + 6*a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c) ^2 + a^2*d)*sin(d*x + c))
Timed out. \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**8*csc(d*x+c)**2/(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (152) = 304\).
Time = 0.04 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.87 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {11340 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3360 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2520 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {315 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {70 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{2}} - \frac {15120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {{\left (\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {450 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1008 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2520 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3360 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {11340 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 70\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{9}}{a^{2} \sin \left (d x + c\right )^{9}}}{322560 \, d} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
1/322560*((11340*sin(d*x + c)/(cos(d*x + c) + 1) - 3360*sin(d*x + c)^3/(co s(d*x + c) + 1)^3 + 2520*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 1008*sin(d* x + c)^5/(cos(d*x + c) + 1)^5 + 450*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 315*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 70*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^2 - 15120*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 + (315*sin(d* x + c)/(cos(d*x + c) + 1) - 450*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1008 *sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 2520*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 3360*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 11340*sin(d*x + c)^8/(c os(d*x + c) + 1)^8 - 70)*(cos(d*x + c) + 1)^9/(a^2*sin(d*x + c)^9))/d
Time = 0.23 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.46 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {15120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {42774 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 11340 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 3360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1008 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 70}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}} - \frac {70 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 315 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 450 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1008 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2520 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3360 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11340 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{18}}}{322560 \, d} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
-1/322560*(15120*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (42774*tan(1/2*d*x + 1/2*c)^9 - 11340*tan(1/2*d*x + 1/2*c)^8 + 3360*tan(1/2*d*x + 1/2*c)^6 - 2 520*tan(1/2*d*x + 1/2*c)^5 + 1008*tan(1/2*d*x + 1/2*c)^4 - 450*tan(1/2*d*x + 1/2*c)^2 + 315*tan(1/2*d*x + 1/2*c) - 70)/(a^2*tan(1/2*d*x + 1/2*c)^9) - (70*a^16*tan(1/2*d*x + 1/2*c)^9 - 315*a^16*tan(1/2*d*x + 1/2*c)^8 + 450* a^16*tan(1/2*d*x + 1/2*c)^7 - 1008*a^16*tan(1/2*d*x + 1/2*c)^5 + 2520*a^16 *tan(1/2*d*x + 1/2*c)^4 - 3360*a^16*tan(1/2*d*x + 1/2*c)^3 + 11340*a^16*ta n(1/2*d*x + 1/2*c))/a^18)/d
Time = 34.70 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.30 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {70\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-70\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+315\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-450\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-11340\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+11340\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+450\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+15120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{322560\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \] Input:
int(cot(c + d*x)^8/(sin(c + d*x)^2*(a + a*sin(c + d*x))^2),x)
Output:
-(70*cos(c/2 + (d*x)/2)^18 - 70*sin(c/2 + (d*x)/2)^18 + 315*cos(c/2 + (d*x )/2)*sin(c/2 + (d*x)/2)^17 - 315*cos(c/2 + (d*x)/2)^17*sin(c/2 + (d*x)/2) - 450*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^16 + 1008*cos(c/2 + (d*x)/2) ^4*sin(c/2 + (d*x)/2)^14 - 2520*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^13 + 3360*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^12 - 11340*cos(c/2 + (d*x) /2)^8*sin(c/2 + (d*x)/2)^10 + 11340*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/ 2)^8 - 3360*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^6 + 2520*cos(c/2 + (d *x)/2)^13*sin(c/2 + (d*x)/2)^5 - 1008*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x )/2)^4 + 450*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d*x)/2)^2 + 15120*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 )/(322560*a^2*d*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^9)
Time = 0.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {-1664 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}+945 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-832 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+630 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+4416 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-7560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+5040 \cos \left (d x +c \right ) \sin \left (d x +c \right )-2240 \cos \left (d x +c \right )-945 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{9}}{20160 \sin \left (d x +c \right )^{9} a^{2} d} \] Input:
int(cot(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c))^2,x)
Output:
( - 1664*cos(c + d*x)*sin(c + d*x)**8 + 945*cos(c + d*x)*sin(c + d*x)**7 - 832*cos(c + d*x)*sin(c + d*x)**6 + 630*cos(c + d*x)*sin(c + d*x)**5 + 441 6*cos(c + d*x)*sin(c + d*x)**4 - 7560*cos(c + d*x)*sin(c + d*x)**3 + 320*c os(c + d*x)*sin(c + d*x)**2 + 5040*cos(c + d*x)*sin(c + d*x) - 2240*cos(c + d*x) - 945*log(tan((c + d*x)/2))*sin(c + d*x)**9)/(20160*sin(c + d*x)**9 *a**2*d)