Integrand size = 29, antiderivative size = 210 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{128 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {\cot ^{11}(c+d x)}{11 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{128 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d} \] Output:
3/128*arctanh(cos(d*x+c))/a^2/d-2/5*cot(d*x+c)^5/a^2/d-5/7*cot(d*x+c)^7/a^ 2/d-4/9*cot(d*x+c)^9/a^2/d-1/11*cot(d*x+c)^11/a^2/d+3/128*cot(d*x+c)*csc(d *x+c)/a^2/d+1/64*cot(d*x+c)*csc(d*x+c)^3/a^2/d+1/80*cot(d*x+c)*csc(d*x+c)^ 5/a^2/d-3/40*cot(d*x+c)*csc(d*x+c)^7/a^2/d+1/5*cot(d*x+c)^3*csc(d*x+c)^7/a ^2/d
Time = 7.75 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (2661120 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\cot (c+d x) \csc ^{10}(c+d x) (-5402624-5752832 \cos (2 (c+d x))+346112 \cos (4 (c+d x))+583168 \cos (6 (c+d x))-104448 \cos (8 (c+d x))+8704 \cos (10 (c+d x))+2457378 \sin (c+d x)+5907132 \sin (3 (c+d x))+656964 \sin (5 (c+d x))-121275 \sin (7 (c+d x))+10395 \sin (9 (c+d x)))\right )}{113541120 a^2 d (1+\sin (c+d x))^2} \] Input:
Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]
Output:
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(2661120*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + Cot[c + d*x]*Csc[c + d*x]^10*(-5402624 - 5752832 *Cos[2*(c + d*x)] + 346112*Cos[4*(c + d*x)] + 583168*Cos[6*(c + d*x)] - 10 4448*Cos[8*(c + d*x)] + 8704*Cos[10*(c + d*x)] + 2457378*Sin[c + d*x] + 59 07132*Sin[3*(c + d*x)] + 656964*Sin[5*(c + d*x)] - 121275*Sin[7*(c + d*x)] + 10395*Sin[9*(c + d*x)])))/(113541120*a^2*d*(1 + Sin[c + d*x])^2)
Time = 0.73 (sec) , antiderivative size = 214, 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.
| \(\displaystyle \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^{12} (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cot ^4(c+d x) \csc ^8(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x)^4 (a-a \sin (c+d x))^2}{\sin (c+d x)^{12}}dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^2 \cot ^4(c+d x) \csc ^8(c+d x)-2 a^2 \cot ^4(c+d x) \csc ^7(c+d x)+a^2 \cot ^4(c+d x) \csc ^6(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}-\frac {4 a^2 \cot ^9(c+d x)}{9 d}-\frac {5 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{5 d}-\frac {3 a^2 \cot (c+d x) \csc ^7(c+d x)}{40 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{80 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{128 d}}{a^4}\) |
Input:
Int[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]
Output:
((3*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (5 *a^2*Cot[c + d*x]^7)/(7*d) - (4*a^2*Cot[c + d*x]^9)/(9*d) - (a^2*Cot[c + d *x]^11)/(11*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(80*d) - ( 3*a^2*Cot[c + d*x]*Csc[c + d*x]^7)/(40*d) + (a^2*Cot[c + d*x]^3*Csc[c + d* x]^7)/(5*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]
Result contains complex when optimal does not.
Time = 17.59 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.30
| method | result | size |
| risch | \(-\frac {10395 \,{\mathrm e}^{21 i \left (d x +c \right )}-110880 \,{\mathrm e}^{19 i \left (d x +c \right )}-2534400 i {\mathrm e}^{8 i \left (d x +c \right )}+535689 \,{\mathrm e}^{17 i \left (d x +c \right )}-8279040 i {\mathrm e}^{14 i \left (d x +c \right )}+6564096 \,{\mathrm e}^{15 i \left (d x +c \right )}-506880 i {\mathrm e}^{6 i \left (d x +c \right )}+8364510 \,{\mathrm e}^{13 i \left (d x +c \right )}-12536832 i {\mathrm e}^{12 i \left (d x +c \right )}+2365440 i {\mathrm e}^{16 i \left (d x +c \right )}-8364510 \,{\mathrm e}^{9 i \left (d x +c \right )}-191488 i {\mathrm e}^{2 i \left (d x +c \right )}-6564096 \,{\mathrm e}^{7 i \left (d x +c \right )}-20579328 i {\mathrm e}^{10 i \left (d x +c \right )}-535689 \,{\mathrm e}^{5 i \left (d x +c \right )}+957440 i {\mathrm e}^{4 i \left (d x +c \right )}+110880 \,{\mathrm e}^{3 i \left (d x +c \right )}+17408 i-10395 \,{\mathrm e}^{i \left (d x +c \right )}}{221760 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d \,a^{2}}\) | \(272\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {27 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+38 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {27}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {22}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {3}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {38}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {7}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}}{2048 d \,a^{2}}\) | \(304\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {27 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+38 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {27}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {22}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {3}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {38}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {7}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}}{2048 d \,a^{2}}\) | \(304\) |
Input:
int(cot(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-1/221760*(10395*exp(21*I*(d*x+c))-110880*exp(19*I*(d*x+c))-2534400*I*exp( 8*I*(d*x+c))+535689*exp(17*I*(d*x+c))-8279040*I*exp(14*I*(d*x+c))+6564096* exp(15*I*(d*x+c))-506880*I*exp(6*I*(d*x+c))+8364510*exp(13*I*(d*x+c))-1253 6832*I*exp(12*I*(d*x+c))+2365440*I*exp(16*I*(d*x+c))-8364510*exp(9*I*(d*x+ c))-191488*I*exp(2*I*(d*x+c))-6564096*exp(7*I*(d*x+c))-20579328*I*exp(10*I *(d*x+c))-535689*exp(5*I*(d*x+c))+957440*I*exp(4*I*(d*x+c))+110880*exp(3*I *(d*x+c))+17408*I-10395*exp(I*(d*x+c)))/d/a^2/(exp(2*I*(d*x+c))-1)^11-3/12 8/d/a^2*ln(exp(I*(d*x+c))-1)+3/128/d/a^2*ln(exp(I*(d*x+c))+1)
Time = 0.10 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.54 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {34816 \, \cos \left (d x + c\right )^{11} - 191488 \, \cos \left (d x + c\right )^{9} + 430848 \, \cos \left (d x + c\right )^{7} - 354816 \, \cos \left (d x + c\right )^{5} - 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 1386 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} + 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, {\left (a^{2} d \cos \left (d x + c\right )^{10} - 5 \, a^{2} d \cos \left (d x + c\right )^{8} + 10 \, a^{2} d \cos \left (d x + c\right )^{6} - 10 \, a^{2} d \cos \left (d x + c\right )^{4} + 5 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/887040*(34816*cos(d*x + c)^11 - 191488*cos(d*x + c)^9 + 430848*cos(d*x + c)^7 - 354816*cos(d*x + c)^5 - 10395*(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*c os(d*x + c) + 1/2)*sin(d*x + c) + 10395*(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)*sin(d*x + c) + 1386*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 + 128*cos(d*x + c)^5 + 70*cos(d*x + c)^3 - 15*cos(d*x + c))*sin(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)*sin(d*x + c))
Timed out. \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**8*csc(d*x+c)**4/(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (190) = 380\).
Time = 0.04 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.26 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
1/7096320*((131670*sin(d*x + c)/(cos(d*x + c) + 1) - 13860*sin(d*x + c)^2/ (cos(d*x + c) + 1)^2 - 25410*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 27720*s in(d*x + c)^4/(cos(d*x + c) + 1)^4 - 18711*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 6930*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1485*sin(d*x + c)^7/(cos (d*x + c) + 1)^7 - 3465*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 2695*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 1386*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 315*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/a^2 - 166320*log(sin(d*x + c) /(cos(d*x + c) + 1))/a^2 + (1386*sin(d*x + c)/(cos(d*x + c) + 1) - 2695*si n(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3465*sin(d*x + c)^3/(cos(d*x + c) + 1) ^3 - 1485*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 6930*sin(d*x + c)^5/(cos(d *x + c) + 1)^5 + 18711*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 27720*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 25410*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 13860*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 131670*sin(d*x + c)^10/(cos(d *x + c) + 1)^10 - 315)*(cos(d*x + c) + 1)^11/(a^2*sin(d*x + c)^11))/d
Time = 0.21 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.72 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {166320 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {502266 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 131670 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 13860 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 25410 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 27720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 18711 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 6930 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1485 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2695 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1386 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 315}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}} - \frac {315 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1386 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 2695 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3465 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1485 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6930 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 18711 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 27720 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 25410 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 13860 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 131670 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{22}}}{7096320 \, d} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
-1/7096320*(166320*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (502266*tan(1/2*d* x + 1/2*c)^11 - 131670*tan(1/2*d*x + 1/2*c)^10 + 13860*tan(1/2*d*x + 1/2*c )^9 + 25410*tan(1/2*d*x + 1/2*c)^8 - 27720*tan(1/2*d*x + 1/2*c)^7 + 18711* tan(1/2*d*x + 1/2*c)^6 - 6930*tan(1/2*d*x + 1/2*c)^5 - 1485*tan(1/2*d*x + 1/2*c)^4 + 3465*tan(1/2*d*x + 1/2*c)^3 - 2695*tan(1/2*d*x + 1/2*c)^2 + 138 6*tan(1/2*d*x + 1/2*c) - 315)/(a^2*tan(1/2*d*x + 1/2*c)^11) - (315*a^20*ta n(1/2*d*x + 1/2*c)^11 - 1386*a^20*tan(1/2*d*x + 1/2*c)^10 + 2695*a^20*tan( 1/2*d*x + 1/2*c)^9 - 3465*a^20*tan(1/2*d*x + 1/2*c)^8 + 1485*a^20*tan(1/2* d*x + 1/2*c)^7 + 6930*a^20*tan(1/2*d*x + 1/2*c)^6 - 18711*a^20*tan(1/2*d*x + 1/2*c)^5 + 27720*a^20*tan(1/2*d*x + 1/2*c)^4 - 25410*a^20*tan(1/2*d*x + 1/2*c)^3 - 13860*a^20*tan(1/2*d*x + 1/2*c)^2 + 131670*a^20*tan(1/2*d*x + 1/2*c))/a^22)/d
Time = 38.73 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.76 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
int(cot(c + d*x)^8/(sin(c + d*x)^4*(a + a*sin(c + d*x))^2),x)
Output:
-(315*cos(c/2 + (d*x)/2)^22 - 315*sin(c/2 + (d*x)/2)^22 + 1386*cos(c/2 + ( d*x)/2)*sin(c/2 + (d*x)/2)^21 - 1386*cos(c/2 + (d*x)/2)^21*sin(c/2 + (d*x) /2) - 2695*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^20 + 3465*cos(c/2 + (d* x)/2)^3*sin(c/2 + (d*x)/2)^19 - 1485*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/ 2)^18 - 6930*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^17 + 18711*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^16 - 27720*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d *x)/2)^15 + 25410*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^14 + 13860*cos(c /2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^13 - 131670*cos(c/2 + (d*x)/2)^10*sin(c /2 + (d*x)/2)^12 + 131670*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^10 - 13 860*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^9 - 25410*cos(c/2 + (d*x)/2)^ 14*sin(c/2 + (d*x)/2)^8 + 27720*cos(c/2 + (d*x)/2)^15*sin(c/2 + (d*x)/2)^7 - 18711*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d*x)/2)^6 + 6930*cos(c/2 + (d*x) /2)^17*sin(c/2 + (d*x)/2)^5 + 1485*cos(c/2 + (d*x)/2)^18*sin(c/2 + (d*x)/2 )^4 - 3465*cos(c/2 + (d*x)/2)^19*sin(c/2 + (d*x)/2)^3 + 2695*cos(c/2 + (d* x)/2)^20*sin(c/2 + (d*x)/2)^2 + 166320*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d *x)/2))*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^11)/(7096320*a^2*d*cos(c/ 2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^11)
Time = 0.22 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {-17408 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10}+10395 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}-8704 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}+6930 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-6528 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+5544 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+68480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-121968 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+4480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+88704 \cos \left (d x +c \right ) \sin \left (d x +c \right )-40320 \cos \left (d x +c \right )-10395 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{11}}{443520 \sin \left (d x +c \right )^{11} a^{2} d} \] Input:
int(cot(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c))^2,x)
Output:
( - 17408*cos(c + d*x)*sin(c + d*x)**10 + 10395*cos(c + d*x)*sin(c + d*x)* *9 - 8704*cos(c + d*x)*sin(c + d*x)**8 + 6930*cos(c + d*x)*sin(c + d*x)**7 - 6528*cos(c + d*x)*sin(c + d*x)**6 + 5544*cos(c + d*x)*sin(c + d*x)**5 + 68480*cos(c + d*x)*sin(c + d*x)**4 - 121968*cos(c + d*x)*sin(c + d*x)**3 + 4480*cos(c + d*x)*sin(c + d*x)**2 + 88704*cos(c + d*x)*sin(c + d*x) - 40 320*cos(c + d*x) - 10395*log(tan((c + d*x)/2))*sin(c + d*x)**11)/(443520*s in(c + d*x)**11*a**2*d)