Integrand size = 27, antiderivative size = 124 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d} \] Output:
3/16*arctanh(cos(d*x+c))/a^2/d+2/3*cot(d*x+c)^3/a^2/d+2/5*cot(d*x+c)^5/a^2 /d+3/16*cot(d*x+c)*csc(d*x+c)/a^2/d-5/24*cot(d*x+c)*csc(d*x+c)^3/a^2/d-1/6 *cot(d*x+c)*csc(d*x+c)^5/a^2/d
Time = 2.25 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.85 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc ^6(c+d x) \left (1500 \cos (c+d x)-130 \cos (3 (c+d x))-90 \cos (5 (c+d x))-450 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+675 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-270 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+45 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+450 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-675 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+270 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-45 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-960 \sin (2 (c+d x))-384 \sin (4 (c+d x))+64 \sin (6 (c+d x))\right )}{7680 a^2 d} \] Input:
Integrate[(Cot[c + d*x]^6*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]
Output:
-1/7680*(Csc[c + d*x]^6*(1500*Cos[c + d*x] - 130*Cos[3*(c + d*x)] - 90*Cos [5*(c + d*x)] - 450*Log[Cos[(c + d*x)/2]] + 675*Cos[2*(c + d*x)]*Log[Cos[( c + d*x)/2]] - 270*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 45*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 450*Log[Sin[(c + d*x)/2]] - 675*Cos[2*(c + d *x)]*Log[Sin[(c + d*x)/2]] + 270*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 45*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 960*Sin[2*(c + d*x)] - 384*Sin [4*(c + d*x)] + 64*Sin[6*(c + d*x)]))/(a^2*d)
Time = 0.60 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, 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.
| \(\displaystyle \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^7 (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cot ^2(c+d x) \csc ^5(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x)^2 (a-a \sin (c+d x))^2}{\sin (c+d x)^7}dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^2 \cot ^2(c+d x) \csc ^5(c+d x)-2 a^2 \cot ^2(c+d x) \csc ^4(c+d x)+a^2 \cot ^2(c+d x) \csc ^3(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{16 d}}{a^4}\) |
Input:
Int[(Cot[c + d*x]^6*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]
Output:
((3*a^2*ArcTanh[Cos[c + d*x]])/(16*d) + (2*a^2*Cot[c + d*x]^3)/(3*d) + (2* a^2*Cot[c + d*x]^5)/(5*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (5* a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5 )/(6*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 = 3.77 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.35
| method | result | size |
| risch | \(-\frac {45 \,{\mathrm e}^{11 i \left (d x +c \right )}-960 i {\mathrm e}^{8 i \left (d x +c \right )}+65 \,{\mathrm e}^{9 i \left (d x +c \right )}+640 i {\mathrm e}^{6 i \left (d x +c \right )}-750 \,{\mathrm e}^{7 i \left (d x +c \right )}-750 \,{\mathrm e}^{5 i \left (d x +c \right )}+384 i {\mathrm e}^{2 i \left (d x +c \right )}+65 \,{\mathrm e}^{3 i \left (d x +c \right )}-64 i+45 \,{\mathrm e}^{i \left (d x +c \right )}}{120 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d \,a^{2}}\) | \(168\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}}{64 d \,a^{2}}\) | \(176\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}}{64 d \,a^{2}}\) | \(176\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-1/120*(45*exp(11*I*(d*x+c))-960*I*exp(8*I*(d*x+c))+65*exp(9*I*(d*x+c))+64 0*I*exp(6*I*(d*x+c))-750*exp(7*I*(d*x+c))-750*exp(5*I*(d*x+c))+384*I*exp(2 *I*(d*x+c))+65*exp(3*I*(d*x+c))-64*I+45*exp(I*(d*x+c)))/a^2/d/(exp(2*I*(d* x+c))-1)^6+3/16/d/a^2*ln(exp(I*(d*x+c))+1)-3/16/d/a^2*ln(exp(I*(d*x+c))-1)
Time = 0.09 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.58 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {90 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 45 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 45 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 64 \, {\left (2 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) - 90 \, \cos \left (d x + c\right )}{480 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
-1/480*(90*cos(d*x + c)^5 - 80*cos(d*x + c)^3 - 45*(cos(d*x + c)^6 - 3*cos (d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2) + 45*(cos( d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c ) + 1/2) - 64*(2*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*sin(d*x + c) - 90*cos( d*x + c))/(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)
\[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\cot ^{6}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)/(a+a*sin(d*x+c))**2,x)
Output:
Integral(cot(c + d*x)**6*csc(c + d*x)/(sin(c + d*x)**2 + 2*sin(c + d*x) + 1), x)/a**2
Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (112) = 224\).
Time = 0.04 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.21 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {240 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {24 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a^{2}} - \frac {360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {{\left (\frac {24 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {45 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {240 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a^{2} \sin \left (d x + c\right )^{6}}}{1920 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
1/1920*((240*sin(d*x + c)/(cos(d*x + c) + 1) - 15*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 40*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 45*sin(d*x + c)^4/( cos(d*x + c) + 1)^4 - 24*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)/a^2 - 360*log(sin(d*x + c)/(cos(d*x + c) + 1)) /a^2 + (24*sin(d*x + c)/(cos(d*x + c) + 1) - 45*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 40*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x + c)^4/(co s(d*x + c) + 1)^4 - 240*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5)*(cos(d*x + c) + 1)^6/(a^2*sin(d*x + c)^6))/d
Time = 0.23 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.74 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {360 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {882 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}} - \frac {5 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 24 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{12}}}{1920 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
-1/1920*(360*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (882*tan(1/2*d*x + 1/2*c )^6 - 240*tan(1/2*d*x + 1/2*c)^5 + 15*tan(1/2*d*x + 1/2*c)^4 + 40*tan(1/2* d*x + 1/2*c)^3 - 45*tan(1/2*d*x + 1/2*c)^2 + 24*tan(1/2*d*x + 1/2*c) - 5)/ (a^2*tan(1/2*d*x + 1/2*c)^6) - (5*a^10*tan(1/2*d*x + 1/2*c)^6 - 24*a^10*ta n(1/2*d*x + 1/2*c)^5 + 45*a^10*tan(1/2*d*x + 1/2*c)^4 - 40*a^10*tan(1/2*d* x + 1/2*c)^3 - 15*a^10*tan(1/2*d*x + 1/2*c)^2 + 240*a^10*tan(1/2*d*x + 1/2 *c))/a^12)/d
Time = 34.04 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.73 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+24\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+360\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \] Input:
int(cot(c + d*x)^6/(sin(c + d*x)*(a + a*sin(c + d*x))^2),x)
Output:
-(5*cos(c/2 + (d*x)/2)^12 - 5*sin(c/2 + (d*x)/2)^12 + 24*cos(c/2 + (d*x)/2 )*sin(c/2 + (d*x)/2)^11 - 24*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2) - 45 *cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 40*cos(c/2 + (d*x)/2)^3*sin( c/2 + (d*x)/2)^9 + 15*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 - 240*cos( c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 + 240*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 - 15*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 - 40*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 + 45*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x) /2)^2 + 360*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^ 6*sin(c/2 + (d*x)/2)^6)/(1920*a^2*d*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2 )^6)
Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {-64 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+45 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-50 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+96 \cos \left (d x +c \right ) \sin \left (d x +c \right )-40 \cos \left (d x +c \right )-45 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}}{240 \sin \left (d x +c \right )^{6} a^{2} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)/(a+a*sin(d*x+c))^2,x)
Output:
( - 64*cos(c + d*x)*sin(c + d*x)**5 + 45*cos(c + d*x)*sin(c + d*x)**4 - 32 *cos(c + d*x)*sin(c + d*x)**3 - 50*cos(c + d*x)*sin(c + d*x)**2 + 96*cos(c + d*x)*sin(c + d*x) - 40*cos(c + d*x) - 45*log(tan((c + d*x)/2))*sin(c + d*x)**6)/(240*sin(c + d*x)**6*a**2*d)