Integrand size = 21, antiderivative size = 124 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\text {arctanh}(\cos (c+d x))}{8 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d} \] Output:
1/8*arctanh(cos(d*x+c))/a^2/d-2/5*cot(d*x+c)^5/a^2/d-1/7*cot(d*x+c)^7/a^2/ d+1/8*cot(d*x+c)*csc(d*x+c)/a^2/d-7/12*cot(d*x+c)*csc(d*x+c)^3/a^2/d+1/3*c ot(d*x+c)*csc(d*x+c)^5/a^2/d
Leaf count is larger than twice the leaf count of optimal. \(251\) vs. \(2(124)=248\).
Time = 3.82 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.02 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc ^7(c+d x) \left (5880 \cos (c+d x)+2184 \cos (3 (c+d x))-168 \cos (5 (c+d x))-216 \cos (7 (c+d x))-3675 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+3675 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-2170 \sin (2 (c+d x))+2205 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-2205 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-3080 \sin (4 (c+d x))-735 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+735 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-210 \sin (6 (c+d x))+105 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-105 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))\right )}{53760 a^2 d} \] Input:
Integrate[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^2,x]
Output:
-1/53760*(Csc[c + d*x]^7*(5880*Cos[c + d*x] + 2184*Cos[3*(c + d*x)] - 168* Cos[5*(c + d*x)] - 216*Cos[7*(c + d*x)] - 3675*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 3675*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 2170*Sin[2*(c + d*x)] + 2205*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 2205*Log[Sin[(c + d*x)/2]] *Sin[3*(c + d*x)] - 3080*Sin[4*(c + d*x)] - 735*Log[Cos[(c + d*x)/2]]*Sin[ 5*(c + d*x)] + 735*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 210*Sin[6*(c + d*x)] + 105*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] - 105*Log[Sin[(c + d*x )/2]]*Sin[7*(c + d*x)]))/(a^2*d)
Time = 0.47 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3188, 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.
| \(\displaystyle \int \frac {\cot ^8(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^8 (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle \frac {\int \left (a^6 \csc ^8(c+d x)-2 a^6 \csc ^7(c+d x)-a^6 \csc ^6(c+d x)+4 a^6 \csc ^5(c+d x)-a^6 \csc ^4(c+d x)-2 a^6 \csc ^3(c+d x)+a^6 \csc ^2(c+d x)\right )dx}{a^8}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^6 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^6 \cot ^7(c+d x)}{7 d}-\frac {2 a^6 \cot ^5(c+d x)}{5 d}+\frac {a^6 \cot (c+d x) \csc ^5(c+d x)}{3 d}-\frac {7 a^6 \cot (c+d x) \csc ^3(c+d x)}{12 d}+\frac {a^6 \cot (c+d x) \csc (c+d x)}{8 d}}{a^8}\) |
Input:
Int[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^2,x]
Output:
((a^6*ArcTanh[Cos[c + d*x]])/(8*d) - (2*a^6*Cot[c + d*x]^5)/(5*d) - (a^6*C ot[c + d*x]^7)/(7*d) + (a^6*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (7*a^6*Cot[ c + d*x]*Csc[c + d*x]^3)/(12*d) + (a^6*Cot[c + d*x]*Csc[c + d*x]^5)/(3*d)) /a^8
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[a^p Int[ExpandIntegrand[Sin[e + f*x]^p*((a + b*Sin[e + f*x])^(m - p/2)/(a - b*Sin[e + f*x])^(p/2)), x], x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])
Time = 5.93 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.63
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {3}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {11}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{128 d \,a^{2}}\) | \(202\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {3}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {11}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{128 d \,a^{2}}\) | \(202\) |
| risch | \(-\frac {840 i {\mathrm e}^{12 i \left (d x +c \right )}+105 \,{\mathrm e}^{13 i \left (d x +c \right )}-3360 i {\mathrm e}^{10 i \left (d x +c \right )}+1540 \,{\mathrm e}^{11 i \left (d x +c \right )}+840 i {\mathrm e}^{8 i \left (d x +c \right )}+1085 \,{\mathrm e}^{9 i \left (d x +c \right )}-6720 i {\mathrm e}^{6 i \left (d x +c \right )}+1176 i {\mathrm e}^{4 i \left (d x +c \right )}-1085 \,{\mathrm e}^{5 i \left (d x +c \right )}-672 i {\mathrm e}^{2 i \left (d x +c \right )}-1540 \,{\mathrm e}^{3 i \left (d x +c \right )}+216 i-105 \,{\mathrm e}^{i \left (d x +c \right )}}{420 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{2}}\) | \(204\) |
Input:
int(cot(d*x+c)^8/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/128/d/a^2*(1/7*tan(1/2*d*x+1/2*c)^7-2/3*tan(1/2*d*x+1/2*c)^6+3/5*tan(1/2 *d*x+1/2*c)^5+2*tan(1/2*d*x+1/2*c)^4-5*tan(1/2*d*x+1/2*c)^3+2*tan(1/2*d*x+ 1/2*c)^2+11*tan(1/2*d*x+1/2*c)-1/7/tan(1/2*d*x+1/2*c)^7-2/tan(1/2*d*x+1/2* c)^2-3/5/tan(1/2*d*x+1/2*c)^5-2/tan(1/2*d*x+1/2*c)^4+2/3/tan(1/2*d*x+1/2*c )^6-11/tan(1/2*d*x+1/2*c)-16*ln(tan(1/2*d*x+1/2*c))+5/tan(1/2*d*x+1/2*c)^3 )
Time = 0.09 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.74 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {432 \, \cos \left (d x + c\right )^{7} - 672 \, \cos \left (d x + c\right )^{5} - 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 70 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^8/(a+a*sin(d*x+c))^2,x, algorithm="fricas")
Output:
-1/1680*(432*cos(d*x + c)^7 - 672*cos(d*x + c)^5 - 105*(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)*sin(d* x + c) + 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*lo g(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 70*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))*sin(d*x + c))/((a^2*d*cos(d*x + c)^6 - 3*a^2*d*c os(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))
\[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\cot ^{8}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(cot(d*x+c)**8/(a+a*sin(d*x+c))**2,x)
Output:
Integral(cot(c + d*x)**8/(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. 314 vs. \(2 (112) = 224\).
Time = 0.04 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.53 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {1155 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {210 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {525 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {210 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {70 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{2}} - \frac {1680 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {{\left (\frac {70 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {210 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {525 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {210 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1155 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 15\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{2} \sin \left (d x + c\right )^{7}}}{13440 \, d} \] Input:
integrate(cot(d*x+c)^8/(a+a*sin(d*x+c))^2,x, algorithm="maxima")
Output:
1/13440*((1155*sin(d*x + c)/(cos(d*x + c) + 1) + 210*sin(d*x + c)^2/(cos(d *x + c) + 1)^2 - 525*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 210*sin(d*x + c )^4/(cos(d*x + c) + 1)^4 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 70*sin (d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7) /a^2 - 1680*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 + (70*sin(d*x + c)/(c os(d*x + c) + 1) - 63*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 210*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 525*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 210* sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1155*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 15)*(cos(d*x + c) + 1)^7/(a^2*sin(d*x + c)^7))/d
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (112) = 224\).
Time = 0.18 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.98 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {1680 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {4356 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 70 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} - \frac {15 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 70 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 63 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 210 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 525 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 210 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1155 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{14}}}{13440 \, d} \] Input:
integrate(cot(d*x+c)^8/(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
-1/13440*(1680*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (4356*tan(1/2*d*x + 1/ 2*c)^7 - 1155*tan(1/2*d*x + 1/2*c)^6 - 210*tan(1/2*d*x + 1/2*c)^5 + 525*ta n(1/2*d*x + 1/2*c)^4 - 210*tan(1/2*d*x + 1/2*c)^3 - 63*tan(1/2*d*x + 1/2*c )^2 + 70*tan(1/2*d*x + 1/2*c) - 15)/(a^2*tan(1/2*d*x + 1/2*c)^7) - (15*a^1 2*tan(1/2*d*x + 1/2*c)^7 - 70*a^12*tan(1/2*d*x + 1/2*c)^6 + 63*a^12*tan(1/ 2*d*x + 1/2*c)^5 + 210*a^12*tan(1/2*d*x + 1/2*c)^4 - 525*a^12*tan(1/2*d*x + 1/2*c)^3 + 210*a^12*tan(1/2*d*x + 1/2*c)^2 + 1155*a^12*tan(1/2*d*x + 1/2 *c))/a^14)/d
Time = 32.89 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.12 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+70\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-70\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+525\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-525\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1680\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{13440\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \] Input:
int(cot(c + d*x)^8/(a + a*sin(c + d*x))^2,x)
Output:
-(15*cos(c/2 + (d*x)/2)^14 - 15*sin(c/2 + (d*x)/2)^14 + 70*cos(c/2 + (d*x) /2)*sin(c/2 + (d*x)/2)^13 - 70*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2) - 63*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 - 210*cos(c/2 + (d*x)/2)^3*s in(c/2 + (d*x)/2)^11 + 525*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 - 21 0*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9 - 1155*cos(c/2 + (d*x)/2)^6*si n(c/2 + (d*x)/2)^8 + 1155*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 + 210* cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5 - 525*cos(c/2 + (d*x)/2)^10*sin( c/2 + (d*x)/2)^4 + 210*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^3 + 63*cos (c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^2 + 1680*log(sin(c/2 + (d*x)/2)/cos( c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7)/(13440*a^2*d*co s(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7)
Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {-216 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+105 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+312 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-490 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+280 \cos \left (d x +c \right ) \sin \left (d x +c \right )-120 \cos \left (d x +c \right )-105 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{7}}{840 \sin \left (d x +c \right )^{7} a^{2} d} \] Input:
int(cot(d*x+c)^8/(a+a*sin(d*x+c))^2,x)
Output:
( - 216*cos(c + d*x)*sin(c + d*x)**6 + 105*cos(c + d*x)*sin(c + d*x)**5 + 312*cos(c + d*x)*sin(c + d*x)**4 - 490*cos(c + d*x)*sin(c + d*x)**3 + 24*c os(c + d*x)*sin(c + d*x)**2 + 280*cos(c + d*x)*sin(c + d*x) - 120*cos(c + d*x) - 105*log(tan((c + d*x)/2))*sin(c + d*x)**7)/(840*sin(c + d*x)**7*a** 2*d)