Integrand size = 21, antiderivative size = 140 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {5 \text {arctanh}(\cos (c+d x))}{16 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d} \] Output:
-5/16*arctanh(cos(d*x+c))/a^3/d-4/3*cot(d*x+c)^3/a^3/d-cot(d*x+c)^5/a^3/d- 1/7*cot(d*x+c)^7/a^3/d-5/16*cot(d*x+c)*csc(d*x+c)/a^3/d+1/8*cot(d*x+c)*csc (d*x+c)^3/a^3/d+1/2*cot(d*x+c)*csc(d*x+c)^5/a^3/d
Time = 2.98 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.79 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\csc ^7(c+d x) \left (-4704 \cos (c+d x)+672 \cos (3 (c+d x))+1120 \cos (5 (c+d x))-160 \cos (7 (c+d x))-3675 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+3675 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+4998 \sin (2 (c+d x))+2205 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-2205 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+504 \sin (4 (c+d x))-735 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+735 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-210 \sin (6 (c+d x))+105 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-105 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))\right )}{21504 a^3 d} \] Input:
Integrate[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]
Output:
(Csc[c + d*x]^7*(-4704*Cos[c + d*x] + 672*Cos[3*(c + d*x)] + 1120*Cos[5*(c + d*x)] - 160*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] + 4998*Sin[2*(c + d*x)] + 2205*L og[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 2205*Log[Sin[(c + d*x)/2]]*Sin[3*( c + d*x)] + 504*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)]))/(21504*a^3*d)
Time = 0.47 (sec) , antiderivative size = 144, 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.
| \(\displaystyle \int \frac {\cot ^8(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^8 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle \frac {\int \left (a^5 \csc ^8(c+d x)-3 a^5 \csc ^7(c+d x)+2 a^5 \csc ^6(c+d x)+2 a^5 \csc ^5(c+d x)-3 a^5 \csc ^4(c+d x)+a^5 \csc ^3(c+d x)\right )dx}{a^8}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {5 a^5 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a^5 \cot ^7(c+d x)}{7 d}-\frac {a^5 \cot ^5(c+d x)}{d}-\frac {4 a^5 \cot ^3(c+d x)}{3 d}+\frac {a^5 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac {a^5 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {5 a^5 \cot (c+d x) \csc (c+d x)}{16 d}}{a^8}\) |
Input:
Int[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]
Output:
((-5*a^5*ArcTanh[Cos[c + d*x]])/(16*d) - (4*a^5*Cot[c + d*x]^3)/(3*d) - (a ^5*Cot[c + d*x]^5)/d - (a^5*Cot[c + d*x]^7)/(7*d) - (5*a^5*Cot[c + d*x]*Cs c[c + d*x])/(16*d) + (a^5*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) + (a^5*Cot[c + d*x]*Csc[c + d*x]^5)/(2*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])
Result contains complex when optimal does not.
Time = 14.86 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.37
| method | result | size |
| risch | \(\frac {105 \,{\mathrm e}^{13 i \left (d x +c \right )}-2016 i {\mathrm e}^{10 i \left (d x +c \right )}-252 \,{\mathrm e}^{11 i \left (d x +c \right )}+5152 i {\mathrm e}^{8 i \left (d x +c \right )}-2499 \,{\mathrm e}^{9 i \left (d x +c \right )}-448 i {\mathrm e}^{6 i \left (d x +c \right )}+1344 i {\mathrm e}^{4 i \left (d x +c \right )}+2499 \,{\mathrm e}^{5 i \left (d x +c \right )}-1120 i {\mathrm e}^{2 i \left (d x +c \right )}+252 \,{\mathrm e}^{3 i \left (d x +c \right )}+160 i-105 \,{\mathrm e}^{i \left (d x +c \right )}}{168 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d \,a^{3}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d \,a^{3}}\) | \(192\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {13}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {29}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+40 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{3}}\) | \(200\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {13}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {29}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+40 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{3}}\) | \(200\) |
Input:
int(cot(d*x+c)^8/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/168*(105*exp(13*I*(d*x+c))-2016*I*exp(10*I*(d*x+c))-252*exp(11*I*(d*x+c) )+5152*I*exp(8*I*(d*x+c))-2499*exp(9*I*(d*x+c))-448*I*exp(6*I*(d*x+c))+134 4*I*exp(4*I*(d*x+c))+2499*exp(5*I*(d*x+c))-1120*I*exp(2*I*(d*x+c))+252*exp (3*I*(d*x+c))+160*I-105*exp(I*(d*x+c)))/d/a^3/(exp(2*I*(d*x+c))-1)^7+5/16/ d/a^3*ln(exp(I*(d*x+c))-1)-5/16/d/a^3*ln(exp(I*(d*x+c))+1)
Time = 0.09 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.61 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {320 \, \cos \left (d x + c\right )^{7} - 1120 \, \cos \left (d x + c\right )^{5} + 896 \, \cos \left (d x + c\right )^{3} - 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 42 \, {\left (5 \, \cos \left (d x + c\right )^{5} - 8 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \, {\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
1/672*(320*cos(d*x + c)^7 - 1120*cos(d*x + c)^5 + 896*cos(d*x + c)^3 - 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*co s(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 42*(5*cos(d* x + c)^5 - 8*cos(d*x + c)^3 - 5*cos(d*x + c))*sin(d*x + c))/((a^3*d*cos(d* x + c)^6 - 3*a^3*d*cos(d*x + c)^4 + 3*a^3*d*cos(d*x + c)^2 - a^3*d)*sin(d* x + c))
\[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cot ^{8}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cot(d*x+c)**8/(a+a*sin(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)**8/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (128) = 256\).
Time = 0.04 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.25 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {609 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {91 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {21 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3}} - \frac {840 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {{\left (\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {105 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {91 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {609 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{3} \sin \left (d x + c\right )^{7}}}{2688 \, d} \] Input:
integrate(cot(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
-1/2688*((609*sin(d*x + c)/(cos(d*x + c) + 1) - 63*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 91*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 105*sin(d*x + c)^4 /(cos(d*x + c) + 1)^4 - 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 21*sin(d* x + c)^6/(cos(d*x + c) + 1)^6 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^3 - 840*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 - (21*sin(d*x + c)/(cos(d* x + c) + 1) - 63*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 105*sin(d*x + c)^3/ (cos(d*x + c) + 1)^3 - 91*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 609*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 3 )*(cos(d*x + c) + 1)^7/(a^3*sin(d*x + c)^7))/d
Time = 0.21 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.74 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {2178 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 609 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 91 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} + \frac {3 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 21 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 63 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 91 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 63 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 609 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{21}}}{2688 \, d} \] Input:
integrate(cot(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/2688*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (2178*tan(1/2*d*x + 1/2*c )^7 - 609*tan(1/2*d*x + 1/2*c)^6 + 63*tan(1/2*d*x + 1/2*c)^5 + 91*tan(1/2* d*x + 1/2*c)^4 - 105*tan(1/2*d*x + 1/2*c)^3 + 63*tan(1/2*d*x + 1/2*c)^2 - 21*tan(1/2*d*x + 1/2*c) + 3)/(a^3*tan(1/2*d*x + 1/2*c)^7) + (3*a^18*tan(1/ 2*d*x + 1/2*c)^7 - 21*a^18*tan(1/2*d*x + 1/2*c)^6 + 63*a^18*tan(1/2*d*x + 1/2*c)^5 - 105*a^18*tan(1/2*d*x + 1/2*c)^4 + 91*a^18*tan(1/2*d*x + 1/2*c)^ 3 + 63*a^18*tan(1/2*d*x + 1/2*c)^2 - 609*a^18*tan(1/2*d*x + 1/2*c))/a^21)/ d
Time = 33.23 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.76 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-21\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+91\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-609\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+609\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-91\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2688\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \] Input:
int(cot(c + d*x)^8/(a + a*sin(c + d*x))^3,x)
Output:
(3*sin(c/2 + (d*x)/2)^14 - 3*cos(c/2 + (d*x)/2)^14 - 21*cos(c/2 + (d*x)/2) *sin(c/2 + (d*x)/2)^13 + 21*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 - 105*cos(c/2 + (d*x)/2)^3*sin( c/2 + (d*x)/2)^11 + 91*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 + 63*cos (c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9 - 609*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 609*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 - 63*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5 - 91*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d* x)/2)^4 + 105*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 + 840*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)/(2688*a^3*d*cos(c/2 + (d* x)/2)^7*sin(c/2 + (d*x)/2)^7)
Time = 0.17 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99 \[ \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {160 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-105 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+80 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+42 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+168 \cos \left (d x +c \right ) \sin \left (d x +c \right )-48 \cos \left (d x +c \right )+105 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{7}}{336 \sin \left (d x +c \right )^{7} a^{3} d} \] Input:
int(cot(d*x+c)^8/(a+a*sin(d*x+c))^3,x)
Output:
(160*cos(c + d*x)*sin(c + d*x)**6 - 105*cos(c + d*x)*sin(c + d*x)**5 + 80* cos(c + d*x)*sin(c + d*x)**4 + 42*cos(c + d*x)*sin(c + d*x)**3 - 192*cos(c + d*x)*sin(c + d*x)**2 + 168*cos(c + d*x)*sin(c + d*x) - 48*cos(c + d*x) + 105*log(tan((c + d*x)/2))*sin(c + d*x)**7)/(336*sin(c + d*x)**7*a**3*d)