Integrand size = 27, antiderivative size = 179 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (a^2-b^2\right )^2 \csc (c+d x)}{a^5 d}-\frac {b \left (2 a^2-b^2\right ) \csc ^2(c+d x)}{2 a^4 d}+\frac {\left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^3 d}+\frac {b \csc ^4(c+d x)}{4 a^2 d}-\frac {\csc ^5(c+d x)}{5 a d}-\frac {b \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^6 d}+\frac {b \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^6 d} \]
-(a^2-b^2)^2*csc(d*x+c)/a^5/d-1/2*b*(2*a^2-b^2)*csc(d*x+c)^2/a^4/d+1/3*(2* a^2-b^2)*csc(d*x+c)^3/a^3/d+1/4*b*csc(d*x+c)^4/a^2/d-1/5*csc(d*x+c)^5/a/d- b*(a^2-b^2)^2*ln(sin(d*x+c))/a^6/d+b*(a^2-b^2)^2*ln(a+b*sin(d*x+c))/a^6/d
Time = 6.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (a^2-b^2\right )^2 \csc (c+d x)}{a^5 d}-\frac {b \left (2 a^2-b^2\right ) \csc ^2(c+d x)}{2 a^4 d}+\frac {\left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^3 d}+\frac {b \csc ^4(c+d x)}{4 a^2 d}-\frac {\csc ^5(c+d x)}{5 a d}-\frac {b \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^6 d}+\frac {b \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^6 d} \]
-(((a^2 - b^2)^2*Csc[c + d*x])/(a^5*d)) - (b*(2*a^2 - b^2)*Csc[c + d*x]^2) /(2*a^4*d) + ((2*a^2 - b^2)*Csc[c + d*x]^3)/(3*a^3*d) + (b*Csc[c + d*x]^4) /(4*a^2*d) - Csc[c + d*x]^5/(5*a*d) - (b*(a^2 - b^2)^2*Log[Sin[c + d*x]])/ (a^6*d) + (b*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^6*d)
Time = 0.39 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3316, 27, 522, 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.
| \(\displaystyle \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5}{\sin (c+d x)^6 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {\int \frac {\csc ^6(c+d x) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{a+b \sin (c+d x)}d(b \sin (c+d x))}{b^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \int \frac {\csc ^6(c+d x) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{b^6 (a+b \sin (c+d x))}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {b \int \left (\frac {\csc ^6(c+d x)}{a b^2}-\frac {\csc ^5(c+d x)}{a^2 b}+\frac {\left (b^4-2 a^2 b^2\right ) \csc ^4(c+d x)}{a^3 b^4}+\frac {\left (2 a^2 b^2-b^4\right ) \csc ^3(c+d x)}{a^4 b^3}+\frac {\left (a^2-b^2\right )^2 \csc ^2(c+d x)}{a^5 b^2}-\frac {\left (a^2-b^2\right )^2 \csc (c+d x)}{a^6 b}+\frac {\left (a^2-b^2\right )^2}{a^6 (a+b \sin (c+d x))}\right )d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (\frac {\csc ^4(c+d x)}{4 a^2}-\frac {\left (a^2-b^2\right )^2 \log (b \sin (c+d x))}{a^6}+\frac {\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^6}-\frac {\left (a^2-b^2\right )^2 \csc (c+d x)}{a^5 b}-\frac {\left (2 a^2-b^2\right ) \csc ^2(c+d x)}{2 a^4}+\frac {\left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^3 b}-\frac {\csc ^5(c+d x)}{5 a b}\right )}{d}\) |
(b*(-(((a^2 - b^2)^2*Csc[c + d*x])/(a^5*b)) - ((2*a^2 - b^2)*Csc[c + d*x]^ 2)/(2*a^4) + ((2*a^2 - b^2)*Csc[c + d*x]^3)/(3*a^3*b) + Csc[c + d*x]^4/(4* a^2) - Csc[c + d*x]^5/(5*a*b) - ((a^2 - b^2)^2*Log[b*Sin[c + d*x]])/a^6 + ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/a^6))/d
3.14.18.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Time = 0.59 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(\frac {-\frac {\frac {\left (\csc ^{5}\left (d x +c \right )\right ) a^{4}}{5}-\frac {b \left (\csc ^{4}\left (d x +c \right )\right ) a^{3}}{4}-\frac {2 a^{4} \left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {a^{2} b^{2} \left (\csc ^{3}\left (d x +c \right )\right )}{3}+a^{3} b \left (\csc ^{2}\left (d x +c \right )\right )-\frac {a \,b^{3} \left (\csc ^{2}\left (d x +c \right )\right )}{2}+\csc \left (d x +c \right ) a^{4}-2 \csc \left (d x +c \right ) a^{2} b^{2}+\csc \left (d x +c \right ) b^{4}}{a^{5}}+\frac {b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a \csc \left (d x +c \right )+b \right )}{a^{6}}}{d}\) | \(160\) |
| default | \(\frac {-\frac {\frac {\left (\csc ^{5}\left (d x +c \right )\right ) a^{4}}{5}-\frac {b \left (\csc ^{4}\left (d x +c \right )\right ) a^{3}}{4}-\frac {2 a^{4} \left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {a^{2} b^{2} \left (\csc ^{3}\left (d x +c \right )\right )}{3}+a^{3} b \left (\csc ^{2}\left (d x +c \right )\right )-\frac {a \,b^{3} \left (\csc ^{2}\left (d x +c \right )\right )}{2}+\csc \left (d x +c \right ) a^{4}-2 \csc \left (d x +c \right ) a^{2} b^{2}+\csc \left (d x +c \right ) b^{4}}{a^{5}}+\frac {b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a \csc \left (d x +c \right )+b \right )}{a^{6}}}{d}\) | \(160\) |
| parallelrisch | \(\frac {960 b \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-960 b \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \left (\left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}-\frac {5 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b}{2}+\left (-\frac {25}{3} a^{4}+\frac {20}{3} a^{2} b^{2}\right ) \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -20 a \,b^{3}\right ) \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (50 a^{4}-140 a^{2} b^{2}+80 b^{4}\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b}{2}+\left (-\frac {25}{3} a^{4}+\frac {20}{3} a^{2} b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -20 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+50 a^{4}-140 a^{2} b^{2}+80 b^{4}\right )\right ) a}{960 a^{6} d}\) | \(303\) |
| norman | \(\frac {-\frac {1}{160 a d}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d a}+\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a^{2} d}+\frac {b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2} d}+\frac {\left (5 a^{2}-4 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 a^{3} d}+\frac {\left (5 a^{2}-4 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 a^{3} d}-\frac {\left (5 a^{4}-14 a^{2} b^{2}+8 b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5} d}-\frac {\left (5 a^{4}-14 a^{2} b^{2}+8 b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5} d}-\frac {b \left (3 a^{2}-2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{4} d}-\frac {b \left (3 a^{2}-2 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{4} d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{a^{6} d}-\frac {b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6} d}\) | \(363\) |
| risch | \(-\frac {2 i \left (15 a^{4} {\mathrm e}^{9 i \left (d x +c \right )}-30 a^{2} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+15 b^{4} {\mathrm e}^{9 i \left (d x +c \right )}-20 a^{4} {\mathrm e}^{7 i \left (d x +c \right )}+100 a^{2} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-60 b^{4} {\mathrm e}^{7 i \left (d x +c \right )}-15 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+45 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+58 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}-140 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+90 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+15 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-45 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-20 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}+100 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-60 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+60 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+30 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+15 a^{4} {\mathrm e}^{i \left (d x +c \right )}-30 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}+15 b^{4} {\mathrm e}^{i \left (d x +c \right )}-30 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-60 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{15 d \,a^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}+\frac {2 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}-\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{6} d}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{2} d}-\frac {2 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{4} d}+\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{6} d}\) | \(558\) |
1/d*(-1/a^5*(1/5*csc(d*x+c)^5*a^4-1/4*b*csc(d*x+c)^4*a^3-2/3*a^4*csc(d*x+c )^3+1/3*a^2*b^2*csc(d*x+c)^3+a^3*b*csc(d*x+c)^2-1/2*a*b^3*csc(d*x+c)^2+csc (d*x+c)*a^4-2*csc(d*x+c)*a^2*b^2+csc(d*x+c)*b^4)+b*(a^4-2*a^2*b^2+b^4)/a^6 *ln(a*csc(d*x+c)+b))
Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (171) = 342\).
Time = 0.42 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.93 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {32 \, a^{5} - 100 \, a^{3} b^{2} + 60 \, a b^{4} + 60 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{4} - 20 \, {\left (4 \, a^{5} - 11 \, a^{3} b^{2} + 6 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} - 60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) + 60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 15 \, {\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - 2 \, {\left (2 \, a^{4} b - a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, {\left (a^{6} d \cos \left (d x + c\right )^{4} - 2 \, a^{6} d \cos \left (d x + c\right )^{2} + a^{6} d\right )} \sin \left (d x + c\right )} \]
-1/60*(32*a^5 - 100*a^3*b^2 + 60*a*b^4 + 60*(a^5 - 2*a^3*b^2 + a*b^4)*cos( d*x + c)^4 - 20*(4*a^5 - 11*a^3*b^2 + 6*a*b^4)*cos(d*x + c)^2 - 60*(a^4*b - 2*a^2*b^3 + b^5 + (a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^4 - 2*(a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^2)*log(b*sin(d*x + c) + a)*sin(d*x + c) + 60 *(a^4*b - 2*a^2*b^3 + b^5 + (a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^4 - 2*( a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^2)*log(1/2*sin(d*x + c))*sin(d*x + c ) + 15*(3*a^4*b - 2*a^2*b^3 - 2*(2*a^4*b - a^2*b^3)*cos(d*x + c)^2)*sin(d* x + c))/((a^6*d*cos(d*x + c)^4 - 2*a^6*d*cos(d*x + c)^2 + a^6*d)*sin(d*x + c))
Timed out. \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{6}} - \frac {60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{6}} + \frac {15 \, a^{3} b \sin \left (d x + c\right ) - 60 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4} - 12 \, a^{4} - 30 \, {\left (2 \, a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )^{3} + 20 \, {\left (2 \, a^{4} - a^{2} b^{2}\right )} \sin \left (d x + c\right )^{2}}{a^{5} \sin \left (d x + c\right )^{5}}}{60 \, d} \]
1/60*(60*(a^4*b - 2*a^2*b^3 + b^5)*log(b*sin(d*x + c) + a)/a^6 - 60*(a^4*b - 2*a^2*b^3 + b^5)*log(sin(d*x + c))/a^6 + (15*a^3*b*sin(d*x + c) - 60*(a ^4 - 2*a^2*b^2 + b^4)*sin(d*x + c)^4 - 12*a^4 - 30*(2*a^3*b - a*b^3)*sin(d *x + c)^3 + 20*(2*a^4 - a^2*b^2)*sin(d*x + c)^2)/(a^5*sin(d*x + c)^5))/d
Time = 0.36 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.40 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{6}} - \frac {60 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{6} b} - \frac {137 \, a^{4} b \sin \left (d x + c\right )^{5} - 274 \, a^{2} b^{3} \sin \left (d x + c\right )^{5} + 137 \, b^{5} \sin \left (d x + c\right )^{5} - 60 \, a^{5} \sin \left (d x + c\right )^{4} + 120 \, a^{3} b^{2} \sin \left (d x + c\right )^{4} - 60 \, a b^{4} \sin \left (d x + c\right )^{4} - 60 \, a^{4} b \sin \left (d x + c\right )^{3} + 30 \, a^{2} b^{3} \sin \left (d x + c\right )^{3} + 40 \, a^{5} \sin \left (d x + c\right )^{2} - 20 \, a^{3} b^{2} \sin \left (d x + c\right )^{2} + 15 \, a^{4} b \sin \left (d x + c\right ) - 12 \, a^{5}}{a^{6} \sin \left (d x + c\right )^{5}}}{60 \, d} \]
-1/60*(60*(a^4*b - 2*a^2*b^3 + b^5)*log(abs(sin(d*x + c)))/a^6 - 60*(a^4*b ^2 - 2*a^2*b^4 + b^6)*log(abs(b*sin(d*x + c) + a))/(a^6*b) - (137*a^4*b*si n(d*x + c)^5 - 274*a^2*b^3*sin(d*x + c)^5 + 137*b^5*sin(d*x + c)^5 - 60*a^ 5*sin(d*x + c)^4 + 120*a^3*b^2*sin(d*x + c)^4 - 60*a*b^4*sin(d*x + c)^4 - 60*a^4*b*sin(d*x + c)^3 + 30*a^2*b^3*sin(d*x + c)^3 + 40*a^5*sin(d*x + c)^ 2 - 20*a^3*b^2*sin(d*x + c)^2 + 15*a^4*b*sin(d*x + c) - 12*a^5)/(a^6*sin(d *x + c)^5))/d
Time = 12.05 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.13 \[ \int \frac {\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {b^2}{8\,a^3}-\frac {5}{16\,a}+\frac {2\,b\,\left (\frac {b}{16\,a^2}+\frac {2\,b\,\left (\frac {5}{32\,a}-\frac {b^2}{8\,a^3}\right )}{a}\right )}{a}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {b}{32\,a^2}+\frac {b\,\left (\frac {5}{32\,a}-\frac {b^2}{8\,a^3}\right )}{a}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5}{96\,a}-\frac {b^2}{24\,a^3}\right )}{d}+\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^4\,b-2\,a^2\,b^3+b^5\right )}{a^6\,d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^4\,b-2\,a^2\,b^3+b^5\right )}{a^6\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {5\,a^4}{3}-\frac {4\,a^2\,b^2}{3}\right )-\frac {a^4}{5}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (10\,a^4-28\,a^2\,b^2+16\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (4\,a\,b^3-6\,a^3\,b\right )+\frac {a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{32\,a^5\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
(tan(c/2 + (d*x)/2)*(b^2/(8*a^3) - 5/(16*a) + (2*b*(b/(16*a^2) + (2*b*(5/( 32*a) - b^2/(8*a^3)))/a))/a))/d - (tan(c/2 + (d*x)/2)^2*(b/(32*a^2) + (b*( 5/(32*a) - b^2/(8*a^3)))/a))/d - tan(c/2 + (d*x)/2)^5/(160*a*d) + (tan(c/2 + (d*x)/2)^3*(5/(96*a) - b^2/(24*a^3)))/d + (log(a + 2*b*tan(c/2 + (d*x)/ 2) + a*tan(c/2 + (d*x)/2)^2)*(a^4*b + b^5 - 2*a^2*b^3))/(a^6*d) + (b*tan(c /2 + (d*x)/2)^4)/(64*a^2*d) - (log(tan(c/2 + (d*x)/2))*(a^4*b + b^5 - 2*a^ 2*b^3))/(a^6*d) + (tan(c/2 + (d*x)/2)^2*((5*a^4)/3 - (4*a^2*b^2)/3) - a^4/ 5 - tan(c/2 + (d*x)/2)^4*(10*a^4 + 16*b^4 - 28*a^2*b^2) + tan(c/2 + (d*x)/ 2)^3*(4*a*b^3 - 6*a^3*b) + (a^3*b*tan(c/2 + (d*x)/2))/2)/(32*a^5*d*tan(c/2 + (d*x)/2)^5)