Integrand size = 29, antiderivative size = 176 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{256 a d}+\frac {\cot ^7(c+d x)}{7 a d}+\frac {\cot ^9(c+d x)}{9 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d} \] Output:
3/256*arctanh(cos(d*x+c))/a/d+1/7*cot(d*x+c)^7/a/d+1/9*cot(d*x+c)^9/a/d+3/ 256*cot(d*x+c)*csc(d*x+c)/a/d+1/128*cot(d*x+c)*csc(d*x+c)^3/a/d-1/32*cot(d *x+c)*csc(d*x+c)^5/a/d+1/16*cot(d*x+c)^3*csc(d*x+c)^5/a/d-1/10*cot(d*x+c)^ 5*csc(d*x+c)^5/a/d
Leaf count is larger than twice the leaf count of optimal. \(386\) vs. \(2(176)=352\).
Time = 4.16 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.19 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^9(c+d x) \left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (2367540 \cos (c+d x)+1307880 \cos (3 (c+d x))+436968 \cos (5 (c+d x))+18270 \cos (7 (c+d x))-1890 \cos (9 (c+d x))-119070 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+198450 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-113400 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+42525 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-9450 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+945 \cos (10 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+119070 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-198450 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+113400 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-42525 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+9450 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-945 \cos (10 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-537600 \sin (2 (c+d x))-522240 \sin (4 (c+d x))-207360 \sin (6 (c+d x))-25600 \sin (8 (c+d x))+2560 \sin (10 (c+d x))\right )}{165150720 a d (1+\csc (c+d x))} \] Input:
Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
Output:
-1/165150720*(Csc[c + d*x]^9*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(2367 540*Cos[c + d*x] + 1307880*Cos[3*(c + d*x)] + 436968*Cos[5*(c + d*x)] + 18 270*Cos[7*(c + d*x)] - 1890*Cos[9*(c + d*x)] - 119070*Log[Cos[(c + d*x)/2] ] + 198450*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 113400*Cos[4*(c + d*x) ]*Log[Cos[(c + d*x)/2]] + 42525*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 9 450*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 945*Cos[10*(c + d*x)]*Log[Cos [(c + d*x)/2]] + 119070*Log[Sin[(c + d*x)/2]] - 198450*Cos[2*(c + d*x)]*Lo g[Sin[(c + d*x)/2]] + 113400*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 4252 5*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 9450*Cos[8*(c + d*x)]*Log[Sin[( c + d*x)/2]] - 945*Cos[10*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 537600*Sin[2* (c + d*x)] - 522240*Sin[4*(c + d*x)] - 207360*Sin[6*(c + d*x)] - 25600*Sin [8*(c + d*x)] + 2560*Sin[10*(c + d*x)]))/(a*d*(1 + Csc[c + d*x]))
Time = 1.09 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {3042, 3318, 3042, 3087, 244, 2009, 3091, 3042, 3091, 3042, 3091, 3042, 4255, 3042, 4255, 3042, 4257}
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.
| \(\displaystyle \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^{11} (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cot ^6(c+d x) \csc ^5(c+d x)dx}{a}-\frac {\int \cot ^6(c+d x) \csc ^4(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^4 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \cot ^6(c+d x) \left (\cot ^2(c+d x)+1\right )d(-\cot (c+d x))}{a d}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \left (\cot ^8(c+d x)+\cot ^6(c+d x)\right )d(-\cot (c+d x))}{a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {1}{2} \int \cot ^4(c+d x) \csc ^5(c+d x)dx-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{2} \int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3}{8} \int \cot ^2(c+d x) \csc ^5(c+d x)dx+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3}{8} \int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3}{8} \left (-\frac {1}{6} \int \csc ^5(c+d x)dx-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3}{8} \left (-\frac {1}{6} \int \csc (c+d x)^5dx-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \int \csc ^3(c+d x)dx\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \int \csc (c+d x)^3dx\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}-\frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}\) |
Input:
Int[(Cot[c + d*x]^8*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
Output:
-((-1/7*Cot[c + d*x]^7 - Cot[c + d*x]^9/9)/(a*d)) + (-1/10*(Cot[c + d*x]^5 *Csc[c + d*x]^5)/d + ((Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) + (3*(-1/6*(Co t[c + d*x]*Csc[c + d*x]^5)/d + ((Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) - (3*( -1/2*ArcTanh[Cos[c + d*x]]/d - (Cot[c + d*x]*Csc[c + d*x])/(2*d)))/4)/6))/ 8)/2)/a
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 3.02 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.43
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{10}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{4}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {12}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {16}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}+\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {2}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {6}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}}{1024 d a}\) | \(252\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{10}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{4}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {12}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {16}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}+\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {2}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {6}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}}{1024 d a}\) | \(252\) |
| risch | \(-\frac {945 \,{\mathrm e}^{19 i \left (d x +c \right )}-107520 i {\mathrm e}^{14 i \left (d x +c \right )}-9135 \,{\mathrm e}^{17 i \left (d x +c \right )}+414720 i {\mathrm e}^{6 i \left (d x +c \right )}-218484 \,{\mathrm e}^{15 i \left (d x +c \right )}-537600 i {\mathrm e}^{12 i \left (d x +c \right )}-653940 \,{\mathrm e}^{13 i \left (d x +c \right )}-161280 i {\mathrm e}^{16 i \left (d x +c \right )}-1183770 \,{\mathrm e}^{11 i \left (d x +c \right )}-1183770 \,{\mathrm e}^{9 i \left (d x +c \right )}+25600 i {\mathrm e}^{2 i \left (d x +c \right )}-653940 \,{\mathrm e}^{7 i \left (d x +c \right )}+322560 i {\mathrm e}^{10 i \left (d x +c \right )}-218484 \,{\mathrm e}^{5 i \left (d x +c \right )}+46080 i {\mathrm e}^{4 i \left (d x +c \right )}-9135 \,{\mathrm e}^{3 i \left (d x +c \right )}-2560 i+945 \,{\mathrm e}^{i \left (d x +c \right )}}{40320 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d a}\) | \(260\) |
Input:
int(cot(d*x+c)^8*csc(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/1024/d/a*(1/10*tan(1/2*d*x+1/2*c)^10-2/9*tan(1/2*d*x+1/2*c)^9-1/4*tan(1/ 2*d*x+1/2*c)^8+6/7*tan(1/2*d*x+1/2*c)^7-1/2*tan(1/2*d*x+1/2*c)^6+2*tan(1/2 *d*x+1/2*c)^4-16/3*tan(1/2*d*x+1/2*c)^3+tan(1/2*d*x+1/2*c)^2+12*tan(1/2*d* x+1/2*c)-12/tan(1/2*d*x+1/2*c)+16/3/tan(1/2*d*x+1/2*c)^3-2/tan(1/2*d*x+1/2 *c)^4-1/10/tan(1/2*d*x+1/2*c)^10+1/4/tan(1/2*d*x+1/2*c)^8+2/9/tan(1/2*d*x+ 1/2*c)^9-12*ln(tan(1/2*d*x+1/2*c))-6/7/tan(1/2*d*x+1/2*c)^7-1/tan(1/2*d*x+ 1/2*c)^2+1/2/tan(1/2*d*x+1/2*c)^6)
Time = 0.10 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.55 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {1890 \, \cos \left (d x + c\right )^{9} - 8820 \, \cos \left (d x + c\right )^{7} - 16128 \, \cos \left (d x + c\right )^{5} + 8820 \, \cos \left (d x + c\right )^{3} - 945 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 945 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2560 \, {\left (2 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right ) - 1890 \, \cos \left (d x + c\right )}{161280 \, {\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/161280*(1890*cos(d*x + c)^9 - 8820*cos(d*x + c)^7 - 16128*cos(d*x + c)^ 5 + 8820*cos(d*x + c)^3 - 945*(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) + 945*(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) - 2 560*(2*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*sin(d*x + c) - 1890*cos(d*x + c) )/(a*d*cos(d*x + c)^10 - 5*a*d*cos(d*x + c)^8 + 10*a*d*cos(d*x + c)^6 - 10 *a*d*cos(d*x + c)^4 + 5*a*d*cos(d*x + c)^2 - a*d)
Timed out. \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**8*csc(d*x+c)**3/(a+a*sin(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (160) = 320\).
Time = 0.04 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.24 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {15120 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1260 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6720 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2520 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {630 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1080 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {315 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {280 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {126 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}}{a} - \frac {15120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {280 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {315 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1080 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {630 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2520 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {6720 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {1260 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {15120 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 126\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{10}}{a \sin \left (d x + c\right )^{10}}}{1290240 \, d} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/1290240*((15120*sin(d*x + c)/(cos(d*x + c) + 1) + 1260*sin(d*x + c)^2/(c os(d*x + c) + 1)^2 - 6720*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2520*sin(d *x + c)^4/(cos(d*x + c) + 1)^4 - 630*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1080*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 315*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 280*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 126*sin(d*x + c)^10/ (cos(d*x + c) + 1)^10)/a - 15120*log(sin(d*x + c)/(cos(d*x + c) + 1))/a + (280*sin(d*x + c)/(cos(d*x + c) + 1) + 315*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1080*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 630*sin(d*x + c)^4/(cos( d*x + c) + 1)^4 - 2520*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 6720*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 1260*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 1 5120*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 126)*(cos(d*x + c) + 1)^10/(a*s in(d*x + c)^10))/d
Time = 0.22 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.72 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {15120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {126 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 280 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 315 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1080 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 630 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2520 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 6720 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1260 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15120 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{10}} - \frac {44286 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 15120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1260 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 630 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1080 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 126}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{1290240 \, d} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/1290240*(15120*log(abs(tan(1/2*d*x + 1/2*c)))/a - (126*a^9*tan(1/2*d*x + 1/2*c)^10 - 280*a^9*tan(1/2*d*x + 1/2*c)^9 - 315*a^9*tan(1/2*d*x + 1/2*c )^8 + 1080*a^9*tan(1/2*d*x + 1/2*c)^7 - 630*a^9*tan(1/2*d*x + 1/2*c)^6 + 2 520*a^9*tan(1/2*d*x + 1/2*c)^4 - 6720*a^9*tan(1/2*d*x + 1/2*c)^3 + 1260*a^ 9*tan(1/2*d*x + 1/2*c)^2 + 15120*a^9*tan(1/2*d*x + 1/2*c))/a^10 - (44286*t an(1/2*d*x + 1/2*c)^10 - 15120*tan(1/2*d*x + 1/2*c)^9 - 1260*tan(1/2*d*x + 1/2*c)^8 + 6720*tan(1/2*d*x + 1/2*c)^7 - 2520*tan(1/2*d*x + 1/2*c)^6 + 63 0*tan(1/2*d*x + 1/2*c)^4 - 1080*tan(1/2*d*x + 1/2*c)^3 + 315*tan(1/2*d*x + 1/2*c)^2 + 280*tan(1/2*d*x + 1/2*c) - 126)/(a*tan(1/2*d*x + 1/2*c)^10))/d
Time = 36.82 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.74 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
int(cot(c + d*x)^8/(sin(c + d*x)^3*(a + a*sin(c + d*x))),x)
Output:
-(126*cos(c/2 + (d*x)/2)^20 - 126*sin(c/2 + (d*x)/2)^20 + 280*cos(c/2 + (d *x)/2)*sin(c/2 + (d*x)/2)^19 - 280*cos(c/2 + (d*x)/2)^19*sin(c/2 + (d*x)/2 ) + 315*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^18 - 1080*cos(c/2 + (d*x)/ 2)^3*sin(c/2 + (d*x)/2)^17 + 630*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^1 6 - 2520*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^14 + 6720*cos(c/2 + (d*x) /2)^7*sin(c/2 + (d*x)/2)^13 - 1260*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) ^12 - 15120*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^11 + 15120*cos(c/2 + ( d*x)/2)^11*sin(c/2 + (d*x)/2)^9 + 1260*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d* x)/2)^8 - 6720*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^7 + 2520*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^6 - 630*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d *x)/2)^4 + 1080*cos(c/2 + (d*x)/2)^17*sin(c/2 + (d*x)/2)^3 - 315*cos(c/2 + (d*x)/2)^18*sin(c/2 + (d*x)/2)^2 + 15120*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^10)/(1290240*a*d*cos(c /2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^10)
Time = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.06 \[ \int \frac {\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-2560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}+945 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-1280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+630 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+19200 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-15624 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-24320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+21168 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+8960 \cos \left (d x +c \right ) \sin \left (d x +c \right )-8064 \cos \left (d x +c \right )-945 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{10}}{80640 \sin \left (d x +c \right )^{10} a d} \] Input:
int(cot(d*x+c)^8*csc(d*x+c)^3/(a+a*sin(d*x+c)),x)
Output:
( - 2560*cos(c + d*x)*sin(c + d*x)**9 + 945*cos(c + d*x)*sin(c + d*x)**8 - 1280*cos(c + d*x)*sin(c + d*x)**7 + 630*cos(c + d*x)*sin(c + d*x)**6 + 19 200*cos(c + d*x)*sin(c + d*x)**5 - 15624*cos(c + d*x)*sin(c + d*x)**4 - 24 320*cos(c + d*x)*sin(c + d*x)**3 + 21168*cos(c + d*x)*sin(c + d*x)**2 + 89 60*cos(c + d*x)*sin(c + d*x) - 8064*cos(c + d*x) - 945*log(tan((c + d*x)/2 ))*sin(c + d*x)**10)/(80640*sin(c + d*x)**10*a*d)