Integrand size = 29, antiderivative size = 194 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \text {arctanh}(\cos (c+d x))}{256 a d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d} \] Output:
-3/256*arctanh(cos(d*x+c))/a/d-1/7*cot(d*x+c)^7/a/d-2/9*cot(d*x+c)^9/a/d-1 /11*cot(d*x+c)^11/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
Time = 5.67 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.96 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-2661120 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{10}(c+d x) (6840320+9973760 \cos (2 (c+d x))+3543040 \cos (4 (c+d x))+343040 \cos (6 (c+d x))-61440 \cos (8 (c+d x))+5120 \cos (10 (c+d x))-3219678 \sin (c+d x)-2608452 \sin (3 (c+d x))-2181564 \sin (5 (c+d x))-121275 \sin (7 (c+d x))+10395 \sin (9 (c+d x)))\right )}{227082240 a d (1+\sin (c+d x))} \] Input:
Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
Output:
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(-2661120*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x]^10*(6840320 + 9973760 *Cos[2*(c + d*x)] + 3543040*Cos[4*(c + d*x)] + 343040*Cos[6*(c + d*x)] - 6 1440*Cos[8*(c + d*x)] + 5120*Cos[10*(c + d*x)] - 3219678*Sin[c + d*x] - 26 08452*Sin[3*(c + d*x)] - 2181564*Sin[5*(c + d*x)] - 121275*Sin[7*(c + d*x) ] + 10395*Sin[9*(c + d*x)])))/(227082240*a*d*(1 + Sin[c + d*x]))
Time = 1.10 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.99, 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.
| \(\displaystyle \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^{12} (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cot ^6(c+d x) \csc ^6(c+d x)dx}{a}-\frac {\int \cot ^6(c+d x) \csc ^5(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^6 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle \frac {\int \cot ^6(c+d x) \left (\cot ^2(c+d x)+1\right )^2d(-\cot (c+d x))}{a d}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \left (\cot ^{10}(c+d x)+2 \cot ^8(c+d x)+\cot ^6(c+d x)\right )d(-\cot (c+d x))}{a d}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {-\frac {1}{2} \int \cot ^4(c+d x) \csc ^5(c+d x)dx-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {-\frac {1}{2} \int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {\frac {1}{2} \left (\frac {3}{8} \int \cot ^2(c+d x) \csc ^5(c+d x)dx+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {\frac {1}{2} \left (\frac {3}{8} \int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {\frac {1}{2} \left (\frac {3}{8} \left (-\frac {1}{6} \int \csc ^5(c+d x)dx-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {\frac {1}{2} \left (\frac {3}{8} \left (-\frac {1}{6} \int \csc (c+d x)^5dx-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \int \csc ^3(c+d x)dx\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \int \csc (c+d x)^3dx\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {1}{11} \cot ^{11}(c+d x)-\frac {2}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}\right )+\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\right )-\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 d}}{a}\) |
Input:
Int[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
Output:
(-1/7*Cot[c + d*x]^7 - (2*Cot[c + d*x]^9)/9 - Cot[c + d*x]^11/11)/(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*(Cot[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]
Result contains complex when optimal does not.
Time = 4.05 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.40
| method | result | size |
| risch | \(\frac {10395 \,{\mathrm e}^{21 i \left (d x +c \right )}-110880 \,{\mathrm e}^{19 i \left (d x +c \right )}+15206400 i {\mathrm e}^{8 i \left (d x +c \right )}-2302839 \,{\mathrm e}^{17 i \left (d x +c \right )}+11827200 i {\mathrm e}^{14 i \left (d x +c \right )}-4790016 \,{\mathrm e}^{15 i \left (d x +c \right )}+3041280 i {\mathrm e}^{6 i \left (d x +c \right )}-5828130 \,{\mathrm e}^{13 i \left (d x +c \right )}+26019840 i {\mathrm e}^{12 i \left (d x +c \right )}+4730880 i {\mathrm e}^{16 i \left (d x +c \right )}+5828130 \,{\mathrm e}^{9 i \left (d x +c \right )}-112640 i {\mathrm e}^{2 i \left (d x +c \right )}+4790016 \,{\mathrm e}^{7 i \left (d x +c \right )}+21288960 i {\mathrm e}^{10 i \left (d x +c \right )}+2302839 \,{\mathrm e}^{5 i \left (d x +c \right )}+563200 i {\mathrm e}^{4 i \left (d x +c \right )}+110880 \,{\mathrm e}^{3 i \left (d x +c \right )}+10240 i-10395 \,{\mathrm e}^{i \left (d x +c \right )}}{443520 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d a}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d a}\) | \(272\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}+\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}+\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}+\frac {5}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {10}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{2048 d a}\) | \(302\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}+\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}+\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}+\frac {5}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {10}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{2048 d a}\) | \(302\) |
Input:
int(cot(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/443520*(10395*exp(21*I*(d*x+c))-110880*exp(19*I*(d*x+c))+15206400*I*exp( 8*I*(d*x+c))-2302839*exp(17*I*(d*x+c))+11827200*I*exp(14*I*(d*x+c))-479001 6*exp(15*I*(d*x+c))+3041280*I*exp(6*I*(d*x+c))-5828130*exp(13*I*(d*x+c))+2 6019840*I*exp(12*I*(d*x+c))+4730880*I*exp(16*I*(d*x+c))+5828130*exp(9*I*(d *x+c))-112640*I*exp(2*I*(d*x+c))+4790016*exp(7*I*(d*x+c))+21288960*I*exp(1 0*I*(d*x+c))+2302839*exp(5*I*(d*x+c))+563200*I*exp(4*I*(d*x+c))+110880*exp (3*I*(d*x+c))+10240*I-10395*exp(I*(d*x+c)))/d/a/(exp(2*I*(d*x+c))-1)^11+3/ 256/d/a*ln(exp(I*(d*x+c))-1)-3/256/d/a*ln(exp(I*(d*x+c))+1)
Time = 0.10 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.56 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {20480 \, \cos \left (d x + c\right )^{11} - 112640 \, \cos \left (d x + c\right )^{9} + 253440 \, \cos \left (d x + c\right )^{7} - 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 1386 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1774080 \, {\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/1774080*(20480*cos(d*x + c)^11 - 112640*cos(d*x + c)^9 + 253440*cos(d*x + c)^7 - 10395*(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 1 0*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d *x + c) + 10395*(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)*sin (d*x + c) + 1386*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 - 128*cos(d*x + c) ^5 + 70*cos(d*x + c)^3 - 15*cos(d*x + c))*sin(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)*sin(d*x + c))
Timed out. \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**8*csc(d*x+c)**4/(a+a*sin(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (176) = 352\).
Time = 0.04 (sec) , antiderivative size = 475, normalized size of antiderivative = 2.45 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/14192640*((69300*sin(d*x + c)/(cos(d*x + c) + 1) + 13860*sin(d*x + c)^2 /(cos(d*x + c) + 1)^2 - 23100*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 27720* sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 6930*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 6930*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 4950*sin(d*x + c)^7/(cos (d*x + c) + 1)^7 - 3465*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 770*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 1386*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 630*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/a - 166320*log(sin(d*x + c)/(c os(d*x + c) + 1))/a - (1386*sin(d*x + c)/(cos(d*x + c) + 1) + 770*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 3465*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 4 950*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 6930*sin(d*x + c)^5/(cos(d*x + c ) + 1)^5 - 6930*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 27720*sin(d*x + c)^7 /(cos(d*x + c) + 1)^7 - 23100*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 13860* sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 69300*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 630)*(cos(d*x + c) + 1)^11/(a*sin(d*x + c)^11))/d
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (176) = 352\).
Time = 0.20 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.86 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {166320 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {630 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1386 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 770 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3465 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 4950 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6930 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27720 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 23100 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 13860 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 69300 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{11}} - \frac {502266 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 69300 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 13860 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 23100 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 27720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6930 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4950 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 770 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1386 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 630}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{14192640 \, d} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/14192640*(166320*log(abs(tan(1/2*d*x + 1/2*c)))/a + (630*a^10*tan(1/2*d* x + 1/2*c)^11 - 1386*a^10*tan(1/2*d*x + 1/2*c)^10 - 770*a^10*tan(1/2*d*x + 1/2*c)^9 + 3465*a^10*tan(1/2*d*x + 1/2*c)^8 - 4950*a^10*tan(1/2*d*x + 1/2 *c)^7 + 6930*a^10*tan(1/2*d*x + 1/2*c)^6 + 6930*a^10*tan(1/2*d*x + 1/2*c)^ 5 - 27720*a^10*tan(1/2*d*x + 1/2*c)^4 + 23100*a^10*tan(1/2*d*x + 1/2*c)^3 - 13860*a^10*tan(1/2*d*x + 1/2*c)^2 - 69300*a^10*tan(1/2*d*x + 1/2*c))/a^1 1 - (502266*tan(1/2*d*x + 1/2*c)^11 - 69300*tan(1/2*d*x + 1/2*c)^10 - 1386 0*tan(1/2*d*x + 1/2*c)^9 + 23100*tan(1/2*d*x + 1/2*c)^8 - 27720*tan(1/2*d* x + 1/2*c)^7 + 6930*tan(1/2*d*x + 1/2*c)^6 + 6930*tan(1/2*d*x + 1/2*c)^5 - 4950*tan(1/2*d*x + 1/2*c)^4 + 3465*tan(1/2*d*x + 1/2*c)^3 - 770*tan(1/2*d *x + 1/2*c)^2 - 1386*tan(1/2*d*x + 1/2*c) + 630)/(a*tan(1/2*d*x + 1/2*c)^1 1))/d
Time = 38.71 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.98 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
int(cot(c + d*x)^8/(sin(c + d*x)^4*(a + a*sin(c + d*x))),x)
Output:
(630*sin(c/2 + (d*x)/2)^22 - 630*cos(c/2 + (d*x)/2)^22 - 1386*cos(c/2 + (d *x)/2)*sin(c/2 + (d*x)/2)^21 + 1386*cos(c/2 + (d*x)/2)^21*sin(c/2 + (d*x)/ 2) - 770*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^20 + 3465*cos(c/2 + (d*x) /2)^3*sin(c/2 + (d*x)/2)^19 - 4950*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2) ^18 + 6930*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^17 + 6930*cos(c/2 + (d* x)/2)^6*sin(c/2 + (d*x)/2)^16 - 27720*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x) /2)^15 + 23100*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^14 - 13860*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^13 - 69300*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^12 + 69300*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^10 + 13860*c os(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^9 - 23100*cos(c/2 + (d*x)/2)^14*si n(c/2 + (d*x)/2)^8 + 27720*cos(c/2 + (d*x)/2)^15*sin(c/2 + (d*x)/2)^7 - 69 30*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d*x)/2)^6 - 6930*cos(c/2 + (d*x)/2)^17 *sin(c/2 + (d*x)/2)^5 + 4950*cos(c/2 + (d*x)/2)^18*sin(c/2 + (d*x)/2)^4 - 3465*cos(c/2 + (d*x)/2)^19*sin(c/2 + (d*x)/2)^3 + 770*cos(c/2 + (d*x)/2)^2 0*sin(c/2 + (d*x)/2)^2 + 166320*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) *cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^11)/(14192640*a*d*cos(c/2 + (d*x )/2)^11*sin(c/2 + (d*x)/2)^11)
Time = 0.24 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.05 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {10240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10}-10395 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}+5120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-6930 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+3840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+171864 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-144640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-232848 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+206080 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+88704 \cos \left (d x +c \right ) \sin \left (d x +c \right )-80640 \cos \left (d x +c \right )+10395 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{11}}{887040 \sin \left (d x +c \right )^{11} a d} \] Input:
int(cot(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c)),x)
Output:
(10240*cos(c + d*x)*sin(c + d*x)**10 - 10395*cos(c + d*x)*sin(c + d*x)**9 + 5120*cos(c + d*x)*sin(c + d*x)**8 - 6930*cos(c + d*x)*sin(c + d*x)**7 + 3840*cos(c + d*x)*sin(c + d*x)**6 + 171864*cos(c + d*x)*sin(c + d*x)**5 - 144640*cos(c + d*x)*sin(c + d*x)**4 - 232848*cos(c + d*x)*sin(c + d*x)**3 + 206080*cos(c + d*x)*sin(c + d*x)**2 + 88704*cos(c + d*x)*sin(c + d*x) - 80640*cos(c + d*x) + 10395*log(tan((c + d*x)/2))*sin(c + d*x)**11)/(887040 *sin(c + d*x)**11*a*d)