Integrand size = 29, antiderivative size = 152 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 \text {arctanh}(\cos (c+d x))}{128 a d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {\cot ^9(c+d x)}{9 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d}+\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d} \] Output:
-5/128*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-5 /128*cot(d*x+c)*csc(d*x+c)/a/d+5/64*cot(d*x+c)*csc(d*x+c)^3/a/d-5/48*cot(d *x+c)^3*csc(d*x+c)^3/a/d+1/8*cot(d*x+c)^5*csc(d*x+c)^3/a/d
Leaf count is larger than twice the leaf count of optimal. \(313\) vs. \(2(152)=304\).
Time = 4.44 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.06 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^9(c+d x) \left (129024 \cos (c+d x)+75264 \cos (3 (c+d x))+23040 \cos (5 (c+d x))+2304 \cos (7 (c+d x))-256 \cos (9 (c+d x))+39690 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-39690 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-36540 \sin (2 (c+d x))-26460 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+26460 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-20916 \sin (4 (c+d x))+11340 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-11340 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-16044 \sin (6 (c+d x))-2835 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+2835 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-630 \sin (8 (c+d x))+315 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))-315 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))\right )}{2064384 a d} \] Input:
Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
Output:
-1/2064384*(Csc[c + d*x]^9*(129024*Cos[c + d*x] + 75264*Cos[3*(c + d*x)] + 23040*Cos[5*(c + d*x)] + 2304*Cos[7*(c + d*x)] - 256*Cos[9*(c + d*x)] + 3 9690*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 39690*Log[Sin[(c + d*x)/2]]*Sin[ c + d*x] - 36540*Sin[2*(c + d*x)] - 26460*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 26460*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 20916*Sin[4*(c + d *x)] + 11340*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 11340*Log[Sin[(c + d *x)/2]]*Sin[5*(c + d*x)] - 16044*Sin[6*(c + d*x)] - 2835*Log[Cos[(c + d*x) /2]]*Sin[7*(c + d*x)] + 2835*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 630* Sin[8*(c + d*x)] + 315*Log[Cos[(c + d*x)/2]]*Sin[9*(c + d*x)] - 315*Log[Si n[(c + d*x)/2]]*Sin[9*(c + d*x)]))/(a*d)
Time = 0.93 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 3318, 3042, 3087, 244, 2009, 3091, 3042, 3091, 3042, 3091, 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.
| \(\displaystyle \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^{10} (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cot ^6(c+d x) \csc ^4(c+d x)dx}{a}-\frac {\int \cot ^6(c+d x) \csc ^3(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^4 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle \frac {\int \cot ^6(c+d x) \left (\cot ^2(c+d x)+1\right )d(-\cot (c+d x))}{a d}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \left (\cot ^8(c+d x)+\cot ^6(c+d x)\right )d(-\cot (c+d x))}{a d}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {-\frac {5}{8} \int \cot ^4(c+d x) \csc ^3(c+d x)dx-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {-\frac {5}{8} \int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {-\frac {5}{8} \left (-\frac {1}{2} \int \cot ^2(c+d x) \csc ^3(c+d x)dx-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {-\frac {5}{8} \left (-\frac {1}{2} \int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \int \csc ^3(c+d x)dx+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \int \csc (c+d x)^3dx+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)}{a d}-\frac {-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}}{a}\) |
Input:
Int[(Cot[c + d*x]^8*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
Output:
(-1/7*Cot[c + d*x]^7 - Cot[c + d*x]^9/9)/(a*d) - (-1/8*(Cot[c + d*x]^5*Csc [c + d*x]^3)/d - (5*(-1/6*(Cot[c + d*x]^3*Csc[c + d*x]^3)/d + ((Cot[c + d* x]*Csc[c + d*x]^3)/(4*d) + (-1/2*ArcTanh[Cos[c + d*x]]/d - (Cot[c + d*x]*C sc[c + d*x])/(2*d))/4)/2))/8)/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 = 2.34 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.50
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{4}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {8}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {1}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}+\frac {3}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}}{512 d a}\) | \(228\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{4}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {8}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {1}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}+\frac {3}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}}{512 d a}\) | \(228\) |
| risch | \(\frac {315 \,{\mathrm e}^{17 i \left (d x +c \right )}-48384 i {\mathrm e}^{8 i \left (d x +c \right )}+8022 \,{\mathrm e}^{15 i \left (d x +c \right )}-80640 i {\mathrm e}^{10 i \left (d x +c \right )}+10458 \,{\mathrm e}^{13 i \left (d x +c \right )}-16128 i {\mathrm e}^{14 i \left (d x +c \right )}+18270 \,{\mathrm e}^{11 i \left (d x +c \right )}-48384 i {\mathrm e}^{6 i \left (d x +c \right )}-26880 i {\mathrm e}^{12 i \left (d x +c \right )}-18270 \,{\mathrm e}^{7 i \left (d x +c \right )}-2304 i {\mathrm e}^{2 i \left (d x +c \right )}-10458 \,{\mathrm e}^{5 i \left (d x +c \right )}-6912 i {\mathrm e}^{4 i \left (d x +c \right )}-8022 \,{\mathrm e}^{3 i \left (d x +c \right )}+256 i-315 \,{\mathrm e}^{i \left (d x +c \right )}}{4032 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d a}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d a}\) | \(238\) |
Input:
int(cot(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/512/d/a*(1/9*tan(1/2*d*x+1/2*c)^9-1/4*tan(1/2*d*x+1/2*c)^8-3/7*tan(1/2*d *x+1/2*c)^7+4/3*tan(1/2*d*x+1/2*c)^6-2*tan(1/2*d*x+1/2*c)^4+8/3*tan(1/2*d* x+1/2*c)^3-4*tan(1/2*d*x+1/2*c)^2-6*tan(1/2*d*x+1/2*c)+2/tan(1/2*d*x+1/2*c )^4+4/tan(1/2*d*x+1/2*c)^2-4/3/tan(1/2*d*x+1/2*c)^6+20*ln(tan(1/2*d*x+1/2* c))+6/tan(1/2*d*x+1/2*c)-8/3/tan(1/2*d*x+1/2*c)^3+1/4/tan(1/2*d*x+1/2*c)^8 -1/9/tan(1/2*d*x+1/2*c)^9+3/7/tan(1/2*d*x+1/2*c)^7)
Time = 0.09 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.64 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {512 \, \cos \left (d x + c\right )^{9} - 2304 \, \cos \left (d x + c\right )^{7} - 315 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 315 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 42 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16128 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/16128*(512*cos(d*x + c)^9 - 2304*cos(d*x + c)^7 - 315*(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*log(1/2*cos(d* x + c) + 1/2)*sin(d*x + c) + 315*(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*co s(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 42*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c)^3 + 15* cos(d*x + c))*sin(d*x + c))/((a*d*cos(d*x + c)^8 - 4*a*d*cos(d*x + c)^6 + 6*a*d*cos(d*x + c)^4 - 4*a*d*cos(d*x + c)^2 + a*d)*sin(d*x + c))
Timed out. \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**8*csc(d*x+c)**2/(a+a*sin(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (138) = 276\).
Time = 0.05 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.34 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {1512 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1008 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {672 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {504 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {336 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {108 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {63 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {28 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a} - \frac {5040 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {{\left (\frac {63 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {108 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {336 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {504 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {672 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1008 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {1512 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 28\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{9}}{a \sin \left (d x + c\right )^{9}}}{129024 \, d} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/129024*((1512*sin(d*x + c)/(cos(d*x + c) + 1) + 1008*sin(d*x + c)^2/(co s(d*x + c) + 1)^2 - 672*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 504*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 336*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 10 8*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 63*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 28*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a - 5040*log(sin(d*x + c)/( cos(d*x + c) + 1))/a - (63*sin(d*x + c)/(cos(d*x + c) + 1) + 108*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 336*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 504 *sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 672*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1008*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 1512*sin(d*x + c)^8/(cos (d*x + c) + 1)^8 - 28)*(cos(d*x + c) + 1)^9/(a*sin(d*x + c)^9))/d
Time = 0.20 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.80 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {5040 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {28 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 63 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 108 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 336 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1008 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1512 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}} - \frac {14258 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1512 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1008 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 672 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 108 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 28}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{129024 \, d} \] Input:
integrate(cot(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/129024*(5040*log(abs(tan(1/2*d*x + 1/2*c)))/a + (28*a^8*tan(1/2*d*x + 1/ 2*c)^9 - 63*a^8*tan(1/2*d*x + 1/2*c)^8 - 108*a^8*tan(1/2*d*x + 1/2*c)^7 + 336*a^8*tan(1/2*d*x + 1/2*c)^6 - 504*a^8*tan(1/2*d*x + 1/2*c)^4 + 672*a^8* tan(1/2*d*x + 1/2*c)^3 - 1008*a^8*tan(1/2*d*x + 1/2*c)^2 - 1512*a^8*tan(1/ 2*d*x + 1/2*c))/a^9 - (14258*tan(1/2*d*x + 1/2*c)^9 - 1512*tan(1/2*d*x + 1 /2*c)^8 - 1008*tan(1/2*d*x + 1/2*c)^7 + 672*tan(1/2*d*x + 1/2*c)^6 - 504*t an(1/2*d*x + 1/2*c)^5 + 336*tan(1/2*d*x + 1/2*c)^3 - 108*tan(1/2*d*x + 1/2 *c)^2 - 63*tan(1/2*d*x + 1/2*c) + 28)/(a*tan(1/2*d*x + 1/2*c)^9))/d
Time = 35.38 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.86 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {28\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-28\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-63\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-108\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}-504\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-1512\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+1512\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+504\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+108\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+5040\,\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 )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{129024\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \] Input:
int(cot(c + d*x)^8/(sin(c + d*x)^2*(a + a*sin(c + d*x))),x)
Output:
(28*sin(c/2 + (d*x)/2)^18 - 28*cos(c/2 + (d*x)/2)^18 - 63*cos(c/2 + (d*x)/ 2)*sin(c/2 + (d*x)/2)^17 + 63*cos(c/2 + (d*x)/2)^17*sin(c/2 + (d*x)/2) - 1 08*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^16 + 336*cos(c/2 + (d*x)/2)^3*s in(c/2 + (d*x)/2)^15 - 504*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^13 + 67 2*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^12 - 1008*cos(c/2 + (d*x)/2)^7*s in(c/2 + (d*x)/2)^11 - 1512*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^10 + 1 512*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^8 + 1008*cos(c/2 + (d*x)/2)^1 1*sin(c/2 + (d*x)/2)^7 - 672*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^6 + 504*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^5 - 336*cos(c/2 + (d*x)/2)^15 *sin(c/2 + (d*x)/2)^3 + 108*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d*x)/2)^2 + 5 040*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^9*sin(c/ 2 + (d*x)/2)^9)/(129024*a*d*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^9)
Time = 0.23 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {256 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-315 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+128 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+2478 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-1920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-2856 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+2432 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+1008 \cos \left (d x +c \right ) \sin \left (d x +c \right )-896 \cos \left (d x +c \right )+315 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{9}}{8064 \sin \left (d x +c \right )^{9} a d} \] Input:
int(cot(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x)
Output:
(256*cos(c + d*x)*sin(c + d*x)**8 - 315*cos(c + d*x)*sin(c + d*x)**7 + 128 *cos(c + d*x)*sin(c + d*x)**6 + 2478*cos(c + d*x)*sin(c + d*x)**5 - 1920*c os(c + d*x)*sin(c + d*x)**4 - 2856*cos(c + d*x)*sin(c + d*x)**3 + 2432*cos (c + d*x)*sin(c + d*x)**2 + 1008*cos(c + d*x)*sin(c + d*x) - 896*cos(c + d *x) + 315*log(tan((c + d*x)/2))*sin(c + d*x)**9)/(8064*sin(c + d*x)**9*a*d )