Integrand size = 11, antiderivative size = 216 \[ \int \frac {\cot (x)}{(1+\cot (x))^{5/2}} \, dx=\frac {1}{4} \sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )-\frac {1}{4} \sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )-\frac {1}{3 (1+\cot (x))^{3/2}}+\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{8 \sqrt {1+\sqrt {2}}}-\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{8 \sqrt {1+\sqrt {2}}} \]
-1/3/(1+cot(x))^(3/2)+1/8*ln(1+cot(x)+2^(1/2)-(1+cot(x))^(1/2)*(2+2*2^(1/2 ))^(1/2))/(1+2^(1/2))^(1/2)-1/8*ln(1+cot(x)+2^(1/2)+(1+cot(x))^(1/2)*(2+2* 2^(1/2))^(1/2))/(1+2^(1/2))^(1/2)+1/4*arctan((-2*(1+cot(x))^(1/2)+(2+2*2^( 1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)-1/4*arctan((2*(1+cot( x))^(1/2)+(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.32 \[ \int \frac {\cot (x)}{(1+\cot (x))^{5/2}} \, dx=-\frac {1}{4} (1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )-\frac {1}{4} (1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )-\frac {1}{3 (1+\cot (x))^{3/2}} \]
-1/4*((1 - I)^(3/2)*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]]) - ((1 + I)^(3/2 )*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]])/4 - 1/(3*(1 + Cot[x])^(3/2))
Time = 0.50 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.182, Rules used = {3042, 25, 4012, 25, 3042, 3966, 484, 1407, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {\cot (x)}{(\cot (x)+1)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\tan \left (x+\frac {\pi }{2}\right )}{\left (1-\tan \left (x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\tan \left (x+\frac {\pi }{2}\right )}{\left (1-\tan \left (x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\frac {1}{2} \int -\frac {1}{\sqrt {\cot (x)+1}}dx-\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {\cot (x)+1}}dx-\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx-\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 3966 |
| \(\displaystyle -\frac {1}{2} \int \frac {1}{\sqrt {\cot (x)+1} \left (\cot ^2(x)+1\right )}d\cot (x)-\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 484 |
| \(\displaystyle -\int \frac {1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}-\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 1407 |
| \(\displaystyle -\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-\sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}-\frac {\int \frac {\sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}-\frac {1}{2} \int -\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}-\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}+\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}-\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\cot (x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {\frac {1}{2} \int \frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\cot (x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {\frac {1}{2} \int \frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
-1/3*1/(1 + Cot[x])^(3/2) - (Sqrt[(1 + Sqrt[2])/(-1 + Sqrt[2])]*ArcTan[(-S qrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/2)/(4*Sqrt[1 + Sqrt[2]]) - (Sqrt[(1 + Sqrt[2])/(-1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqr t[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] + Log[1 + Sqrt[2] + C ot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/2)/(4*Sqrt[1 + Sqrt[2]])
3.1.50.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2* d Subst[Int[1/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Su bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c , d, n}, x] && NeQ[a^2 + b^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Leaf count of result is larger than twice the leaf count of optimal. \(443\) vs. \(2(152)=304\).
Time = 0.03 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.06
| method | result | size |
| derivativedivides | \(-\frac {1}{3 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{16}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{8 \sqrt {-2+2 \sqrt {2}}}+\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{16}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{8 \sqrt {-2+2 \sqrt {2}}}+\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{2 \sqrt {-2+2 \sqrt {2}}}\) | \(444\) |
| default | \(-\frac {1}{3 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{16}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{8 \sqrt {-2+2 \sqrt {2}}}+\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{16}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{8 \sqrt {-2+2 \sqrt {2}}}+\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{2 \sqrt {-2+2 \sqrt {2}}}\) | \(444\) |
-1/3/(1+cot(x))^(3/2)-1/16*(2+2*2^(1/2))^(1/2)*2^(1/2)*ln(1+cot(x)+2^(1/2) -(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2))+1/8*(2+2*2^(1/2))^(1/2)*ln(1+cot(x) +2^(1/2)-(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2))-1/8*2^(1/2)*(2+2*2^(1/2))/( -2+2*2^(1/2))^(1/2)*arctan((2*(1+cot(x))^(1/2)-(2+2*2^(1/2))^(1/2))/(-2+2* 2^(1/2))^(1/2))+1/4*(2+2*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+cot(x) )^(1/2)-(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))-1/2/(-2+2*2^(1/2))^(1/2 )*arctan((2*(1+cot(x))^(1/2)-(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))*2^ (1/2)+1/16*(2+2*2^(1/2))^(1/2)*2^(1/2)*ln(1+cot(x)+2^(1/2)+(1+cot(x))^(1/2 )*(2+2*2^(1/2))^(1/2))-1/8*(2+2*2^(1/2))^(1/2)*ln(1+cot(x)+2^(1/2)+(1+cot( x))^(1/2)*(2+2*2^(1/2))^(1/2))-1/8*2^(1/2)*(2+2*2^(1/2))/(-2+2*2^(1/2))^(1 /2)*arctan((2*(1+cot(x))^(1/2)+(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))+ 1/4*(2+2*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+cot(x))^(1/2)+(2+2*2^( 1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))-1/2/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+c ot(x))^(1/2)+(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))*2^(1/2)
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.06 \[ \int \frac {\cot (x)}{(1+\cot (x))^{5/2}} \, dx=-\frac {3 \, \sqrt {i - 1} {\left (\sqrt {2} \sin \left (2 \, x\right ) + \sqrt {2}\right )} \log \left (-\left (i - 1\right ) \, \sqrt {2} \sqrt {i - 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - 3 \, \sqrt {i - 1} {\left (\sqrt {2} \sin \left (2 \, x\right ) + \sqrt {2}\right )} \log \left (\left (i - 1\right ) \, \sqrt {2} \sqrt {i - 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) + 3 \, \sqrt {-i - 1} {\left (\sqrt {2} \sin \left (2 \, x\right ) + \sqrt {2}\right )} \log \left (\left (i + 1\right ) \, \sqrt {2} \sqrt {-i - 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - 3 \, \sqrt {-i - 1} {\left (\sqrt {2} \sin \left (2 \, x\right ) + \sqrt {2}\right )} \log \left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {-i - 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - 4 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{24 \, {\left (\sin \left (2 \, x\right ) + 1\right )}} \]
-1/24*(3*sqrt(I - 1)*(sqrt(2)*sin(2*x) + sqrt(2))*log(-(I - 1)*sqrt(2)*sqr t(I - 1) + 2*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))) - 3*sqrt(I - 1)*(sq rt(2)*sin(2*x) + sqrt(2))*log((I - 1)*sqrt(2)*sqrt(I - 1) + 2*sqrt((cos(2* x) + sin(2*x) + 1)/sin(2*x))) + 3*sqrt(-I - 1)*(sqrt(2)*sin(2*x) + sqrt(2) )*log((I + 1)*sqrt(2)*sqrt(-I - 1) + 2*sqrt((cos(2*x) + sin(2*x) + 1)/sin( 2*x))) - 3*sqrt(-I - 1)*(sqrt(2)*sin(2*x) + sqrt(2))*log(-(I + 1)*sqrt(2)* sqrt(-I - 1) + 2*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))) - 4*sqrt((cos(2 *x) + sin(2*x) + 1)/sin(2*x))*(cos(2*x) - 1))/(sin(2*x) + 1)
\[ \int \frac {\cot (x)}{(1+\cot (x))^{5/2}} \, dx=\int \frac {\cot {\left (x \right )}}{\left (\cot {\left (x \right )} + 1\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\cot (x)}{(1+\cot (x))^{5/2}} \, dx=\int { \frac {\cot \left (x\right )}{{\left (\cot \left (x\right ) + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\cot (x)}{(1+\cot (x))^{5/2}} \, dx=\int { \frac {\cot \left (x\right )}{{\left (\cot \left (x\right ) + 1\right )}^{\frac {5}{2}}} \,d x } \]
Time = 13.10 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.10 \[ \int \frac {\cot (x)}{(1+\cot (x))^{5/2}} \, dx=\mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}+1}+\frac {4\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}+1}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}-2\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\right )-\mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}-1}-\frac {4\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}-1}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}+2\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\right )-\frac {1}{3\,{\left (\mathrm {cot}\left (x\right )+1\right )}^{3/2}} \]
atanh((4*2^(1/2)*(- 2^(1/2)/64 - 1/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(2^(1 /2)/64 - 1/64)^(1/2)*(- 2^(1/2)/64 - 1/64)^(1/2) + 1) + (4*2^(1/2)*(2^(1/2 )/64 - 1/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(2^(1/2)/64 - 1/64)^(1/2)*(- 2^ (1/2)/64 - 1/64)^(1/2) + 1))*(2*(- 2^(1/2)/64 - 1/64)^(1/2) - 2*(2^(1/2)/6 4 - 1/64)^(1/2)) - atanh((4*2^(1/2)*(- 2^(1/2)/64 - 1/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(2^(1/2)/64 - 1/64)^(1/2)*(- 2^(1/2)/64 - 1/64)^(1/2) - 1) - (4*2^(1/2)*(2^(1/2)/64 - 1/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(2^(1/2)/64 - 1/64)^(1/2)*(- 2^(1/2)/64 - 1/64)^(1/2) - 1))*(2*(- 2^(1/2)/64 - 1/64)^( 1/2) + 2*(2^(1/2)/64 - 1/64)^(1/2)) - 1/(3*(cot(x) + 1)^(3/2))