Integrand size = 13, antiderivative size = 143 \[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{5/2}} \, dx=\frac {1}{4} \sqrt {-1+\sqrt {2}} \arctan \left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right )+\frac {1}{4} \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right )+\frac {1}{3 (1+\cot (x))^{3/2}}-\frac {1}{\sqrt {1+\cot (x)}} \]
1/3/(1+cot(x))^(3/2)-1/(1+cot(x))^(1/2)+1/4*arctan((3+cot(x)*(1-2^(1/2))-2 *2^(1/2))/(1+cot(x))^(1/2)/(-14+10*2^(1/2))^(1/2))*(2^(1/2)-1)^(1/2)+1/4*a rctanh((3+2*2^(1/2)+cot(x)*(1+2^(1/2)))/(1+cot(x))^(1/2)/(14+10*2^(1/2))^( 1/2))*(1+2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.54 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.43 \[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{5/2}} \, dx=\frac {4-(1+i) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\left (\frac {1}{2}-\frac {i}{2}\right ) (1+\cot (x))\right )-(1-i) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\left (\frac {1}{2}+\frac {i}{2}\right ) (1+\cot (x))\right )}{6 (1+\cot (x))^{3/2}} \]
(4 - (1 + I)*Hypergeometric2F1[-3/2, 1, -1/2, (1/2 - I/2)*(1 + Cot[x])] - (1 - I)*Hypergeometric2F1[-3/2, 1, -1/2, (1/2 + I/2)*(1 + Cot[x])])/(6*(1 + Cot[x])^(3/2))
Time = 0.58 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.19, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.077, Rules used = {3042, 4025, 25, 3042, 4012, 27, 3042, 25, 4019, 25, 3042, 4018, 216, 220}
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.
| \(\displaystyle \int \frac {\cot ^2(x)}{(\cot (x)+1)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan \left (x+\frac {\pi }{2}\right )^2}{\left (1-\tan \left (x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle \frac {1}{2} \int -\frac {1-\cot (x)}{(\cot (x)+1)^{3/2}}dx+\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3 (\cot (x)+1)^{3/2}}-\frac {1}{2} \int \frac {1-\cot (x)}{(\cot (x)+1)^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3 (\cot (x)+1)^{3/2}}-\frac {1}{2} \int \frac {\tan \left (x+\frac {\pi }{2}\right )+1}{\left (1-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int -\frac {2 \cot (x)}{\sqrt {\cot (x)+1}}dx-\frac {2}{\sqrt {\cot (x)+1}}\right )+\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {\cot (x)}{\sqrt {\cot (x)+1}}dx-\frac {2}{\sqrt {\cot (x)+1}}\right )+\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\int -\frac {\tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx-\frac {2}{\sqrt {\cot (x)+1}}\right )+\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {\tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx-\frac {2}{\sqrt {\cot (x)+1}}\right )+\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 4019 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int -\frac {1-\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}+\frac {\int -\frac {1-\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-\frac {2}{\sqrt {\cot (x)+1}}\right )+\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1-\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-\frac {\int \frac {1-\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-\frac {2}{\sqrt {\cot (x)+1}}\right )+\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {1-\left (-1-\sqrt {2}\right ) \tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1-\left (-1+\sqrt {2}\right ) \tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx}{2 \sqrt {2}}-\frac {2}{\sqrt {\cot (x)+1}}\right )+\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 4018 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\left (3-2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3\right )^2}{\cot (x)+1}-2 \left (7-5 \sqrt {2}\right )}d\left (-\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {\cot (x)+1}}\right )}{\sqrt {2}}+\frac {\left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3\right )^2}{\cot (x)+1}-2 \left (7+5 \sqrt {2}\right )}d\left (-\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {\cot (x)+1}}\right )}{\sqrt {2}}-\frac {2}{\sqrt {\cot (x)+1}}\right )+\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3\right )^2}{\cot (x)+1}-2 \left (7+5 \sqrt {2}\right )}d\left (-\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {\cot (x)+1}}\right )}{\sqrt {2}}+\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\cot (x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7}}-\frac {2}{\sqrt {\cot (x)+1}}\right )+\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\cot (x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7}}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\cot (x)+1}}\right )}{2 \sqrt {7+5 \sqrt {2}}}-\frac {2}{\sqrt {\cot (x)+1}}\right )+\frac {1}{3 (\cot (x)+1)^{3/2}}\) |
1/(3*(1 + Cot[x])^(3/2)) + (((3 - 2*Sqrt[2])*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Cot[x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])])/(2*Sqrt[-7 + 5*Sqrt[2]]) + ((3 + 2*Sqrt[2])*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Co t[x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])])/(2*Sqrt[7 + 5*Sqrt[2]]) - 2/Sqrt[1 + Cot[x]])/2
3.1.49.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 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 ]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[-2*(d^2/f) Subst[Int[1/(2*b*c*d - 4*a*d^2 + x^2), x], x, (b*c - 2*a*d - b*d*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0 ] && NeQ[c^2 + d^2, 0] && EqQ[2*a*c*d - b*(c^2 - d^2), 0]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( f_.)*(x_)]], x_Symbol] :> With[{q = Rt[a^2 + b^2, 2]}, Simp[1/(2*q) Int[( a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]], x], x] - Simp[1/(2*q) Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f *x])/Sqrt[a + b*Tan[e + f*x]], x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && N eQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2*a*c*d - b*(c^2 - d^2), 0] && NiceSqrtQ[a^2 + b^2]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*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^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 0.04 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.36
| method | result | size |
| derivativedivides | \(\frac {1}{3 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}-\frac {1}{\sqrt {1+\cot \left (x \right )}}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{8}+\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{8}\) | \(195\) |
| default | \(\frac {1}{3 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}-\frac {1}{\sqrt {1+\cot \left (x \right )}}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{8}+\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{8}\) | \(195\) |
1/3/(1+cot(x))^(3/2)-1/(1+cot(x))^(1/2)-1/8*2^(1/2)*(1/2*(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))+2*(2^(1/2)-1) /(-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)))+1/8*2^(1/2)*(1/2*(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))+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)))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.66 \[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{5/2}} \, dx=\frac {3 \, \sqrt {i + 1} {\left (\sqrt {2} \sin \left (2 \, x\right ) + \sqrt {2}\right )} \log \left (-\left (i - 1\right ) \, \sqrt {2} \sqrt {i + 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - 3 \, \sqrt {i + 1} {\left (\sqrt {2} \sin \left (2 \, x\right ) + \sqrt {2}\right )} \log \left (\left (i - 1\right ) \, \sqrt {2} \sqrt {i + 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) + 3 \, \sqrt {-i + 1} {\left (\sqrt {2} \sin \left (2 \, x\right ) + \sqrt {2}\right )} \log \left (\left (i + 1\right ) \, \sqrt {2} \sqrt {-i + 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - 3 \, \sqrt {-i + 1} {\left (\sqrt {2} \sin \left (2 \, x\right ) + \sqrt {2}\right )} \log \left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {-i + 1} + 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) + 4 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} {\left (2 \, \cos \left (2 \, x\right ) - 3 \, \sin \left (2 \, x\right ) - 2\right )}}{24 \, {\left (\sin \left (2 \, x\right ) + 1\right )}} \]
1/24*(3*sqrt(I + 1)*(sqrt(2)*sin(2*x) + sqrt(2))*log(-(I - 1)*sqrt(2)*sqrt (I + 1) + 2*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))) - 3*sqrt(I + 1)*(sqr t(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)*s qrt(-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))*(2*cos(2*x) - 3*sin(2*x) - 2))/(sin(2*x) + 1)
\[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{5/2}} \, dx=\int \frac {\cot ^{2}{\left (x \right )}}{\left (\cot {\left (x \right )} + 1\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{5/2}} \, dx=\int { \frac {\cot \left (x\right )^{2}}{{\left (\cot \left (x\right ) + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{5/2}} \, dx=\int { \frac {\cot \left (x\right )^{2}}{{\left (\cot \left (x\right ) + 1\right )}^{\frac {5}{2}}} \,d x } \]
Time = 13.18 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.69 \[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{5/2}} \, dx=\mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}-1}-\frac {4\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}-1}\right )\,\left (2\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}+2\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\right )-\mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}+1}+\frac {4\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}+1}\right )\,\left (2\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}-2\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\right )-\frac {\mathrm {cot}\left (x\right )+\frac {2}{3}}{{\left (\mathrm {cot}\left (x\right )+1\right )}^{3/2}} \]
atanh((4*2^(1/2)*(1/64 - 2^(1/2)/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(1/64 - 2^(1/2)/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*(1/64 - 2^(1/2)/64)^(1/2)*(2^(1/2)/ 64 + 1/64)^(1/2) - 1))*(2*(1/64 - 2^(1/2)/64)^(1/2) + 2*(2^(1/2)/64 + 1/64 )^(1/2)) - atanh((4*2^(1/2)*(1/64 - 2^(1/2)/64)^(1/2)*(cot(x) + 1)^(1/2))/ (64*(1/64 - 2^(1/2)/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*(1/64 - 2^(1/2)/64)^(1/2 )*(2^(1/2)/64 + 1/64)^(1/2) + 1))*(2*(1/64 - 2^(1/2)/64)^(1/2) - 2*(2^(1/2 )/64 + 1/64)^(1/2)) - (cot(x) + 2/3)/(cot(x) + 1)^(3/2)