Integrand size = 13, antiderivative size = 223 \[ \int \cot ^2(x) \sqrt {1+\cot (x)} \, dx=-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )-\frac {2}{3} (1+\cot (x))^{3/2}+\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}} \]
-2/3*(1+cot(x))^(3/2)-1/2*arctan((-2*(1+cot(x))^(1/2)+(2+2*2^(1/2))^(1/2)) /(-2+2*2^(1/2))^(1/2))*(2+2*2^(1/2))^(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+2*2^(1/2))^(1/2)+1/2*ln(1+co t(x)+2^(1/2)-(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(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*2^(1/2))^( 1/2)
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.31 \[ \int \cot ^2(x) \sqrt {1+\cot (x)} \, dx=-i \sqrt {1-i} \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )+i \sqrt {1+i} \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )-\frac {2}{3} (1+\cot (x))^{3/2} \]
(-I)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]] + I*Sqrt[1 + I]*Arc Tanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]] - (2*(1 + Cot[x])^(3/2))/3
Time = 0.54 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4026, 25, 3042, 3966, 483, 1447, 1475, 1083, 217, 1478, 25, 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.
| \(\displaystyle \int \cot ^2(x) \sqrt {\cot (x)+1} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {1-\tan \left (x+\frac {\pi }{2}\right )} \tan \left (x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 4026 |
| \(\displaystyle \int -\sqrt {\cot (x)+1}dx-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \sqrt {\cot (x)+1}dx-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}dx-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 3966 |
| \(\displaystyle \int \frac {\sqrt {\cot (x)+1}}{\cot ^2(x)+1}d\cot (x)-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 483 |
| \(\displaystyle 2 \int \frac {\cot (x)+1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 1447 |
| \(\displaystyle 2 \left (\frac {1}{2} \int \frac {\cot (x)+\sqrt {2}+1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}-\frac {1}{2} \int \frac {-\cot (x)+\sqrt {2}-1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}\right )-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 1475 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}+\frac {1}{2} \int \frac {1}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}\right )-\frac {1}{2} \int \frac {-\cot (x)+\sqrt {2}-1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}\right )-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (-\int \frac {1}{-\cot (x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}\right )-\int \frac {1}{-\cot (x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}\right )\right )-\frac {1}{2} \int \frac {-\cot (x)+\sqrt {2}-1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}\right )-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \int \frac {-\cot (x)+\sqrt {2}-1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}\right )-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 1478 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\int -\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )\right )-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\int \frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )\right )-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \left (\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )\right )-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
(-2*(1 + Cot[x])^(3/2))/3 + 2*((ArcTan[(-Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]]/Sqrt[2*(-1 + Sqrt[2])] + ArcTan[(Sqrt[2 *(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]]/Sqrt[2*(-1 + Sqrt[2])])/2 + (Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[2*(1 + Sqrt[2])]) - Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[2*(1 + Sqrt[2])]))/2)
3.1.41.3.1 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[Sqrt[(c_) + (d_.)*(x_)]/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[2*d Subst[Int[x^2/(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[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a/c, 2]}, Simp[1/2 Int[(q + x^2)/(a + b*x^2 + c*x^4), x], x] - Simp[1/2 Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && LtQ[b ^2 - 4*a*c, 0] && PosQ[a*c]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^ 2, x], x], x] + Simp[e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; F reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] , 0]))
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[-2*(d/e) - b/c, 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ [c*d^2 - a*e^2, 0] && !GtQ[b^2 - 4*a*c, 0]
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_)])^2, x_Symbol] :> Simp[d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*( m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e + f* x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ [m, -1] && !(EqQ[m, 2] && EqQ[a, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(355\) vs. \(2(160)=320\).
Time = 0.15 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.60
| method | result | size |
| derivativedivides | \(-\frac {2 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}{3}+\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}\) | \(356\) |
| default | \(-\frac {2 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}{3}+\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}\) | \(356\) |
-2/3*(1+cot(x))^(3/2)+1/4*(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))+1/2*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))^(1/2)*ln(1+cot(x)+2^(1/2)-(1+cot(x))^(1/2)*(2+2*2 ^(1/2))^(1/2))-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))^(1/2)* 2^(1/2)*ln(1+cot(x)+2^(1/2)+(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2))+1/2*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))^(1/2)*ln(1+cot(x)+2^( 1/2)+(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2))-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))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.81 \[ \int \cot ^2(x) \sqrt {1+\cot (x)} \, dx=\frac {3 \, \sqrt {i - 1} \log \left (i \, \sqrt {i - 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) \sin \left (2 \, x\right ) - 3 \, \sqrt {i - 1} \log \left (-i \, \sqrt {i - 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) \sin \left (2 \, x\right ) - 3 \, \sqrt {-i - 1} \log \left (i \, \sqrt {-i - 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) \sin \left (2 \, x\right ) + 3 \, \sqrt {-i - 1} \log \left (-i \, \sqrt {-i - 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) \sin \left (2 \, x\right ) - 4 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )}}{6 \, \sin \left (2 \, x\right )} \]
1/6*(3*sqrt(I - 1)*log(I*sqrt(I - 1) + sqrt((cos(2*x) + sin(2*x) + 1)/sin( 2*x)))*sin(2*x) - 3*sqrt(I - 1)*log(-I*sqrt(I - 1) + sqrt((cos(2*x) + sin( 2*x) + 1)/sin(2*x)))*sin(2*x) - 3*sqrt(-I - 1)*log(I*sqrt(-I - 1) + sqrt(( cos(2*x) + sin(2*x) + 1)/sin(2*x)))*sin(2*x) + 3*sqrt(-I - 1)*log(-I*sqrt( -I - 1) + sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x)))*sin(2*x) - 4*sqrt((cos (2*x) + sin(2*x) + 1)/sin(2*x))*(cos(2*x) + sin(2*x) + 1))/sin(2*x)
\[ \int \cot ^2(x) \sqrt {1+\cot (x)} \, dx=\int \sqrt {\cot {\left (x \right )} + 1} \cot ^{2}{\left (x \right )}\, dx \]
\[ \int \cot ^2(x) \sqrt {1+\cot (x)} \, dx=\int { \sqrt {\cot \left (x\right ) + 1} \cot \left (x\right )^{2} \,d x } \]
\[ \int \cot ^2(x) \sqrt {1+\cot (x)} \, dx=\int { \sqrt {\cot \left (x\right ) + 1} \cot \left (x\right )^{2} \,d x } \]
Time = 12.98 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.53 \[ \int \cot ^2(x) \sqrt {1+\cot (x)} \, dx=\mathrm {atanh}\left (4\,\sqrt {\mathrm {cot}\left (x\right )+1}\,{\left (\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}+\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}+2\,\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right )-\frac {2\,{\left (\mathrm {cot}\left (x\right )+1\right )}^{3/2}}{3}+\mathrm {atanh}\left (4\,\sqrt {\mathrm {cot}\left (x\right )+1}\,{\left (\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}-\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}-2\,\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right ) \]
atanh(4*(cot(x) + 1)^(1/2)*((- 2^(1/2)/8 - 1/8)^(1/2) + (2^(1/2)/8 - 1/8)^ (1/2))^3)*(2*(- 2^(1/2)/8 - 1/8)^(1/2) + 2*(2^(1/2)/8 - 1/8)^(1/2)) - (2*( cot(x) + 1)^(3/2))/3 + atanh(4*(cot(x) + 1)^(1/2)*((- 2^(1/2)/8 - 1/8)^(1/ 2) - (2^(1/2)/8 - 1/8)^(1/2))^3)*(2*(- 2^(1/2)/8 - 1/8)^(1/2) - 2*(2^(1/2) /8 - 1/8)^(1/2))