Integrand size = 11, antiderivative size = 178 \[ \int \frac {\cot (x)}{(1+\cot (x))^{3/2}} \, dx=\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}}{1+\sqrt {2}+\cot (x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {1}{\sqrt {1+\cot (x)}} \] Output:
1/4*(2+2*2^(1/2))^(1/2)*arctan(((2+2*2^(1/2))^(1/2)-2*(1+cot(x))^(1/2))/(- 2+2*2^(1/2))^(1/2))-1/4*(2+2*2^(1/2))^(1/2)*arctan(((2+2*2^(1/2))^(1/2)+2* (1+cot(x))^(1/2))/(-2+2*2^(1/2))^(1/2))+1/2*arctanh((2+2*2^(1/2))^(1/2)*(1 +cot(x))^(1/2)/(1+2^(1/2)+cot(x)))/(2+2*2^(1/2))^(1/2)-1/(1+cot(x))^(1/2)
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.40 \[ \int \frac {\cot (x)}{(1+\cot (x))^{3/2}} \, dx=\frac {1}{2} i \sqrt {1-i} \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )-\frac {1}{2} i \sqrt {1+i} \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )-\frac {1}{\sqrt {1+\cot (x)}} \] Input:
Integrate[Cot[x]/(1 + Cot[x])^(3/2),x]
Output:
(I/2)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]] - (I/2)*Sqrt[1 + I ]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]] - 1/Sqrt[1 + Cot[x]]
Time = 0.51 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.29, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.273, Rules used = {3042, 25, 4012, 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.
| \(\displaystyle \int \frac {\cot (x)}{(\cot (x)+1)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\tan \left (x+\frac {\pi }{2}\right )}{\left (1-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\tan \left (x+\frac {\pi }{2}\right )}{\left (1-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\frac {1}{2} \int -\sqrt {\cot (x)+1}dx-\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \int \sqrt {\cot (x)+1}dx-\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}dx-\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 3966 |
| \(\displaystyle -\frac {1}{2} \int \frac {\sqrt {\cot (x)+1}}{\cot ^2(x)+1}d\cot (x)-\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 483 |
| \(\displaystyle -\int \frac {\cot (x)+1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}-\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 1447 |
| \(\displaystyle \frac {1}{2} \int \frac {-\cot (x)+\sqrt {2}-1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}-\frac {1}{2} \int \frac {\cot (x)+\sqrt {2}+1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}-\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 1475 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}-\frac {1}{2} \int \frac {1}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}\right )+\frac {1}{2} \int \frac {-\cot (x)+\sqrt {2}-1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}-\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {1}{-\cot (x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}\right )+\int \frac {1}{-\cot (x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}\right )\right )+\frac {1}{2} \int \frac {-\cot (x)+\sqrt {2}-1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}-\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \int \frac {-\cot (x)+\sqrt {2}-1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}+\frac {1}{2} \left (-\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}-\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 1478 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\int -\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{2} \left (-\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}-\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\int \frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{2} \left (-\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}-\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}-\frac {\arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{\sqrt {\cot (x)+1}}+\frac {1}{2} \left (\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )\) |
Input:
Int[Cot[x]/(1 + Cot[x])^(3/2),x]
Output:
(-(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 - 1/Sqrt[1 + C ot[x]] + (-1/2*Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + C ot[x]]]/Sqrt[2*(1 + Sqrt[2])] + Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqr t[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[2*(1 + Sqrt[2])]))/2
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_)]), 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. \(355\) vs. \(2(126)=252\).
Time = 0.14 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.00
| method | result | size |
| derivativedivides | \(\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (\cot \left (x \right )+1+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{8}-\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{8}+\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (\cot \left (x \right )+1-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{8}-\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{8}+\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {1}{\sqrt {1+\cot \left (x \right )}}\) | \(356\) |
| default | \(\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (\cot \left (x \right )+1+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{8}-\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{8}+\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (\cot \left (x \right )+1-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{8}-\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{8}+\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {1}{\sqrt {1+\cot \left (x \right )}}\) | \(356\) |
Input:
int(cot(x)/(1+cot(x))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/8*(2+2*2^(1/2))^(1/2)*2^(1/2)*ln(cot(x)+1+(1+cot(x))^(1/2)*(2+2*2^(1/2)) ^(1/2)+2^(1/2))-1/4*2^(1/2)*(2+2*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan(((2+ 2*2^(1/2))^(1/2)+2*(1+cot(x))^(1/2))/(-2+2*2^(1/2))^(1/2))-1/8*(2+2*2^(1/2 ))^(1/2)*ln(cot(x)+1+(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2)+2^(1/2))+1/4*(2+ 2*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan(((2+2*2^(1/2))^(1/2)+2*(1+cot(x))^( 1/2))/(-2+2*2^(1/2))^(1/2))-1/8*(2+2*2^(1/2))^(1/2)*2^(1/2)*ln(cot(x)+1-(1 +cot(x))^(1/2)*(2+2*2^(1/2))^(1/2)+2^(1/2))-1/4*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))+1/8*(2+2*2^(1/2))^(1/2)*ln(cot(x)+1-(1+cot(x))^(1/2)*(2+2*2^( 1/2))^(1/2)+2^(1/2))+1/4*(2+2*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))-1/(1+cot(x))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (126) = 252\).
Time = 0.09 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.06 \[ \int \frac {\cot (x)}{(1+\cot (x))^{3/2}} \, dx=-\frac {2 \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )} \arctan \left (2 \, {\left (\sqrt {2} + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} + 2 \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) + 2 \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )} \arctan \left (-2 \, {\left (\sqrt {2} + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} + 2 \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )} \log \left (\frac {2 \, {\left (\sqrt {2} + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) + {\left (\sqrt {2} + 1\right )} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}\right ) + \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )} \log \left (-\frac {2 \, {\left (\sqrt {2} + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) - {\left (\sqrt {2} + 1\right )} \sin \left (2 \, x\right ) - \cos \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 )}} \sin \left (2 \, x\right )}{4 \, {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )}} \] Input:
integrate(cot(x)/(1+cot(x))^(3/2),x, algorithm="fricas")
Output:
-1/4*(2*sqrt(1/2*sqrt(2) + 1/2)*(cos(2*x) + sin(2*x) + 1)*arctan(2*(sqrt(2 ) + 1)*sqrt(1/2*sqrt(2) + 1/2)*sqrt(1/2*sqrt(2) - 1/2) + 2*sqrt(1/2*sqrt(2 ) + 1/2)*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))) + 2*sqrt(1/2*sqrt(2) + 1/2)*(cos(2*x) + sin(2*x) + 1)*arctan(-2*(sqrt(2) + 1)*sqrt(1/2*sqrt(2) + 1/2)*sqrt(1/2*sqrt(2) - 1/2) + 2*sqrt(1/2*sqrt(2) + 1/2)*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))) - sqrt(1/2*sqrt(2) - 1/2)*(cos(2*x) + sin(2*x) + 1)*log((2*(sqrt(2) + 1)*sqrt(1/2*sqrt(2) - 1/2)*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))*sin(2*x) + (sqrt(2) + 1)*sin(2*x) + cos(2*x) + 1)/sin(2*x)) + sqrt(1/2*sqrt(2) - 1/2)*(cos(2*x) + sin(2*x) + 1)*log(-(2*(sqrt(2) + 1) *sqrt(1/2*sqrt(2) - 1/2)*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))*sin(2*x) - (sqrt(2) + 1)*sin(2*x) - cos(2*x) - 1)/sin(2*x)) + 4*sqrt((cos(2*x) + s in(2*x) + 1)/sin(2*x))*sin(2*x))/(cos(2*x) + sin(2*x) + 1)
\[ \int \frac {\cot (x)}{(1+\cot (x))^{3/2}} \, dx=\int \frac {\cot {\left (x \right )}}{\left (\cot {\left (x \right )} + 1\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(x)/(1+cot(x))**(3/2),x)
Output:
Integral(cot(x)/(cot(x) + 1)**(3/2), x)
\[ \int \frac {\cot (x)}{(1+\cot (x))^{3/2}} \, dx=\int { \frac {\cot \left (x\right )}{{\left (\cot \left (x\right ) + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(cot(x)/(1+cot(x))^(3/2),x, algorithm="maxima")
Output:
integrate(cot(x)/(cot(x) + 1)^(3/2), x)
\[ \int \frac {\cot (x)}{(1+\cot (x))^{3/2}} \, dx=\int { \frac {\cot \left (x\right )}{{\left (\cot \left (x\right ) + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(cot(x)/(1+cot(x))^(3/2),x, algorithm="giac")
Output:
integrate(cot(x)/(cot(x) + 1)^(3/2), x)
Time = 9.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.68 \[ \int \frac {\cot (x)}{(1+\cot (x))^{3/2}} \, dx=-\mathrm {atanh}\left (32\,\sqrt {\mathrm {cot}\left (x\right )+1}\,{\left (\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}+\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}+2\,\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )-\frac {1}{\sqrt {\mathrm {cot}\left (x\right )+1}}-\mathrm {atanh}\left (32\,\sqrt {\mathrm {cot}\left (x\right )+1}\,{\left (\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}-\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}-2\,\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right ) \] Input:
int(cot(x)/(cot(x) + 1)^(3/2),x)
Output:
- atanh(32*(cot(x) + 1)^(1/2)*((- 2^(1/2)/32 - 1/32)^(1/2) + (2^(1/2)/32 - 1/32)^(1/2))^3)*(2*(- 2^(1/2)/32 - 1/32)^(1/2) + 2*(2^(1/2)/32 - 1/32)^(1 /2)) - 1/(cot(x) + 1)^(1/2) - atanh(32*(cot(x) + 1)^(1/2)*((- 2^(1/2)/32 - 1/32)^(1/2) - (2^(1/2)/32 - 1/32)^(1/2))^3)*(2*(- 2^(1/2)/32 - 1/32)^(1/2 ) - 2*(2^(1/2)/32 - 1/32)^(1/2))
\[ \int \frac {\cot (x)}{(1+\cot (x))^{3/2}} \, dx=\int \frac {\sqrt {\cot \left (x \right )+1}\, \cot \left (x \right )}{\cot \left (x \right )^{2}+2 \cot \left (x \right )+1}d x \] Input:
int(cot(x)/(1+cot(x))^(3/2),x)
Output:
int((sqrt(cot(x) + 1)*cot(x))/(cot(x)**2 + 2*cot(x) + 1),x)