Integrand size = 11, antiderivative size = 172 \[ \int \cot (x) (1+\cot (x))^{3/2} \, dx=-\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}}{1+\sqrt {2}+\cot (x)}\right )}{\sqrt {1+\sqrt {2}}}-2 \sqrt {1+\cot (x)}-\frac {2}{3} (1+\cot (x))^{3/2} \] Output:
-(1+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^(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))+arctanh((2+2*2^(1/2))^(1/2)*(1+cot(x))^(1/2)/ (1+2^(1/2)+cot(x)))/(1+2^(1/2))^(1/2)-2*(1+cot(x))^(1/2)-2/3*(1+cot(x))^(3 /2)
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.39 \[ \int \cot (x) (1+\cot (x))^{3/2} \, dx=(1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )+(1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )-\frac {2}{3} \sqrt {1+\cot (x)} (4+\cot (x)) \] Input:
Integrate[Cot[x]*(1 + Cot[x])^(3/2),x]
Output:
(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]] + (1 + I)^(3/2)*ArcTan h[Sqrt[1 + Cot[x]]/Sqrt[1 + I]] - (2*Sqrt[1 + Cot[x]]*(4 + Cot[x]))/3
Time = 0.61 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.48, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.364, Rules used = {3042, 25, 4011, 3042, 4011, 27, 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.
| \(\displaystyle \int \cot (x) (\cot (x)+1)^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\left (1-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2} \tan \left (x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \left (1-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2} \tan \left (x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle -\int (1-\cot (x)) \sqrt {\cot (x)+1}dx-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \sqrt {1-\tan \left (x+\frac {\pi }{2}\right )} \left (\tan \left (x+\frac {\pi }{2}\right )+1\right )dx-\frac {2}{3} (\cot (x)+1)^{3/2}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle -\int \frac {2}{\sqrt {\cot (x)+1}}dx-\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \int \frac {1}{\sqrt {\cot (x)+1}}dx-\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -2 \int \frac {1}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx-\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 3966 |
| \(\displaystyle 2 \int \frac {1}{\sqrt {\cot (x)+1} \left (\cot ^2(x)+1\right )}d\cot (x)-\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 484 |
| \(\displaystyle 4 \int \frac {1}{(\cot (x)+1)^2-2 (\cot (x)+1)+2}d\sqrt {\cot (x)+1}-\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 1407 |
| \(\displaystyle 4 \left (\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-\sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\int \frac {\sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )-\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 4 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}-\frac {1}{2} \int -\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )-\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}+\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )-\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 4 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\cot (x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\cot (x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}\right )-\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 4 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1}d\sqrt {\cot (x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}\right )-\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 4 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\cot (x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )-\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}\) |
Input:
Int[Cot[x]*(1 + Cot[x])^(3/2),x]
Output:
-2*Sqrt[1 + Cot[x]] - (2*(1 + Cot[x])^(3/2))/3 + 4*((Sqrt[(1 + Sqrt[2])/(- 1 + Sqrt[2])]*ArcTan[(-Sqrt[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 + 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]]))
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[(-(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[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(451\) vs. \(2(126)=252\).
Time = 0.13 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.63
| method | result | size |
| derivativedivides | \(-\frac {2 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\cot \left (x \right )}-\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (\cot \left (x \right )+1+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{4}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (\cot \left (x \right )+1-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{4}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}}\) | \(452\) |
| default | \(-\frac {2 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\cot \left (x \right )}-\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (\cot \left (x \right )+1+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{4}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (\cot \left (x \right )+1-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{4}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {\sqrt {2}\, \left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}}\) | \(452\) |
Input:
int(cot(x)*(1+cot(x))^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(1+cot(x))^(3/2)-2*(1+cot(x))^(1/2)-1/4*(2+2*2^(1/2))^(1/2)*2^(1/2)*l n(cot(x)+1+(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2)+2^(1/2))+1/2*(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))+1/2*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))-(2+2*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arcta n(((2+2*2^(1/2))^(1/2)+2*(1+cot(x))^(1/2))/(-2+2*2^(1/2))^(1/2))+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))*2^(1/2)+1/4*(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))-1/2*(2+2*2^(1/2))^(1/2)*ln(cot(x)+1-(1+c ot(x))^(1/2)*(2+2*2^(1/2))^(1/2)+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))-(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))+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)
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (126) = 252\).
Time = 0.09 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.82 \[ \int \cot (x) (1+\cot (x))^{3/2} \, dx=\frac {6 \, \sqrt {\sqrt {2} + 1} \arctan \left ({\left (\sqrt {2} + 1\right )}^{\frac {3}{2}} \sqrt {\sqrt {2} - 1} + \sqrt {2} \sqrt {\sqrt {2} + 1} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) \sin \left (2 \, x\right ) + 6 \, \sqrt {\sqrt {2} + 1} \arctan \left (-{\left (\sqrt {2} + 1\right )}^{\frac {3}{2}} \sqrt {\sqrt {2} - 1} + \sqrt {2} \sqrt {\sqrt {2} + 1} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) \sin \left (2 \, x\right ) + 3 \, \sqrt {\sqrt {2} - 1} \log \left (\frac {{\left (\sqrt {2} + 2\right )} \sqrt {\sqrt {2} - 1} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) + {\left (\sqrt {2} + 1\right )} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}\right ) \sin \left (2 \, x\right ) - 3 \, \sqrt {\sqrt {2} - 1} \log \left (-\frac {{\left (\sqrt {2} + 2\right )} \sqrt {\sqrt {2} - 1} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) - {\left (\sqrt {2} + 1\right )} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 1}{\sin \left (2 \, x\right )}\right ) \sin \left (2 \, x\right ) - 4 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} {\left (\cos \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right ) + 1\right )}}{6 \, \sin \left (2 \, x\right )} \] Input:
integrate(cot(x)*(1+cot(x))^(3/2),x, algorithm="fricas")
Output:
1/6*(6*sqrt(sqrt(2) + 1)*arctan((sqrt(2) + 1)^(3/2)*sqrt(sqrt(2) - 1) + sq rt(2)*sqrt(sqrt(2) + 1)*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x)))*sin(2*x) + 6*sqrt(sqrt(2) + 1)*arctan(-(sqrt(2) + 1)^(3/2)*sqrt(sqrt(2) - 1) + sqr t(2)*sqrt(sqrt(2) + 1)*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x)))*sin(2*x) + 3*sqrt(sqrt(2) - 1)*log(((sqrt(2) + 2)*sqrt(sqrt(2) - 1)*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))*sin(2*x) - 3*sqrt(sqrt(2) - 1)*log(-((sqrt(2) + 2)*sqrt(sqrt(2 ) - 1)*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))*sin(2*x) - (sqrt(2) + 1)*s in(2*x) - cos(2*x) - 1)/sin(2*x))*sin(2*x) - 4*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))*(cos(2*x) + 4*sin(2*x) + 1))/sin(2*x)
\[ \int \cot (x) (1+\cot (x))^{3/2} \, dx=\int \left (\cot {\left (x \right )} + 1\right )^{\frac {3}{2}} \cot {\left (x \right )}\, dx \] Input:
integrate(cot(x)*(1+cot(x))**(3/2),x)
Output:
Integral((cot(x) + 1)**(3/2)*cot(x), x)
\[ \int \cot (x) (1+\cot (x))^{3/2} \, dx=\int { {\left (\cot \left (x\right ) + 1\right )}^{\frac {3}{2}} \cot \left (x\right ) \,d x } \] Input:
integrate(cot(x)*(1+cot(x))^(3/2),x, algorithm="maxima")
Output:
integrate((cot(x) + 1)^(3/2)*cot(x), x)
\[ \int \cot (x) (1+\cot (x))^{3/2} \, dx=\int { {\left (\cot \left (x\right ) + 1\right )}^{\frac {3}{2}} \cot \left (x\right ) \,d x } \] Input:
integrate(cot(x)*(1+cot(x))^(3/2),x, algorithm="giac")
Output:
integrate((cot(x) + 1)^(3/2)*cot(x), x)
Time = 9.46 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.48 \[ \int \cot (x) (1+\cot (x))^{3/2} \, dx=-\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}-64}-\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}-64}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}+\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}+64}+\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}+64}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}-\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}\right )-2\,\sqrt {\mathrm {cot}\left (x\right )+1}-\frac {2\,{\left (\mathrm {cot}\left (x\right )+1\right )}^{3/2}}{3} \] Input:
int(cot(x)*(cot(x) + 1)^(3/2),x)
Output:
atan((2^(1/2)*(- 2^(1/2)/4 - 1/4)^(1/2)*(cot(x) + 1)^(1/2)*64i)/(256*(2^(1 /2)/4 - 1/4)^(1/2)*(- 2^(1/2)/4 - 1/4)^(1/2) + 64) + (2^(1/2)*(2^(1/2)/4 - 1/4)^(1/2)*(cot(x) + 1)^(1/2)*64i)/(256*(2^(1/2)/4 - 1/4)^(1/2)*(- 2^(1/2 )/4 - 1/4)^(1/2) + 64))*((- 2^(1/2)/4 - 1/4)^(1/2)*2i - (2^(1/2)/4 - 1/4)^ (1/2)*2i) - atan((2^(1/2)*(- 2^(1/2)/4 - 1/4)^(1/2)*(cot(x) + 1)^(1/2)*64i )/(256*(2^(1/2)/4 - 1/4)^(1/2)*(- 2^(1/2)/4 - 1/4)^(1/2) - 64) - (2^(1/2)* (2^(1/2)/4 - 1/4)^(1/2)*(cot(x) + 1)^(1/2)*64i)/(256*(2^(1/2)/4 - 1/4)^(1/ 2)*(- 2^(1/2)/4 - 1/4)^(1/2) - 64))*((- 2^(1/2)/4 - 1/4)^(1/2)*2i + (2^(1/ 2)/4 - 1/4)^(1/2)*2i) - 2*(cot(x) + 1)^(1/2) - (2*(cot(x) + 1)^(3/2))/3
\[ \int \cot (x) (1+\cot (x))^{3/2} \, dx=\int \sqrt {\cot \left (x \right )+1}\, \cot \left (x \right )d x +\int \sqrt {\cot \left (x \right )+1}\, \cot \left (x \right )^{2}d x \] Input:
int(cot(x)*(1+cot(x))^(3/2),x)
Output:
int(sqrt(cot(x) + 1)*cot(x),x) + int(sqrt(cot(x) + 1)*cot(x)**2,x)