Integrand size = 13, antiderivative size = 139 \[ \int \cot ^2(x) (1+\cot (x))^{3/2} \, dx=-\sqrt {-1+\sqrt {2}} \arctan \left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right )+2 \sqrt {1+\cot (x)}-\frac {2}{5} (1+\cot (x))^{5/2} \] Output:
-(2^(1/2)-1)^(1/2)*arctan((3-2*2^(1/2)+(1-2^(1/2))*cot(x))/(-14+10*2^(1/2) )^(1/2)/(1+cot(x))^(1/2))-(1+2^(1/2))^(1/2)*arctanh((3+2*2^(1/2)+(1+2^(1/2 ))*cot(x))/(14+10*2^(1/2))^(1/2)/(1+cot(x))^(1/2))+2*(1+cot(x))^(1/2)-2/5* (1+cot(x))^(5/2)
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.54 \[ \int \cot ^2(x) (1+\cot (x))^{3/2} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )}{\sqrt {1-i}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )}{\sqrt {1+i}}+2 \sqrt {1+\cot (x)}-\frac {2}{5} (1+\cot (x))^{5/2} \] Input:
Integrate[Cot[x]^2*(1 + Cot[x])^(3/2),x]
Output:
(-2*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]])/Sqrt[1 - I] - (2*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]])/Sqrt[1 + I] + 2*Sqrt[1 + Cot[x]] - (2*(1 + Cot[x] )^(5/2))/5
Time = 0.64 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.21, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.077, Rules used = {3042, 4026, 25, 3042, 3963, 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.
| \(\displaystyle \int \cot ^2(x) (\cot (x)+1)^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (1-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2} \tan \left (x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 4026 |
| \(\displaystyle \int -(\cot (x)+1)^{3/2}dx-\frac {2}{5} (\cot (x)+1)^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int (\cot (x)+1)^{3/2}dx-\frac {2}{5} (\cot (x)+1)^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \left (1-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {2}{5} (\cot (x)+1)^{5/2}\) |
| \(\Big \downarrow \) 3963 |
| \(\displaystyle -\int \frac {2 \cot (x)}{\sqrt {\cot (x)+1}}dx-\frac {2}{5} (\cot (x)+1)^{5/2}+2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \int \frac {\cot (x)}{\sqrt {\cot (x)+1}}dx-\frac {2}{5} (\cot (x)+1)^{5/2}+2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -2 \int -\frac {\tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx-\frac {2}{5} (\cot (x)+1)^{5/2}+2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \int \frac {\tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx-\frac {2}{5} (\cot (x)+1)^{5/2}+2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 4019 |
| \(\displaystyle 2 \left (\frac {\int -\frac {1-\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-\frac {\int -\frac {1-\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}\right )-\frac {2}{5} (\cot (x)+1)^{5/2}+2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\int \frac {1-\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-\frac {\int \frac {1-\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}\right )-\frac {2}{5} (\cot (x)+1)^{5/2}+2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 \left (\frac {\int \frac {1-\left (-1-\sqrt {2}\right ) \tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx}{2 \sqrt {2}}-\frac {\int \frac {1-\left (-1+\sqrt {2}\right ) \tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx}{2 \sqrt {2}}\right )-\frac {2}{5} (\cot (x)+1)^{5/2}+2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 4018 |
| \(\displaystyle 2 \left (\frac {\left (3-2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3\right )^2}{\cot (x)+1}-2 \left (7-5 \sqrt {2}\right )}d\left (-\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {\cot (x)+1}}\right )}{\sqrt {2}}-\frac {\left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3\right )^2}{\cot (x)+1}-2 \left (7+5 \sqrt {2}\right )}d\left (-\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {\cot (x)+1}}\right )}{\sqrt {2}}\right )-\frac {2}{5} (\cot (x)+1)^{5/2}+2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 2 \left (-\frac {\left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3\right )^2}{\cot (x)+1}-2 \left (7+5 \sqrt {2}\right )}d\left (-\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {\cot (x)+1}}\right )}{\sqrt {2}}-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\cot (x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7}}\right )-\frac {2}{5} (\cot (x)+1)^{5/2}+2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle 2 \left (-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\cot (x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7}}-\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\cot (x)+1}}\right )}{2 \sqrt {7+5 \sqrt {2}}}\right )-\frac {2}{5} (\cot (x)+1)^{5/2}+2 \sqrt {\cot (x)+1}\) |
Input:
Int[Cot[x]^2*(1 + Cot[x])^(3/2),x]
Output:
2*(-1/2*((3 - 2*Sqrt[2])*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Cot[x])/(Sq rt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])])/Sqrt[-7 + 5*Sqrt[2]] - ((3 + 2* Sqrt[2])*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Cot[x])/(Sqrt[2*(7 + 5*Sqr t[2])]*Sqrt[1 + Cot[x]])])/(2*Sqrt[7 + 5*Sqrt[2]])) + 2*Sqrt[1 + Cot[x]] - (2*(1 + Cot[x])^(5/2))/5
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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] + Int[(a^2 - b^2 + 2*a*b*Tan[c + d *x])*(a + b*Tan[c + d*x])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[n, 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[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])
Time = 0.15 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.42
| method | result | size |
| derivativedivides | \(-\frac {2 \left (1+\cot \left (x \right )\right )^{\frac {5}{2}}}{5}+2 \sqrt {1+\cot \left (x \right )}-\frac {\sqrt {2}\, \left (-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}\) | \(197\) |
| default | \(-\frac {2 \left (1+\cot \left (x \right )\right )^{\frac {5}{2}}}{5}+2 \sqrt {1+\cot \left (x \right )}-\frac {\sqrt {2}\, \left (-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (\cot \left (x \right )+1+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\cot \left (x \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}\) | \(197\) |
Input:
int(cot(x)^2*(1+cot(x))^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/5*(1+cot(x))^(5/2)+2*(1+cot(x))^(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))+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)))-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))+2*(1-2^(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)))
Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (100) = 200\).
Time = 0.08 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.34 \[ \int \cot ^2(x) (1+\cot (x))^{3/2} \, dx=\frac {10 \, \sqrt {\sqrt {2} - 1} {\left (\cos \left (2 \, x\right ) - 1\right )} \arctan \left ({\left (\sqrt {2} + 1\right )}^{\frac {3}{2}} \sqrt {\sqrt {2} - 1} + {\left (\sqrt {2} + 2\right )} \sqrt {\sqrt {2} - 1} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) + 10 \, \sqrt {\sqrt {2} - 1} {\left (\cos \left (2 \, x\right ) - 1\right )} \arctan \left (-{\left (\sqrt {2} + 1\right )}^{\frac {3}{2}} \sqrt {\sqrt {2} - 1} + {\left (\sqrt {2} + 2\right )} \sqrt {\sqrt {2} - 1} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - 5 \, \sqrt {\sqrt {2} + 1} {\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (\frac {\sqrt {2} \sqrt {\sqrt {2} + 1} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) + {\left (\sqrt {2} + 1\right )} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}\right ) + 5 \, \sqrt {\sqrt {2} + 1} {\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (-\frac {\sqrt {2} \sqrt {\sqrt {2} + 1} \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) - {\left (\sqrt {2} + 1\right )} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 1}{\sin \left (2 \, x\right )}\right ) + 4 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} {\left (5 \, \cos \left (2 \, x\right ) + 2 \, \sin \left (2 \, x\right ) - 3\right )}}{10 \, {\left (\cos \left (2 \, x\right ) - 1\right )}} \] Input:
integrate(cot(x)^2*(1+cot(x))^(3/2),x, algorithm="fricas")
Output:
1/10*(10*sqrt(sqrt(2) - 1)*(cos(2*x) - 1)*arctan((sqrt(2) + 1)^(3/2)*sqrt( sqrt(2) - 1) + (sqrt(2) + 2)*sqrt(sqrt(2) - 1)*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))) + 10*sqrt(sqrt(2) - 1)*(cos(2*x) - 1)*arctan(-(sqrt(2) + 1) ^(3/2)*sqrt(sqrt(2) - 1) + (sqrt(2) + 2)*sqrt(sqrt(2) - 1)*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))) - 5*sqrt(sqrt(2) + 1)*(cos(2*x) - 1)*log((sqrt( 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)) + 5*sqrt(sqrt(2) + 1)*(cos (2*x) - 1)*log(-(sqrt(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)) + 4* sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))*(5*cos(2*x) + 2*sin(2*x) - 3))/(c os(2*x) - 1)
\[ \int \cot ^2(x) (1+\cot (x))^{3/2} \, dx=\int \left (\cot {\left (x \right )} + 1\right )^{\frac {3}{2}} \cot ^{2}{\left (x \right )}\, dx \] Input:
integrate(cot(x)**2*(1+cot(x))**(3/2),x)
Output:
Integral((cot(x) + 1)**(3/2)*cot(x)**2, x)
Timed out. \[ \int \cot ^2(x) (1+\cot (x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cot(x)^2*(1+cot(x))^(3/2),x, algorithm="maxima")
Output:
Timed out
\[ \int \cot ^2(x) (1+\cot (x))^{3/2} \, dx=\int { {\left (\cot \left (x\right ) + 1\right )}^{\frac {3}{2}} \cot \left (x\right )^{2} \,d x } \] Input:
integrate(cot(x)^2*(1+cot(x))^(3/2),x, algorithm="giac")
Output:
integrate((cot(x) + 1)^(3/2)*cot(x)^2, x)
Time = 10.08 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.83 \[ \int \cot ^2(x) (1+\cot (x))^{3/2} \, dx=\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}-64}-\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}-64}\right )\,\left (\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,2{}\mathrm {i}+\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}\,2{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}+64}+\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}+64}\right )\,\left (\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,2{}\mathrm {i}-\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}\,2{}\mathrm {i}\right )+2\,\sqrt {\mathrm {cot}\left (x\right )+1}-\frac {2\,{\left (\mathrm {cot}\left (x\right )+1\right )}^{5/2}}{5} \] Input:
int(cot(x)^2*(cot(x) + 1)^(3/2),x)
Output:
atan((2^(1/2)*(1/4 - 2^(1/2)/4)^(1/2)*(cot(x) + 1)^(1/2)*64i)/(256*(1/4 - 2^(1/2)/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*(1/4 - 2^(1/2)/4)^(1/2)*(2^(1/2)/4 + 1/4)^(1/2) - 64))*((1/4 - 2^(1/2)/4)^(1/2)*2i + (2^(1/2)/4 + 1/4)^(1/2)*2i ) - atan((2^(1/2)*(1/4 - 2^(1/2)/4)^(1/2)*(cot(x) + 1)^(1/2)*64i)/(256*(1/ 4 - 2^(1/2)/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*(1/4 - 2^(1/2)/4)^(1/2)*(2^(1/2)/ 4 + 1/4)^(1/2) + 64))*((1/4 - 2^(1/2)/4)^(1/2)*2i - (2^(1/2)/4 + 1/4)^(1/2 )*2i) + 2*(cot(x) + 1)^(1/2) - (2*(cot(x) + 1)^(5/2))/5
\[ \int \cot ^2(x) (1+\cot (x))^{3/2} \, dx=\int \sqrt {\cot \left (x \right )+1}\, \cot \left (x \right )^{3}d x +\int \sqrt {\cot \left (x \right )+1}\, \cot \left (x \right )^{2}d x \] Input:
int(cot(x)^2*(1+cot(x))^(3/2),x)
Output:
int(sqrt(cot(x) + 1)*cot(x)**3,x) + int(sqrt(cot(x) + 1)*cot(x)**2,x)