Integrand size = 11, antiderivative size = 135 \[ \int \cot (x) \sqrt {1+\cot (x)} \, dx=\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \cot (x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\cot (x)}}\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \cot (x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\cot (x)}}\right )-2 \sqrt {1+\cot (x)} \]
-2*(1+cot(x))^(1/2)+1/2*arctan(1/2*(4+cot(x)*(2-2^(1/2))-3*2^(1/2))/(1+cot (x))^(1/2)/(-7+5*2^(1/2))^(1/2))*(-2+2*2^(1/2))^(1/2)+1/2*arctanh(1/2*(4+3 *2^(1/2)+cot(x)*(2+2^(1/2)))/(1+cot(x))^(1/2)/(7+5*2^(1/2))^(1/2))*(2+2*2^ (1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.45 \[ \int \cot (x) \sqrt {1+\cot (x)} \, dx=\sqrt {1-i} \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )+\sqrt {1+i} \text {arctanh}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )-2 \sqrt {1+\cot (x)} \]
Sqrt[1 - I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]] + Sqrt[1 + I]*ArcTanh[Sq rt[1 + Cot[x]]/Sqrt[1 + I]] - 2*Sqrt[1 + Cot[x]]
Time = 0.53 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {3042, 25, 4011, 3042, 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.
| \(\displaystyle \int \cot (x) \sqrt {\cot (x)+1} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )} \tan \left (x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \sqrt {1-\tan \left (x+\frac {\pi }{2}\right )} \tan \left (x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle -\int \frac {1-\cot (x)}{\sqrt {\cot (x)+1}}dx-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {\tan \left (x+\frac {\pi }{2}\right )+1}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 4019 |
| \(\displaystyle -\frac {\int \frac {\left (2-\sqrt {2}\right ) \cot (x)+\sqrt {2}}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\left (2-\sqrt {2}\right ) \cot (x)+\sqrt {2}}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sqrt {2}-\left (-2-\sqrt {2}\right ) \tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2}-\left (2-\sqrt {2}\right ) \tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx}{2 \sqrt {2}}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 4018 |
| \(\displaystyle \sqrt {2} \left (3-2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4\right )^2}{\cot (x)+1}-4 \left (7-5 \sqrt {2}\right )}d\frac {\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4}{\sqrt {\cot (x)+1}}+\sqrt {2} \left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4\right )^2}{\cot (x)+1}-4 \left (7+5 \sqrt {2}\right )}d\left (-\frac {\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4}{\sqrt {\cot (x)+1}}\right )-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \sqrt {2} \left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4\right )^2}{\cot (x)+1}-4 \left (7+5 \sqrt {2}\right )}d\left (-\frac {\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4}{\sqrt {\cot (x)+1}}\right )+\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\cot (x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )}}-2 \sqrt {\cot (x)+1}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\cot (x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )}}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\cot (x)+1}}\right )}{\sqrt {2 \left (7+5 \sqrt {2}\right )}}-2 \sqrt {\cot (x)+1}\) |
((3 - 2*Sqrt[2])*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Cot[x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x]])])/Sqrt[2*(-7 + 5*Sqrt[2])] + ((3 + 2*Sqrt[2 ])*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Cot[x])/(2*Sqrt[7 + 5*Sqrt[2]]*S qrt[1 + Cot[x]])])/Sqrt[2*(7 + 5*Sqrt[2])] - 2*Sqrt[1 + Cot[x]]
3.1.42.3.1 Defintions of rubi rules used
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[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]
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]
Time = 0.04 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.29
| method | result | size |
| derivativedivides | \(-2 \sqrt {1+\cot \left (x \right )}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}+\frac {\left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\) | \(174\) |
| default | \(-2 \sqrt {1+\cot \left (x \right )}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}+\frac {\left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\) | \(174\) |
-2*(1+cot(x))^(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))+(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))+1/4*(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^(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 = 142, normalized size of antiderivative = 1.05 \[ \int \cot (x) \sqrt {1+\cot (x)} \, dx=\frac {1}{2} \, \sqrt {i + 1} \log \left (\sqrt {i + 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - \frac {1}{2} \, \sqrt {i + 1} \log \left (-\sqrt {i + 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) + \frac {1}{2} \, \sqrt {-i + 1} \log \left (\sqrt {-i + 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - \frac {1}{2} \, \sqrt {-i + 1} \log \left (-\sqrt {-i + 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - 2 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} \]
1/2*sqrt(I + 1)*log(sqrt(I + 1) + sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x)) ) - 1/2*sqrt(I + 1)*log(-sqrt(I + 1) + sqrt((cos(2*x) + sin(2*x) + 1)/sin( 2*x))) + 1/2*sqrt(-I + 1)*log(sqrt(-I + 1) + sqrt((cos(2*x) + sin(2*x) + 1 )/sin(2*x))) - 1/2*sqrt(-I + 1)*log(-sqrt(-I + 1) + sqrt((cos(2*x) + sin(2 *x) + 1)/sin(2*x))) - 2*sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))
\[ \int \cot (x) \sqrt {1+\cot (x)} \, dx=\int \sqrt {\cot {\left (x \right )} + 1} \cot {\left (x \right )}\, dx \]
\[ \int \cot (x) \sqrt {1+\cot (x)} \, dx=\int { \sqrt {\cot \left (x\right ) + 1} \cot \left (x\right ) \,d x } \]
\[ \int \cot (x) \sqrt {1+\cot (x)} \, dx=\int { \sqrt {\cot \left (x\right ) + 1} \cot \left (x\right ) \,d x } \]
Time = 12.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.56 \[ \int \cot (x) \sqrt {1+\cot (x)} \, dx=\mathrm {atanh}\left (\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{4\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{4\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}-\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}\right )\,\left (2\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}+2\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}\right )-\mathrm {atanh}\left (\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{4\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}-\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{4\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}\right )\,\left (2\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}-2\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}\right )-2\,\sqrt {\mathrm {cot}\left (x\right )+1} \]
atanh((cot(x) + 1)^(1/2)/(4*(1/8 - 2^(1/2)/8)^(1/2)) + (cot(x) + 1)^(1/2)/ (4*(2^(1/2)/8 + 1/8)^(1/2)) - (2^(1/2)*(cot(x) + 1)^(1/2))/(8*(1/8 - 2^(1/ 2)/8)^(1/2)) + (2^(1/2)*(cot(x) + 1)^(1/2))/(8*(2^(1/2)/8 + 1/8)^(1/2)))*( 2*(1/8 - 2^(1/2)/8)^(1/2) + 2*(2^(1/2)/8 + 1/8)^(1/2)) - atanh((cot(x) + 1 )^(1/2)/(4*(2^(1/2)/8 + 1/8)^(1/2)) - (cot(x) + 1)^(1/2)/(4*(1/8 - 2^(1/2) /8)^(1/2)) + (2^(1/2)*(cot(x) + 1)^(1/2))/(8*(1/8 - 2^(1/2)/8)^(1/2)) + (2 ^(1/2)*(cot(x) + 1)^(1/2))/(8*(2^(1/2)/8 + 1/8)^(1/2)))*(2*(1/8 - 2^(1/2)/ 8)^(1/2) - 2*(2^(1/2)/8 + 1/8)^(1/2)) - 2*(cot(x) + 1)^(1/2)