Integrand size = 13, antiderivative size = 139 \[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx=\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \cot (x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\cot (x)}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \cot (x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\cot (x)}}\right )+\frac {1}{\sqrt {1+\cot (x)}} \]
1/(1+cot(x))^(1/2)+1/4*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)+1/4*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 higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.45 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.45 \[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx=\frac {4-(1+i) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\left (\frac {1}{2}-\frac {i}{2}\right ) (1+\cot (x))\right )-(1-i) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\left (\frac {1}{2}+\frac {i}{2}\right ) (1+\cot (x))\right )}{2 \sqrt {1+\cot (x)}} \]
(4 - (1 + I)*Hypergeometric2F1[-1/2, 1, 1/2, (1/2 - I/2)*(1 + Cot[x])] - ( 1 - I)*Hypergeometric2F1[-1/2, 1, 1/2, (1/2 + I/2)*(1 + Cot[x])])/(2*Sqrt[ 1 + Cot[x]])
Time = 0.51 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 4025, 25, 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.
| \(\displaystyle \int \frac {\cot ^2(x)}{(\cot (x)+1)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan \left (x+\frac {\pi }{2}\right )^2}{\left (1-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle \frac {1}{2} \int -\frac {1-\cot (x)}{\sqrt {\cot (x)+1}}dx+\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{\sqrt {\cot (x)+1}}-\frac {1}{2} \int \frac {1-\cot (x)}{\sqrt {\cot (x)+1}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{\sqrt {\cot (x)+1}}-\frac {1}{2} \int \frac {\tan \left (x+\frac {\pi }{2}\right )+1}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4019 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-\frac {\int \frac {\left (2-\sqrt {2}\right ) \cot (x)+\sqrt {2}}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}\right )+\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\left (2-\sqrt {2}\right ) \cot (x)+\sqrt {2}}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \cot (x)}{\sqrt {\cot (x)+1}}dx}{2 \sqrt {2}}\right )+\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\left (-2-\sqrt {2}\right ) \tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2}-\left (2-\sqrt {2}\right ) \tan \left (x+\frac {\pi }{2}\right )}{\sqrt {1-\tan \left (x+\frac {\pi }{2}\right )}}dx}{2 \sqrt {2}}\right )+\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 4018 |
| \(\displaystyle \frac {1}{2} \left (\sqrt {2} \left (3-2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4\right )^2}{\cot (x)+1}-4 \left (7-5 \sqrt {2}\right )}d\frac {\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4}{\sqrt {\cot (x)+1}}+\sqrt {2} \left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4\right )^2}{\cot (x)+1}-4 \left (7+5 \sqrt {2}\right )}d\left (-\frac {\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4}{\sqrt {\cot (x)+1}}\right )\right )+\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2} \left (\sqrt {2} \left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4\right )^2}{\cot (x)+1}-4 \left (7+5 \sqrt {2}\right )}d\left (-\frac {\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4}{\sqrt {\cot (x)+1}}\right )+\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\cot (x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )}}\right )+\frac {1}{\sqrt {\cot (x)+1}}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\cot (x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )}}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\cot (x)+1}}\right )}{\sqrt {2 \left (7+5 \sqrt {2}\right )}}\right )+\frac {1}{\sqrt {\cot (x)+1}}\) |
(((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]]* Sqrt[1 + Cot[x]])])/Sqrt[2*(7 + 5*Sqrt[2])])/2 + 1/Sqrt[1 + Cot[x]]
3.1.47.3.1 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[((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[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*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^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 0.04 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.24
| method | result | size |
| derivativedivides | \(\frac {1}{\sqrt {1+\cot \left (x \right )}}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}+\frac {\left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}\) | \(173\) |
| default | \(\frac {1}{\sqrt {1+\cot \left (x \right )}}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}+\frac {\left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}\) | \(173\) |
1/(1+cot(x))^(1/2)-1/8*(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^(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))+1/8*(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^(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 = 0.26 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.42 \[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx=\frac {\sqrt {i + 1} {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )} \log \left (\sqrt {i + 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - \sqrt {i + 1} {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )} \log \left (-\sqrt {i + 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) + \sqrt {-i + 1} {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )} \log \left (\sqrt {-i + 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) - \sqrt {-i + 1} {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )} \log \left (-\sqrt {-i + 1} + \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}}\right ) + 4 \, \sqrt {\frac {\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}} \sin \left (2 \, x\right )}{4 \, {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )}} \]
1/4*(sqrt(I + 1)*(cos(2*x) + sin(2*x) + 1)*log(sqrt(I + 1) + sqrt((cos(2*x ) + sin(2*x) + 1)/sin(2*x))) - sqrt(I + 1)*(cos(2*x) + sin(2*x) + 1)*log(- sqrt(I + 1) + sqrt((cos(2*x) + sin(2*x) + 1)/sin(2*x))) + sqrt(-I + 1)*(co s(2*x) + sin(2*x) + 1)*log(sqrt(-I + 1) + sqrt((cos(2*x) + sin(2*x) + 1)/s in(2*x))) - sqrt(-I + 1)*(cos(2*x) + sin(2*x) + 1)*log(-sqrt(-I + 1) + sqr t((cos(2*x) + sin(2*x) + 1)/sin(2*x))) + 4*sqrt((cos(2*x) + sin(2*x) + 1)/ sin(2*x))*sin(2*x))/(cos(2*x) + sin(2*x) + 1)
\[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx=\int \frac {\cot ^{2}{\left (x \right )}}{\left (\cot {\left (x \right )} + 1\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx=\int { \frac {\cot \left (x\right )^{2}}{{\left (\cot \left (x\right ) + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx=\int { \frac {\cot \left (x\right )^{2}}{{\left (\cot \left (x\right ) + 1\right )}^{\frac {3}{2}}} \,d x } \]
Time = 13.08 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.50 \[ \int \frac {\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx=\frac {1}{\sqrt {\mathrm {cot}\left (x\right )+1}}-\mathrm {atanh}\left (\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}-\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{16\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{16\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}\right )\,\left (2\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}-2\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}\right )+\mathrm {atanh}\left (\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}-\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{16\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{16\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}\right )\,\left (2\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}+2\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}\right ) \]
1/(cot(x) + 1)^(1/2) - atanh((cot(x) + 1)^(1/2)/(8*(2^(1/2)/32 + 1/32)^(1/ 2)) - (cot(x) + 1)^(1/2)/(8*(1/32 - 2^(1/2)/32)^(1/2)) + (2^(1/2)*(cot(x) + 1)^(1/2))/(16*(1/32 - 2^(1/2)/32)^(1/2)) + (2^(1/2)*(cot(x) + 1)^(1/2))/ (16*(2^(1/2)/32 + 1/32)^(1/2)))*(2*(1/32 - 2^(1/2)/32)^(1/2) - 2*(2^(1/2)/ 32 + 1/32)^(1/2)) + atanh((cot(x) + 1)^(1/2)/(8*(1/32 - 2^(1/2)/32)^(1/2)) + (cot(x) + 1)^(1/2)/(8*(2^(1/2)/32 + 1/32)^(1/2)) - (2^(1/2)*(cot(x) + 1 )^(1/2))/(16*(1/32 - 2^(1/2)/32)^(1/2)) + (2^(1/2)*(cot(x) + 1)^(1/2))/(16 *(2^(1/2)/32 + 1/32)^(1/2)))*(2*(1/32 - 2^(1/2)/32)^(1/2) + 2*(2^(1/2)/32 + 1/32)^(1/2))