3.1.50 \(\int \frac {\cot (x)}{(1+\cot (x))^{5/2}} \, dx\) [50]

Optimal. Leaf size=216 \[ \frac {1}{4} \sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )-\frac {1}{4} \sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )-\frac {1}{3 (1+\cot (x))^{3/2}}+\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{8 \sqrt {1+\sqrt {2}}}-\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{8 \sqrt {1+\sqrt {2}}} \]

[Out]

-1/3/(1+cot(x))^(3/2)+1/8*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)-1/8*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)+1/4*arctan((-2*(1+cot(x))^(1/2)+(2+2*2
^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)-1/4*arctan((2*(1+cot(x))^(1/2)+(2+2*2^(1/2))^(1/2))/(-2
+2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3610, 21, 3566, 722, 1108, 648, 632, 210, 642} \begin {gather*} \frac {1}{4} \sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{4} \sqrt {1+\sqrt {2}} \text {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}{3 (\cot (x)+1)^{3/2}}+\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{8 \sqrt {1+\sqrt {2}}}-\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{8 \sqrt {1+\sqrt {2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[x]/(1 + Cot[x])^(5/2),x]

[Out]

(Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/4 - (Sqrt[1 +
Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/4 - 1/(3*(1 + Cot[x])^(3
/2)) + Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(8*Sqrt[1 + Sqrt[2]]) - Log[1 + Sqrt
[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(8*Sqrt[1 + Sqrt[2]])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 210

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])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 722

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^2 + a*e^2 - 2*c
*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1108

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]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[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]

Rule 3566

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[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]

Rule 3610

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] + Dist[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]

Rubi steps

\begin {align*} \int \frac {\cot (x)}{(1+\cot (x))^{5/2}} \, dx &=-\frac {1}{3 (1+\cot (x))^{3/2}}-\frac {1}{2} \int \frac {-1-\cot (x)}{(1+\cot (x))^{3/2}} \, dx\\ &=-\frac {1}{3 (1+\cot (x))^{3/2}}+\frac {1}{2} \int \frac {1}{\sqrt {1+\cot (x)}} \, dx\\ &=-\frac {1}{3 (1+\cot (x))^{3/2}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+x} \left (1+x^2\right )} \, dx,x,\cot (x)\right )\\ &=-\frac {1}{3 (1+\cot (x))^{3/2}}-\text {Subst}\left (\int \frac {1}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\cot (x)}\right )\\ &=-\frac {1}{3 (1+\cot (x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}\\ &=-\frac {1}{3 (1+\cot (x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{4 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {-\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{8 \sqrt {1+\sqrt {2}}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{8 \sqrt {1+\sqrt {2}}}\\ &=-\frac {1}{3 (1+\cot (x))^{3/2}}+\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{8 \sqrt {1+\sqrt {2}}}-\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {\text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}\right )}{2 \sqrt {2}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{4 \sqrt {-1+\sqrt {2}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{4 \sqrt {-1+\sqrt {2}}}-\frac {1}{3 (1+\cot (x))^{3/2}}+\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{8 \sqrt {1+\sqrt {2}}}-\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{8 \sqrt {1+\sqrt {2}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.26, size = 69, normalized size = 0.32 \begin {gather*} -\frac {1}{4} (1-i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )-\frac {1}{4} (1+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )-\frac {1}{3 (1+\cot (x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/(1 + Cot[x])^(5/2),x]

[Out]

-1/4*((1 - I)^(3/2)*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]]) - ((1 + I)^(3/2)*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 +
I]])/4 - 1/(3*(1 + Cot[x])^(3/2))

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Maple [A]
time = 0.25, size = 283, normalized size = 1.31

method	result	size
derivativedivides	\(-\frac {1}{3 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{16}-\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{16}+\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}\)	\(283\)
default	\(-\frac {1}{3 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{16}-\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{16}+\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}\)	\(283\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(1+cot(x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3/(1+cot(x))^(3/2)-1/16*(-(2+2*2^(1/2))^(1/2)*2^(1/2)+2*(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))-1/4*(2*2^(1/2)-1/2*(-(2+2*2^(1/2))^(1/2)*2^(1/2)+2*(2+2*2^(1/2))^(1/2))*(2+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))+1/16*(-(
2+2*2^(1/2))^(1/2)*2^(1/2)+2*(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/
4*(-2*2^(1/2)+1/2*(-(2+2*2^(1/2))^(1/2)*2^(1/2)+2*(2+2*2^(1/2))^(1/2))*(2+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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(1+cot(x))^(5/2),x, algorithm="maxima")

[Out]

integrate(cot(x)/(cot(x) + 1)^(5/2), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(1+cot(x))^(5/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot {\left (x \right )}}{\left (\cot {\left (x \right )} + 1\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(1+cot(x))**(5/2),x)

[Out]

Integral(cot(x)/(cot(x) + 1)**(5/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(1+cot(x))^(5/2),x, algorithm="giac")

[Out]

integrate(cot(x)/(cot(x) + 1)^(5/2), x)

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Mupad [B]
time = 0.71, size = 238, normalized size = 1.10 \begin {gather*} \mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}+1}+\frac {4\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}+1}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}-2\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\right )-\mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}-1}-\frac {4\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}-1}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}+2\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\right )-\frac {1}{3\,{\left (\mathrm {cot}\left (x\right )+1\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(cot(x) + 1)^(5/2),x)

[Out]

atanh((4*2^(1/2)*(- 2^(1/2)/64 - 1/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(2^(1/2)/64 - 1/64)^(1/2)*(- 2^(1/2)/64 -
1/64)^(1/2) + 1) + (4*2^(1/2)*(2^(1/2)/64 - 1/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(2^(1/2)/64 - 1/64)^(1/2)*(-
2^(1/2)/64 - 1/64)^(1/2) + 1))*(2*(- 2^(1/2)/64 - 1/64)^(1/2) - 2*(2^(1/2)/64 - 1/64)^(1/2)) - atanh((4*2^(1/2
)*(- 2^(1/2)/64 - 1/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(2^(1/2)/64 - 1/64)^(1/2)*(- 2^(1/2)/64 - 1/64)^(1/2) -
1) - (4*2^(1/2)*(2^(1/2)/64 - 1/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(2^(1/2)/64 - 1/64)^(1/2)*(- 2^(1/2)/64 - 1/
64)^(1/2) - 1))*(2*(- 2^(1/2)/64 - 1/64)^(1/2) + 2*(2^(1/2)/64 - 1/64)^(1/2)) - 1/(3*(cot(x) + 1)^(3/2))

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