3.50 \(\int e^{-3 \coth ^{-1}(a x)} x^3 \, dx\)

Optimal. Leaf size=136 \[ \frac{1}{4} x^4 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{x^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{19 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a^2}-\frac{6 x \sqrt{1-\frac{1}{a^2 x^2}}}{a^3}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^3 \left (a+\frac{1}{x}\right )}+\frac{51 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a^4} \]

[Out]

(-4*Sqrt[1 - 1/(a^2*x^2)])/(a^3*(a + x^(-1))) - (6*Sqrt[1 - 1/(a^2*x^2)]*x)/a^3 + (19*Sqrt[1 - 1/(a^2*x^2)]*x^
2)/(8*a^2) - (Sqrt[1 - 1/(a^2*x^2)]*x^3)/a + (Sqrt[1 - 1/(a^2*x^2)]*x^4)/4 + (51*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]
])/(8*a^4)

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Rubi [A]  time = 1.01928, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6169, 6742, 266, 51, 63, 208, 271, 264, 651} \[ \frac{1}{4} x^4 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{x^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{19 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a^2}-\frac{6 x \sqrt{1-\frac{1}{a^2 x^2}}}{a^3}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^3 \left (a+\frac{1}{x}\right )}+\frac{51 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/E^(3*ArcCoth[a*x]),x]

[Out]

(-4*Sqrt[1 - 1/(a^2*x^2)])/(a^3*(a + x^(-1))) - (6*Sqrt[1 - 1/(a^2*x^2)]*x)/a^3 + (19*Sqrt[1 - 1/(a^2*x^2)]*x^
2)/(8*a^2) - (Sqrt[1 - 1/(a^2*x^2)]*x^3)/a + (Sqrt[1 - 1/(a^2*x^2)]*x^4)/4 + (51*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]
])/(8*a^4)

Rule 6169

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/
a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

\begin{align*} \int e^{-3 \coth ^{-1}(a x)} x^3 \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^5 \left (1+\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{x^5 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3}{a x^4 \sqrt{1-\frac{x^2}{a^2}}}+\frac{4}{a^2 x^3 \sqrt{1-\frac{x^2}{a^2}}}-\frac{4}{a^3 x^2 \sqrt{1-\frac{x^2}{a^2}}}+\frac{4}{a^4 x \sqrt{1-\frac{x^2}{a^2}}}-\frac{4}{a^4 (a+x) \sqrt{1-\frac{x^2}{a^2}}}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^4}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^4}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^3}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}-\operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^3 \left (a+\frac{1}{x}\right )}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}} x}{a^3}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^3}{a}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a^4}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^3}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a^2}\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^3 \left (a+\frac{1}{x}\right )}-\frac{6 \sqrt{1-\frac{1}{a^2 x^2}} x}{a^3}+\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{a^2}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^3}{a}+\frac{1}{4} \sqrt{1-\frac{1}{a^2 x^2}} x^4-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a^4}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{8 a^2}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a^2}\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^3 \left (a+\frac{1}{x}\right )}-\frac{6 \sqrt{1-\frac{1}{a^2 x^2}} x}{a^3}+\frac{19 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{8 a^2}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^3}{a}+\frac{1}{4} \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a^4}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{16 a^4}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a^2}\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^3 \left (a+\frac{1}{x}\right )}-\frac{6 \sqrt{1-\frac{1}{a^2 x^2}} x}{a^3}+\frac{19 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{8 a^2}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^3}{a}+\frac{1}{4} \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{6 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a^4}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a^2}\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^3 \left (a+\frac{1}{x}\right )}-\frac{6 \sqrt{1-\frac{1}{a^2 x^2}} x}{a^3}+\frac{19 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{8 a^2}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^3}{a}+\frac{1}{4} \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{51 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0882918, size = 83, normalized size = 0.61 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^4 x^4-6 a^3 x^3+11 a^2 x^2-29 a x-80\right )}{a x+1}+51 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{8 a^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/E^(3*ArcCoth[a*x]),x]

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-80 - 29*a*x + 11*a^2*x^2 - 6*a^3*x^3 + 2*a^4*x^4))/(1 + a*x) + 51*Log[(1 + Sqrt[
1 - 1/(a^2*x^2)])*x])/(8*a^4)

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Maple [B]  time = 0.138, size = 539, normalized size = 4. \begin{align*}{\frac{1}{8\,{a}^{4} \left ( ax-1 \right ) } \left ( 2\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+4\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+21\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-8\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}+2\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+42\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-21\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-16\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-72\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+72\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}+21\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-42\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}+8\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-144\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+144\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}-21\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a-72\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+72\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*((a*x-1)/(a*x+1))^(3/2),x)

[Out]

1/8*(2*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^3*a^3+4*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^2*a^2+21*(a^2*x^2-1)^(1/2)*(a^2
)^(1/2)*x^3*a^3-8*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2+2*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*x*a+42*(a^2*x^2-
1)^(1/2)*(a^2)^(1/2)*x^2*a^2-21*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3-16*(a^2)^(1/2)*(
(a*x-1)*(a*x+1))^(3/2)*x*a-72*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2+72*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a
*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3+21*(a^2)^(1/2)*(a^2*x^2-1)^(1/2)*x*a-42*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(
1/2))/(a^2)^(1/2))*x*a^2+8*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-144*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a+144
*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x*a^2-21*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))
/(a^2)^(1/2))*a-72*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)+72*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^
2)^(1/2)))/a^4*((a*x-1)/(a*x+1))^(3/2)/(a^2)^(1/2)/((a*x-1)*(a*x+1))^(1/2)/(a*x-1)

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Maxima [A]  time = 1.04308, size = 301, normalized size = 2.21 \begin{align*} -\frac{1}{8} \, a{\left (\frac{2 \,{\left (77 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 149 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 123 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 35 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{4 \,{\left (a x - 1\right )} a^{5}}{a x + 1} - \frac{6 \,{\left (a x - 1\right )}^{2} a^{5}}{{\left (a x + 1\right )}^{2}} + \frac{4 \,{\left (a x - 1\right )}^{3} a^{5}}{{\left (a x + 1\right )}^{3}} - \frac{{\left (a x - 1\right )}^{4} a^{5}}{{\left (a x + 1\right )}^{4}} - a^{5}} - \frac{51 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{5}} + \frac{51 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{5}} + \frac{32 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

-1/8*a*(2*(77*((a*x - 1)/(a*x + 1))^(7/2) - 149*((a*x - 1)/(a*x + 1))^(5/2) + 123*((a*x - 1)/(a*x + 1))^(3/2)
- 35*sqrt((a*x - 1)/(a*x + 1)))/(4*(a*x - 1)*a^5/(a*x + 1) - 6*(a*x - 1)^2*a^5/(a*x + 1)^2 + 4*(a*x - 1)^3*a^5
/(a*x + 1)^3 - (a*x - 1)^4*a^5/(a*x + 1)^4 - a^5) - 51*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^5 + 51*log(sqrt((a
*x - 1)/(a*x + 1)) - 1)/a^5 + 32*sqrt((a*x - 1)/(a*x + 1))/a^5)

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Fricas [A]  time = 1.47619, size = 227, normalized size = 1.67 \begin{align*} \frac{{\left (2 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 11 \, a^{2} x^{2} - 29 \, a x - 80\right )} \sqrt{\frac{a x - 1}{a x + 1}} + 51 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 51 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{8 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")

[Out]

1/8*((2*a^4*x^4 - 6*a^3*x^3 + 11*a^2*x^2 - 29*a*x - 80)*sqrt((a*x - 1)/(a*x + 1)) + 51*log(sqrt((a*x - 1)/(a*x
+ 1)) + 1) - 51*log(sqrt((a*x - 1)/(a*x + 1)) - 1))/a^4

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*((a*x-1)/(a*x+1))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")

[Out]

undef