Integrand size = 18, antiderivative size = 129 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^4(x)}} \, dx=-\frac {x^4 \text {sech}^2(x)}{4 \sqrt {a \text {sech}^4(x)}}+\frac {x^3 \log \left (1-e^{2 x}\right ) \text {sech}^2(x)}{\sqrt {a \text {sech}^4(x)}}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right ) \text {sech}^2(x)}{2 \sqrt {a \text {sech}^4(x)}}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 x}\right ) \text {sech}^2(x)}{2 \sqrt {a \text {sech}^4(x)}}+\frac {3 \operatorname {PolyLog}\left (4,e^{2 x}\right ) \text {sech}^2(x)}{4 \sqrt {a \text {sech}^4(x)}} \]
-1/4*x^4*sech(x)^2/(a*sech(x)^4)^(1/2)+x^3*ln(1-exp(2*x))*sech(x)^2/(a*sec h(x)^4)^(1/2)+3/2*x^2*polylog(2,exp(2*x))*sech(x)^2/(a*sech(x)^4)^(1/2)-3/ 2*x*polylog(3,exp(2*x))*sech(x)^2/(a*sech(x)^4)^(1/2)+3/4*polylog(4,exp(2* x))*sech(x)^2/(a*sech(x)^4)^(1/2)
Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.53 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^4(x)}} \, dx=-\frac {\left (x^4-4 x^3 \log \left (1-e^{2 x}\right )-6 x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )+6 x \operatorname {PolyLog}\left (3,e^{2 x}\right )-3 \operatorname {PolyLog}\left (4,e^{2 x}\right )\right ) \text {sech}^2(x)}{4 \sqrt {a \text {sech}^4(x)}} \]
-1/4*((x^4 - 4*x^3*Log[1 - E^(2*x)] - 6*x^2*PolyLog[2, E^(2*x)] + 6*x*Poly Log[3, E^(2*x)] - 3*PolyLog[4, E^(2*x)])*Sech[x]^2)/Sqrt[a*Sech[x]^4]
Result contains complex when optimal does not.
Time = 0.89 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.71, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {7271, 3042, 26, 4199, 25, 2620, 3011, 7163, 2720, 7143}
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.
| \(\displaystyle \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^4(x)}} \, dx\) |
| \(\Big \downarrow \) 7271 |
| \(\displaystyle \frac {\text {sech}^2(x) \int x^3 \coth (x)dx}{\sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^2(x) \int -i x^3 \tan \left (i x+\frac {\pi }{2}\right )dx}{\sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \text {sech}^2(x) \int x^3 \tan \left (i x+\frac {\pi }{2}\right )dx}{\sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle -\frac {i \text {sech}^2(x) \left (2 i \int -\frac {e^{2 x} x^3}{1-e^{2 x}}dx-\frac {i x^4}{4}\right )}{\sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \text {sech}^2(x) \left (-2 i \int \frac {e^{2 x} x^3}{1-e^{2 x}}dx-\frac {i x^4}{4}\right )}{\sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {i \text {sech}^2(x) \left (-2 i \left (\frac {3}{2} \int x^2 \log \left (1-e^{2 x}\right )dx-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )-\frac {i x^4}{4}\right )}{\sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {i \text {sech}^2(x) \left (-2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )-\frac {i x^4}{4}\right )}{\sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle -\frac {i \text {sech}^2(x) \left (-2 i \left (\frac {3}{2} \left (-\frac {1}{2} \int \operatorname {PolyLog}\left (3,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )-\frac {i x^4}{4}\right )}{\sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {i \text {sech}^2(x) \left (-2 i \left (\frac {3}{2} \left (-\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (3,e^{2 x}\right )de^{2 x}-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )-\frac {i x^4}{4}\right )}{\sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {i \text {sech}^2(x) \left (-2 i \left (\frac {3}{2} \left (-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )+\frac {1}{2} x \operatorname {PolyLog}\left (3,e^{2 x}\right )-\frac {\operatorname {PolyLog}\left (4,e^{2 x}\right )}{4}\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )-\frac {i x^4}{4}\right )}{\sqrt {a \text {sech}^4(x)}}\) |
((-I)*((-1/4*I)*x^4 - (2*I)*(-1/2*(x^3*Log[1 - E^(2*x)]) + (3*(-1/2*(x^2*P olyLog[2, E^(2*x)]) + (x*PolyLog[3, E^(2*x)])/2 - PolyLog[4, E^(2*x)]/4))/ 2))*Sech[x]^2)/Sqrt[a*Sech[x]^4]
3.9.47.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ .)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp [2*I Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(328\) vs. \(2(107)=214\).
Time = 0.11 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.55
| method | result | size |
| risch | \(-\frac {{\mathrm e}^{2 x} x^{4}}{4 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {{\mathrm e}^{2 x} x^{3} \ln \left (1+{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {3 \,{\mathrm e}^{2 x} x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {6 \,{\mathrm e}^{2 x} x \operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {6 \,{\mathrm e}^{2 x} \operatorname {polylog}\left (4, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {{\mathrm e}^{2 x} x^{3} \ln \left (1-{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {3 \,{\mathrm e}^{2 x} x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {6 \,{\mathrm e}^{2 x} x \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {6 \,{\mathrm e}^{2 x} \operatorname {polylog}\left (4, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}\) | \(329\) |
-1/4/(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)/(1+exp(2*x))^2*exp(2*x)*x^4+1/(a*ex p(4*x)/(1+exp(2*x))^4)^(1/2)/(1+exp(2*x))^2*exp(2*x)*x^3*ln(1+exp(x))+3/(a *exp(4*x)/(1+exp(2*x))^4)^(1/2)/(1+exp(2*x))^2*exp(2*x)*x^2*polylog(2,-exp (x))-6/(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)/(1+exp(2*x))^2*exp(2*x)*x*polylog (3,-exp(x))+6/(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)/(1+exp(2*x))^2*exp(2*x)*po lylog(4,-exp(x))+1/(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)/(1+exp(2*x))^2*exp(2* x)*x^3*ln(1-exp(x))+3/(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)/(1+exp(2*x))^2*exp (2*x)*x^2*polylog(2,exp(x))-6/(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)/(1+exp(2*x ))^2*exp(2*x)*x*polylog(3,exp(x))+6/(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)/(1+e xp(2*x))^2*exp(2*x)*polylog(4,exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (106) = 212\).
Time = 0.28 (sec) , antiderivative size = 427, normalized size of antiderivative = 3.31 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^4(x)}} \, dx=\frac {{\left (24 \, \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (2 \, x\right )} {\rm polylog}\left (4, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 24 \, \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (2 \, x\right )} {\rm polylog}\left (4, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) - 24 \, {\left (x e^{\left (4 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} + x\right )} \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} e^{\left (2 \, x\right )} {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 24 \, {\left (x e^{\left (4 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} + x\right )} \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} e^{\left (2 \, x\right )} {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\left (x^{4} e^{\left (4 \, x\right )} + 2 \, x^{4} e^{\left (2 \, x\right )} + x^{4} - 12 \, {\left (x^{2} e^{\left (4 \, x\right )} + 2 \, x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 12 \, {\left (x^{2} e^{\left (4 \, x\right )} + 2 \, x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - 4 \, {\left (x^{3} e^{\left (4 \, x\right )} + 2 \, x^{3} e^{\left (2 \, x\right )} + x^{3}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - 4 \, {\left (x^{3} e^{\left (4 \, x\right )} + 2 \, x^{3} e^{\left (2 \, x\right )} + x^{3}\right )} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right )\right )} \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{4 \, a} \]
1/4*(24*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*(e^(4*x) + 2*e^(2*x) + 1)*e^(2*x)*polylog(4, cosh(x) + sinh(x)) + 24*sqrt(a/(e^(8* x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*(e^(4*x) + 2*e^(2*x) + 1)*e^( 2*x)*polylog(4, -cosh(x) - sinh(x)) - 24*(x*e^(4*x) + 2*x*e^(2*x) + x)*sqr t(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(3, cosh(x) + sinh(x)) - 24*(x*e^(4*x) + 2*x*e^(2*x) + x)*sqrt(a/(e^(8*x) + 4* e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(3, -cosh(x) - sinh(x )) - (x^4*e^(4*x) + 2*x^4*e^(2*x) + x^4 - 12*(x^2*e^(4*x) + 2*x^2*e^(2*x) + x^2)*dilog(cosh(x) + sinh(x)) - 12*(x^2*e^(4*x) + 2*x^2*e^(2*x) + x^2)*d ilog(-cosh(x) - sinh(x)) - 4*(x^3*e^(4*x) + 2*x^3*e^(2*x) + x^3)*log(cosh( x) + sinh(x) + 1) - 4*(x^3*e^(4*x) + 2*x^3*e^(2*x) + x^3)*log(-cosh(x) - s inh(x) + 1))*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^( 2*x))*e^(-2*x)/a
\[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^4(x)}} \, dx=\int \frac {x^{3} \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}}{\sqrt {a \operatorname {sech}^{4}{\left (x \right )}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.67 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^4(x)}} \, dx=-\frac {x^{4}}{4 \, \sqrt {a}} + \frac {x^{3} \log \left (e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (-e^{x}\right ) - 6 \, x {\rm Li}_{3}(-e^{x}) + 6 \, {\rm Li}_{4}(-e^{x})}{\sqrt {a}} + \frac {x^{3} \log \left (-e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (e^{x}\right ) - 6 \, x {\rm Li}_{3}(e^{x}) + 6 \, {\rm Li}_{4}(e^{x})}{\sqrt {a}} \]
-1/4*x^4/sqrt(a) + (x^3*log(e^x + 1) + 3*x^2*dilog(-e^x) - 6*x*polylog(3, -e^x) + 6*polylog(4, -e^x))/sqrt(a) + (x^3*log(-e^x + 1) + 3*x^2*dilog(e^x ) - 6*x*polylog(3, e^x) + 6*polylog(4, e^x))/sqrt(a)
\[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^4(x)}} \, dx=\int { \frac {x^{3} \operatorname {csch}\left (x\right ) \operatorname {sech}\left (x\right )}{\sqrt {a \operatorname {sech}\left (x\right )^{4}}} \,d x } \]
Timed out. \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^4(x)}} \, dx=\int \frac {x^3}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^4}}} \,d x \]