\(\int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x} \, dx\) [443]

Optimal result

Integrand size = 24, antiderivative size = 174 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x} \, dx=4 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)+\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2-2 \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2-2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )+2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+2 i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right ) \] Output:

4*arctan((-a*x+1)^(1/2)/(a*x+1)^(1/2))*arctanh(a*x)+(-a^2*x^2+1)^(1/2)*arc 
tanh(a*x)^2-2*arctanh((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^2-2*arctanh 
(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2,(a*x 
+1)/(-a^2*x^2+1)^(1/2))+2*I*polylog(2,-I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))-2*I 
*polylog(2,I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))+2*polylog(3,-(a*x+1)/(-a^2*x^2+ 
1)^(1/2))-2*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x} \, dx=\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2+\text {arctanh}(a x)^2 \log \left (1-e^{-\text {arctanh}(a x)}\right )+2 i \text {arctanh}(a x) \log \left (1-i e^{-\text {arctanh}(a x)}\right )-2 i \text {arctanh}(a x) \log \left (1+i e^{-\text {arctanh}(a x)}\right )-\text {arctanh}(a x)^2 \log \left (1+e^{-\text {arctanh}(a x)}\right )+2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )+2 i \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-2 i \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )-2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{-\text {arctanh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,e^{-\text {arctanh}(a x)}\right ) \] Input:

Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x,x]
 

Output:

Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2 + ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x] 
)] + (2*I)*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] - (2*I)*ArcTanh[a*x]*Log 
[1 + I/E^ArcTanh[a*x]] - ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 2*Arc 
Tanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] + (2*I)*PolyLog[2, (-I)/E^ArcTanh 
[a*x]] - (2*I)*PolyLog[2, I/E^ArcTanh[a*x]] - 2*ArcTanh[a*x]*PolyLog[2, E^ 
(-ArcTanh[a*x])] + 2*PolyLog[3, -E^(-ArcTanh[a*x])] - 2*PolyLog[3, E^(-Arc 
Tanh[a*x])]
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.22, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6576, 6556, 6512, 6582, 3042, 26, 4670, 3011, 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.

\(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x} \, dx\)

\(\Big \downarrow \) 6576

\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}}dx-a^2 \int \frac {x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx\)

\(\Big \downarrow \) 6556

\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}}dx-a^2 \left (\frac {2 \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a^2}\right )\)

\(\Big \downarrow \) 6512

\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}}dx-a^2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a^2}+\frac {2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )}{a}\right )\)

\(\Big \downarrow \) 6582

\(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a x}d\text {arctanh}(a x)-a^2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a^2}+\frac {2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )}{a}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \int i \text {arctanh}(a x)^2 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)-a^2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a^2}+\frac {2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )}{a}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \text {arctanh}(a x)^2 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)-a^2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a^2}+\frac {2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )}{a}\right )\)

\(\Big \downarrow \) 4670

\(\displaystyle i \left (2 i \int \text {arctanh}(a x) \log \left (1-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-2 i \int \text {arctanh}(a x) \log \left (1+e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )-a^2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a^2}+\frac {2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )}{a}\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle i \left (-2 i \left (\int \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\int \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )-a^2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a^2}+\frac {2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )}{a}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle i \left (-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )-a^2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a^2}+\frac {2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )}{a}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle i \left (-2 i \left (\operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )-a^2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a^2}+\frac {2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )}{a}\right )\)

Input:

Int[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x,x]
 

Output:

-(a^2*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/a^2) + (2*((-2*ArcTan[Sqrt[1 - 
 a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, ((-I)*Sqrt[1 - a*x])/ 
Sqrt[1 + a*x]])/a + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a))/a) 
) + I*((2*I)*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - (2*I)*(-(ArcTanh[a*x 
]*PolyLog[2, -E^ArcTanh[a*x]]) + PolyLog[3, -E^ArcTanh[a*x]]) + (2*I)*(-(A 
rcTanh[a*x]*PolyLog[2, E^ArcTanh[a*x]]) + PolyLog[3, E^ArcTanh[a*x]]))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2720

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]]]
 

rule 3011

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6512

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol 
] :> Simp[-2*(a + b*ArcTanh[c*x])*(ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*S 
qrt[d])), x] + (-Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/( 
c*Sqrt[d])), x] + Simp[I*b*(PolyLog[2, I*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/(c* 
Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 
0]
 

rule 6556

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q 
_.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(2*e*(q 
+ 1))), x] + Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTanh[c* 
x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[p, 0] && NeQ[q, -1]
 

rule 6576

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(q_.), x_Symbol] :> Simp[d   Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a 
+ b*ArcTanh[c*x])^p, x], x] - Simp[c^2*(d/f^2)   Int[(f*x)^(m + 2)*(d + e*x 
^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, 
x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ 
[p, 1] && IntegerQ[q]))
 

rule 6582

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 
]), x_Symbol] :> Simp[1/Sqrt[d]   Subst[Int[(a + b*x)^p*Csch[x], x], x, Arc 
Tanh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 
 0] && GtQ[d, 0]
 

rule 7143

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]
 

Maple [F]

\[\int \frac {\sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )^{2}}{x}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{x} \,d x } \] Input:

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

Output:

integral(sqrt(-a^2*x^2 + 1)*arctanh(a*x)^2/x, x)
 

Sympy [F]

\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{2}{\left (a x \right )}}{x}\, dx \] Input:

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

Output:

Integral(sqrt(-(a*x - 1)*(a*x + 1))*atanh(a*x)**2/x, x)
 

Maxima [F]

\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{x} \,d x } \] Input:

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

Output:

integrate(sqrt(-a^2*x^2 + 1)*arctanh(a*x)^2/x, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\sqrt {1-a^2\,x^2}}{x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x} \, dx=\int \frac {\sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )^{2}}{x}d x \] Input:

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

Output:

int((sqrt( - a**2*x**2 + 1)*atanh(a*x)**2)/x,x)