\(\int x^3 \text {arctanh}(1-d-d \coth (a+b x)) \, dx\) [307]

Optimal result

Integrand size = 19, antiderivative size = 165 \[ \int x^3 \text {arctanh}(1-d-d \coth (a+b x)) \, dx=\frac {b x^5}{20}+\frac {1}{4} x^4 \text {arctanh}(1-d-d \coth (a+b x))-\frac {1}{8} x^4 \log \left (1-(1-d) e^{2 a+2 b x}\right )-\frac {x^3 \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{4 b}+\frac {3 x^2 \operatorname {PolyLog}\left (3,(1-d) e^{2 a+2 b x}\right )}{8 b^2}-\frac {3 x \operatorname {PolyLog}\left (4,(1-d) e^{2 a+2 b x}\right )}{8 b^3}+\frac {3 \operatorname {PolyLog}\left (5,(1-d) e^{2 a+2 b x}\right )}{16 b^4} \] Output:

1/20*b*x^5-1/4*x^4*arctanh(-1+d+d*coth(b*x+a))-1/8*x^4*ln(1-(1-d)*exp(2*b* 
x+2*a))-1/4*x^3*polylog(2,(1-d)*exp(2*b*x+2*a))/b+3/8*x^2*polylog(3,(1-d)* 
exp(2*b*x+2*a))/b^2-3/8*x*polylog(4,(1-d)*exp(2*b*x+2*a))/b^3+3/16*polylog 
(5,(1-d)*exp(2*b*x+2*a))/b^4
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.92 \[ \int x^3 \text {arctanh}(1-d-d \coth (a+b x)) \, dx=\frac {4 b^4 x^4 \text {arctanh}(1-d-d \coth (a+b x))-2 b^4 x^4 \log \left (1+\frac {e^{-2 (a+b x)}}{-1+d}\right )+4 b^3 x^3 \operatorname {PolyLog}\left (2,-\frac {e^{-2 (a+b x)}}{-1+d}\right )+6 b^2 x^2 \operatorname {PolyLog}\left (3,-\frac {e^{-2 (a+b x)}}{-1+d}\right )+6 b x \operatorname {PolyLog}\left (4,-\frac {e^{-2 (a+b x)}}{-1+d}\right )+3 \operatorname {PolyLog}\left (5,-\frac {e^{-2 (a+b x)}}{-1+d}\right )}{16 b^4} \] Input:

Integrate[x^3*ArcTanh[1 - d - d*Coth[a + b*x]],x]
 

Output:

(4*b^4*x^4*ArcTanh[1 - d - d*Coth[a + b*x]] - 2*b^4*x^4*Log[1 + 1/((-1 + d 
)*E^(2*(a + b*x)))] + 4*b^3*x^3*PolyLog[2, -(1/((-1 + d)*E^(2*(a + b*x)))) 
] + 6*b^2*x^2*PolyLog[3, -(1/((-1 + d)*E^(2*(a + b*x))))] + 6*b*x*PolyLog[ 
4, -(1/((-1 + d)*E^(2*(a + b*x))))] + 3*PolyLog[5, -(1/((-1 + d)*E^(2*(a + 
 b*x))))])/(16*b^4)
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.30, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6795, 2615, 2620, 3011, 7163, 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.

\(\displaystyle \int x^3 \text {arctanh}(d (-\coth (a+b x))-d+1) \, dx\)

\(\Big \downarrow \) 6795

\(\displaystyle \frac {1}{4} b \int \frac {x^4}{1-(1-d) e^{2 a+2 b x}}dx+\frac {1}{4} x^4 \text {arctanh}(d (-\coth (a+b x))-d+1)\)

\(\Big \downarrow \) 2615

\(\displaystyle \frac {1}{4} b \left ((1-d) \int \frac {e^{2 a+2 b x} x^4}{1-(1-d) e^{2 a+2 b x}}dx+\frac {x^5}{5}\right )+\frac {1}{4} x^4 \text {arctanh}(d (-\coth (a+b x))-d+1)\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {1}{4} b \left ((1-d) \left (\frac {2 \int x^3 \log \left (1-(1-d) e^{2 a+2 b x}\right )dx}{b (1-d)}-\frac {x^4 \log \left (1-(1-d) e^{2 a+2 b x}\right )}{2 b (1-d)}\right )+\frac {x^5}{5}\right )+\frac {1}{4} x^4 \text {arctanh}(d (-\coth (a+b x))-d+1)\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {1}{4} b \left ((1-d) \left (\frac {2 \left (\frac {3 \int x^2 \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )dx}{2 b}-\frac {x^3 \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{2 b}\right )}{b (1-d)}-\frac {x^4 \log \left (1-(1-d) e^{2 a+2 b x}\right )}{2 b (1-d)}\right )+\frac {x^5}{5}\right )+\frac {1}{4} x^4 \text {arctanh}(d (-\coth (a+b x))-d+1)\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {1}{4} b \left ((1-d) \left (\frac {2 \left (\frac {3 \left (\frac {x^2 \operatorname {PolyLog}\left (3,(1-d) e^{2 a+2 b x}\right )}{2 b}-\frac {\int x \operatorname {PolyLog}\left (3,(1-d) e^{2 a+2 b x}\right )dx}{b}\right )}{2 b}-\frac {x^3 \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{2 b}\right )}{b (1-d)}-\frac {x^4 \log \left (1-(1-d) e^{2 a+2 b x}\right )}{2 b (1-d)}\right )+\frac {x^5}{5}\right )+\frac {1}{4} x^4 \text {arctanh}(d (-\coth (a+b x))-d+1)\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {1}{4} b \left ((1-d) \left (\frac {2 \left (\frac {3 \left (\frac {x^2 \operatorname {PolyLog}\left (3,(1-d) e^{2 a+2 b x}\right )}{2 b}-\frac {\frac {x \operatorname {PolyLog}\left (4,(1-d) e^{2 a+2 b x}\right )}{2 b}-\frac {\int \operatorname {PolyLog}\left (4,(1-d) e^{2 a+2 b x}\right )dx}{2 b}}{b}\right )}{2 b}-\frac {x^3 \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{2 b}\right )}{b (1-d)}-\frac {x^4 \log \left (1-(1-d) e^{2 a+2 b x}\right )}{2 b (1-d)}\right )+\frac {x^5}{5}\right )+\frac {1}{4} x^4 \text {arctanh}(d (-\coth (a+b x))-d+1)\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {1}{4} b \left ((1-d) \left (\frac {2 \left (\frac {3 \left (\frac {x^2 \operatorname {PolyLog}\left (3,(1-d) e^{2 a+2 b x}\right )}{2 b}-\frac {\frac {x \operatorname {PolyLog}\left (4,(1-d) e^{2 a+2 b x}\right )}{2 b}-\frac {\int e^{-2 a-2 b x} \operatorname {PolyLog}\left (4,(1-d) e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}}{b}\right )}{2 b}-\frac {x^3 \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{2 b}\right )}{b (1-d)}-\frac {x^4 \log \left (1-(1-d) e^{2 a+2 b x}\right )}{2 b (1-d)}\right )+\frac {x^5}{5}\right )+\frac {1}{4} x^4 \text {arctanh}(d (-\coth (a+b x))-d+1)\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {1}{4} x^4 \text {arctanh}(d (-\coth (a+b x))-d+1)+\frac {1}{4} b \left ((1-d) \left (\frac {2 \left (\frac {3 \left (\frac {x^2 \operatorname {PolyLog}\left (3,(1-d) e^{2 a+2 b x}\right )}{2 b}-\frac {\frac {x \operatorname {PolyLog}\left (4,(1-d) e^{2 a+2 b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (5,(1-d) e^{2 a+2 b x}\right )}{4 b^2}}{b}\right )}{2 b}-\frac {x^3 \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{2 b}\right )}{b (1-d)}-\frac {x^4 \log \left (1-(1-d) e^{2 a+2 b x}\right )}{2 b (1-d)}\right )+\frac {x^5}{5}\right )\)

Input:

Int[x^3*ArcTanh[1 - d - d*Coth[a + b*x]],x]
 

Output:

(x^4*ArcTanh[1 - d - d*Coth[a + b*x]])/4 + (b*(x^5/5 + (1 - d)*(-1/2*(x^4* 
Log[1 - (1 - d)*E^(2*a + 2*b*x)])/(b*(1 - d)) + (2*(-1/2*(x^3*PolyLog[2, ( 
1 - d)*E^(2*a + 2*b*x)])/b + (3*((x^2*PolyLog[3, (1 - d)*E^(2*a + 2*b*x)]) 
/(2*b) - ((x*PolyLog[4, (1 - d)*E^(2*a + 2*b*x)])/(2*b) - PolyLog[5, (1 - 
d)*E^(2*a + 2*b*x)]/(4*b^2))/b))/(2*b)))/(b*(1 - d)))))/4
 

Defintions of rubi rules used

rule 2615

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x 
_))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ 
b/a   Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] 
, x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2620

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]
 

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 6795

Int[ArcTanh[(c_.) + Coth[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(ArcTanh[c + d*Coth[a + b*x]]/(f*( 
m + 1))), x] + Simp[b/(f*(m + 1))   Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a 
 + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c 
- d)^2, 1]
 

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]
 

rule 7163

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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 3.31 (sec) , antiderivative size = 1753, normalized size of antiderivative = 10.62

method	result	size
risch	\(\text {Expression too large to display}\)	\(1753\)

Input:

int(-x^3*arctanh(-1+d+d*coth(b*x+a)),x,method=_RETURNVERBOSE)
 

Output:

1/8/b^4*a^4/(d-1)*ln(d*exp(2*b*x+2*a)-exp(2*b*x+2*a)+1)+1/20*b*x^5-1/4*x^4 
*ln(exp(b*x+a))-1/16*(-I*Pi*csgn(I/(exp(2*b*x+2*a)-1)*(d*exp(2*b*x+2*a)-ex 
p(2*b*x+2*a)+1))^3-2*I*Pi+2*ln(d)-I*Pi*csgn(I/(exp(2*b*x+2*a)-1)*d*exp(2*b 
*x+2*a))^3+I*Pi*csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b* 
x+2*a)-1))^2+I*Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)-1))^2*csgn(I*exp(2 
*b*x+2*a))+I*Pi*csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*(d*exp(2*b*x+2*a)-exp(2* 
b*x+2*a)+1))*csgn(I/(exp(2*b*x+2*a)-1)*(d*exp(2*b*x+2*a)-exp(2*b*x+2*a)+1) 
)-I*Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-I*Pi*csgn(I/(exp(2*b*x+ 
2*a)-1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)-1))*csgn(I*exp(2*b*x+2*a))- 
I*Pi*csgn(I/(exp(2*b*x+2*a)-1))*csgn(I/(exp(2*b*x+2*a)-1)*(d*exp(2*b*x+2*a 
)-exp(2*b*x+2*a)+1))^2+I*Pi*csgn(I/(exp(2*b*x+2*a)-1)*d*exp(2*b*x+2*a))^2* 
csgn(I*d)+2*I*Pi*csgn(I/(exp(2*b*x+2*a)-1)*(d*exp(2*b*x+2*a)-exp(2*b*x+2*a 
)+1))^2-I*Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)-1))^3-I*Pi*csgn(I*(d*ex 
p(2*b*x+2*a)-exp(2*b*x+2*a)+1))*csgn(I/(exp(2*b*x+2*a)-1)*(d*exp(2*b*x+2*a 
)-exp(2*b*x+2*a)+1))^2-I*Pi*csgn(I*exp(2*b*x+2*a))^3+2*I*Pi*csgn(I*exp(b*x 
+a))*csgn(I*exp(2*b*x+2*a))^2-I*Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)-1 
))*csgn(I/(exp(2*b*x+2*a)-1)*d*exp(2*b*x+2*a))*csgn(I*d)+I*Pi*csgn(I*exp(2 
*b*x+2*a)/(exp(2*b*x+2*a)-1))*csgn(I/(exp(2*b*x+2*a)-1)*d*exp(2*b*x+2*a))^ 
2)*x^4-1/2/b^3*a^3/(d-1)*ln(1-exp(b*x+a)*(1-d)^(1/2))*x-1/2/b^4*a^3/(d-1)* 
dilog(1+exp(b*x+a)*(1-d)^(1/2))-1/2/b^4*a^3/(d-1)*dilog(1-exp(b*x+a)*(1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (135) = 270\).

Time = 0.11 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.73 \[ \int x^3 \text {arctanh}(1-d-d \coth (a+b x)) \, dx =\text {Too large to display} \] Input:

integrate(-x^3*arctanh(-1+d+d*coth(b*x+a)),x, algorithm="fricas")
 

Output:

1/40*(2*b^5*x^5 - 5*b^4*x^4*log(-(d*cosh(b*x + a) + d*sinh(b*x + a))/(d*co 
sh(b*x + a) + (d - 2)*sinh(b*x + a))) - 20*b^3*x^3*dilog(1/2*sqrt(-4*d + 4 
)*(cosh(b*x + a) + sinh(b*x + a))) - 20*b^3*x^3*dilog(-1/2*sqrt(-4*d + 4)* 
(cosh(b*x + a) + sinh(b*x + a))) - 5*a^4*log(2*(d - 1)*cosh(b*x + a) + 2*( 
d - 1)*sinh(b*x + a) + sqrt(-4*d + 4)) - 5*a^4*log(2*(d - 1)*cosh(b*x + a) 
 + 2*(d - 1)*sinh(b*x + a) - sqrt(-4*d + 4)) + 60*b^2*x^2*polylog(3, 1/2*s 
qrt(-4*d + 4)*(cosh(b*x + a) + sinh(b*x + a))) + 60*b^2*x^2*polylog(3, -1/ 
2*sqrt(-4*d + 4)*(cosh(b*x + a) + sinh(b*x + a))) - 120*b*x*polylog(4, 1/2 
*sqrt(-4*d + 4)*(cosh(b*x + a) + sinh(b*x + a))) - 120*b*x*polylog(4, -1/2 
*sqrt(-4*d + 4)*(cosh(b*x + a) + sinh(b*x + a))) - 5*(b^4*x^4 - a^4)*log(1 
/2*sqrt(-4*d + 4)*(cosh(b*x + a) + sinh(b*x + a)) + 1) - 5*(b^4*x^4 - a^4) 
*log(-1/2*sqrt(-4*d + 4)*(cosh(b*x + a) + sinh(b*x + a)) + 1) + 120*polylo 
g(5, 1/2*sqrt(-4*d + 4)*(cosh(b*x + a) + sinh(b*x + a))) + 120*polylog(5, 
-1/2*sqrt(-4*d + 4)*(cosh(b*x + a) + sinh(b*x + a))))/b^4
 

Sympy [F]

\[ \int x^3 \text {arctanh}(1-d-d \coth (a+b x)) \, dx=- \int x^{3} \operatorname {atanh}{\left (d \coth {\left (a + b x \right )} + d - 1 \right )}\, dx \] Input:

integrate(-x**3*atanh(-1+d+d*coth(b*x+a)),x)
 

Output:

-Integral(x**3*atanh(d*coth(a + b*x) + d - 1), x)
 

Maxima [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.90 \[ \int x^3 \text {arctanh}(1-d-d \coth (a+b x)) \, dx=-\frac {1}{4} \, x^{4} \operatorname {artanh}\left (d \coth \left (b x + a\right ) + d - 1\right ) + \frac {1}{40} \, {\left (\frac {2 \, x^{5}}{d} - \frac {5 \, {\left (2 \, b^{4} x^{4} \log \left ({\left (d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 4 \, b^{3} x^{3} {\rm Li}_2\left (-{\left (d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}\right ) - 6 \, b^{2} x^{2} {\rm Li}_{3}(-{\left (d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}) + 6 \, b x {\rm Li}_{4}(-{\left (d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}) - 3 \, {\rm Li}_{5}(-{\left (d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )})\right )}}{b^{5} d}\right )} b d \] Input:

integrate(-x^3*arctanh(-1+d+d*coth(b*x+a)),x, algorithm="maxima")
 

Output:

-1/4*x^4*arctanh(d*coth(b*x + a) + d - 1) + 1/40*(2*x^5/d - 5*(2*b^4*x^4*l 
og((d - 1)*e^(2*b*x + 2*a) + 1) + 4*b^3*x^3*dilog(-(d - 1)*e^(2*b*x + 2*a) 
) - 6*b^2*x^2*polylog(3, -(d - 1)*e^(2*b*x + 2*a)) + 6*b*x*polylog(4, -(d 
- 1)*e^(2*b*x + 2*a)) - 3*polylog(5, -(d - 1)*e^(2*b*x + 2*a)))/(b^5*d))*b 
*d
 

Giac [F]

\[ \int x^3 \text {arctanh}(1-d-d \coth (a+b x)) \, dx=\int { -x^{3} \operatorname {artanh}\left (d \coth \left (b x + a\right ) + d - 1\right ) \,d x } \] Input:

integrate(-x^3*arctanh(-1+d+d*coth(b*x+a)),x, algorithm="giac")
 

Output:

integrate(-x^3*arctanh(d*coth(b*x + a) + d - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \text {arctanh}(1-d-d \coth (a+b x)) \, dx=\int -x^3\,\mathrm {atanh}\left (d+d\,\mathrm {coth}\left (a+b\,x\right )-1\right ) \,d x \] Input:

int(-x^3*atanh(d + d*coth(a + b*x) - 1),x)
 

Output:

int(-x^3*atanh(d + d*coth(a + b*x) - 1), x)
 

Reduce [F]

\[ \int x^3 \text {arctanh}(1-d-d \coth (a+b x)) \, dx=-\left (\int \mathit {atanh} \left (\coth \left (b x +a \right ) d +d -1\right ) x^{3}d x \right ) \] Input:

int(-x^3*atanh(-1+d+d*coth(b*x+a)),x)
 

Output:

 - int(atanh(coth(a + b*x)*d + d - 1)*x**3,x)