\(\int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^2} \, dx\) [19]

Optimal result

Integrand size = 18, antiderivative size = 517 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^2} \, dx=-\frac {(a+b \text {arctanh}(c x))^3}{e (d+e x)}+\frac {3 b c (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{2 e (c d+e)}-\frac {3 b c (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{2 (c d-e) e}+\frac {3 b c (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d^2-e^2}-\frac {3 b c (a+b \text {arctanh}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{c^2 d^2-e^2}+\frac {3 b^2 c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 e (c d+e)}+\frac {3 b^2 c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 (c d-e) e}-\frac {3 b^2 c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^2 d^2-e^2}+\frac {3 b^2 c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{c^2 d^2-e^2}-\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{4 e (c d+e)}+\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{4 (c d-e) e}-\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 \left (c^2 d^2-e^2\right )}+\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 \left (c^2 d^2-e^2\right )} \] Output:

-(a+b*arctanh(c*x))^3/e/(e*x+d)+3/2*b*c*(a+b*arctanh(c*x))^2*ln(2/(-c*x+1) 
)/e/(c*d+e)-3/2*b*c*(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/(c*d-e)/e+3*b*c*(a+ 
b*arctanh(c*x))^2*ln(2/(c*x+1))/(c^2*d^2-e^2)-3*b*c*(a+b*arctanh(c*x))^2*l 
n(2*c*(e*x+d)/(c*d+e)/(c*x+1))/(c^2*d^2-e^2)+3/2*b^2*c*(a+b*arctanh(c*x))* 
polylog(2,1-2/(-c*x+1))/e/(c*d+e)+3/2*b^2*c*(a+b*arctanh(c*x))*polylog(2,1 
-2/(c*x+1))/(c*d-e)/e-3*b^2*c*(a+b*arctanh(c*x))*polylog(2,1-2/(c*x+1))/(c 
^2*d^2-e^2)+3*b^2*c*(a+b*arctanh(c*x))*polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c* 
x+1))/(c^2*d^2-e^2)-3/4*b^3*c*polylog(3,1-2/(-c*x+1))/e/(c*d+e)+3/4*b^3*c* 
polylog(3,1-2/(c*x+1))/(c*d-e)/e-3*b^3*c*polylog(3,1-2/(c*x+1))/(2*c^2*d^2 
-2*e^2)+3*b^3*c*polylog(3,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/(2*c^2*d^2-2*e^2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 11.53 (sec) , antiderivative size = 1107, normalized size of antiderivative = 2.14 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcTanh[c*x])^3/(d + e*x)^2,x]
 

Output:

-(a^3/(e*(d + e*x))) - (3*a^2*b*ArcTanh[c*x])/(e*(d + e*x)) - (3*a^2*b*c*L 
og[1 - c*x])/(2*e*(c*d + e)) + (3*a^2*b*c*Log[1 + c*x])/(2*c*d*e - 2*e^2) 
- (3*a^2*b*c*Log[d + e*x])/(c^2*d^2 - e^2) + (3*a*b^2*(-(ArcTanh[c*x]^2/(S 
qrt[1 - (c^2*d^2)/e^2]*e*E^ArcTanh[(c*d)/e])) + (x*ArcTanh[c*x]^2)/(d + e* 
x) + (c*d*(I*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*ArcTanh[c*x]*Log[1 - E^(-2 
*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - I*Pi*(ArcTanh[c*x] - Log[1 - c^2*x^ 
2]/2) - 2*ArcTanh[(c*d)/e]*(ArcTanh[c*x] + Log[1 - E^(-2*(ArcTanh[(c*d)/e] 
 + ArcTanh[c*x]))] - Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) + PolyL 
og[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(c^2*d^2 - e^2)))/d + (b 
^3*((x*ArcTanh[c*x]^3)/(d + e*x) + (3*(6*c*d*ArcTanh[c*x]^3 - 2*e*ArcTanh[ 
c*x]^3 + (4*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] + 
 (6*I)*c*d*Pi*ArcTanh[c*x]*Log[(E^(-ArcTanh[c*x]) + E^ArcTanh[c*x])/2] + 6 
*c*d*ArcTanh[c*x]^2*Log[1 - (Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e 
]] + 6*c*d*ArcTanh[c*x]^2*Log[1 + (Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c* 
d) + e]] - 6*c*d*ArcTanh[c*x]^2*Log[1 - E^(ArcTanh[(c*d)/e] + ArcTanh[c*x] 
)] - 6*c*d*ArcTanh[c*x]^2*Log[1 + E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 6 
*c*d*ArcTanh[c*x]^2*Log[1 - E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 12* 
c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[(I/2)*E^(-ArcTanh[(c*d)/e] - ArcTanh 
[c*x])*(-1 + E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x])))] - 6*c*d*ArcTanh[c*x 
]^2*Log[(e*(-1 + E^(2*ArcTanh[c*x])) + c*d*(1 + E^(2*ArcTanh[c*x])))/(2...
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 487, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6480, 2009}

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.

\(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^2} \, dx\)

\(\Big \downarrow \) 6480

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*ArcTanh[c*x])^3/(d + e*x)^2,x]
 

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6480

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S 
ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])^p/(e*(q + 1))), x] - 
 Simp[b*c*(p/(e*(q + 1)))   Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1 
), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] 
 && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
 

Maple [C] (warning: unable to verify)

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

Time = 2.63 (sec) , antiderivative size = 3154, normalized size of antiderivative = 6.10

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3154\)
default	\(\text {Expression too large to display}\)	\(3154\)
parts	\(\text {Expression too large to display}\)	\(3162\)

Input:

int((a+b*arctanh(c*x))^3/(e*x+d)^2,x,method=_RETURNVERBOSE)
 

Output:

1/c*(-a^3*c^2/(c*e*x+c*d)/e+b^3*c^2*(-1/(c*e*x+c*d)/e*arctanh(c*x)^3+3/e*( 
-1/2*I/(c*d+e)/(c*d-e)*Pi*e*arctanh(c*x)^2+1/3/(c*d-e)*arctanh(c*x)^3-e/(c 
*d+e)^2/(c*d-e)*d*c*arctanh(c*x)^2*ln(1-(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c 
*d+e))-1/2*I/(c*d+e)/(c*d-e)*Pi*csgn(I*(d*c*(1-(c*x+1)^2/(c^2*x^2-1))+e*(- 
(c*x+1)^2/(c^2*x^2-1)-1))/(1-(c*x+1)^2/(c^2*x^2-1)))^3*e*arctanh(c*x)^2+1/ 
2*I/(c*d+e)/(c*d-e)*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))^2*e*arctanh(c*x)^ 
2+1/2*I/(c*d+e)/(c*d-e)*Pi*c*d*arctanh(c*x)^2-1/2*I/(c*d+e)/(c*d-e)*Pi*csg 
n(I/(1-(c*x+1)^2/(c^2*x^2-1)))^3*e*arctanh(c*x)^2-e/(c*d+e)^2/(c*d-e)*d*c* 
arctanh(c*x)*polylog(2,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-1/4*I/(c*d 
+e)/(c*d-e)*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3*e 
*arctanh(c*x)^2-1/4*I/(c*d+e)/(c*d-e)*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*e 
*arctanh(c*x)^2+1/4*I/(c*d+e)/(c*d-e)*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-( 
c*x+1)^2/(c^2*x^2-1)))*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(c*x+1)^2/ 
(c^2*x^2-1))*c*d*arctanh(c*x)^2-arctanh(c*x)^2*e/(c*d+e)/(c*d-e)*ln(c*e*x+ 
c*d)+1/2*e^2/(c*d+e)^2/(c*d-e)*polylog(3,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(- 
c*d+e))+arctanh(c*x)^2/(2*c*d-2*e)*ln(c*x+1)-arctanh(c*x)^2/(2*c*d+2*e)*ln 
(c*x-1)-arctanh(c*x)^2/(c*d-e)*ln((c*x+1)/(-c^2*x^2+1)^(1/2))+1/4*I/(c*d+e 
)/(c*d-e)*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn( 
I/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*e*arctanh(c*x)^ 
2+1/4*I/(c*d+e)/(c*d-e)*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c...
 

Fricas [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:

integrate((a+b*arctanh(c*x))^3/(e*x+d)^2,x, algorithm="fricas")
 

Output:

integral((b^3*arctanh(c*x)^3 + 3*a*b^2*arctanh(c*x)^2 + 3*a^2*b*arctanh(c* 
x) + a^3)/(e^2*x^2 + 2*d*e*x + d^2), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{\left (d + e x\right )^{2}}\, dx \] Input:

integrate((a+b*atanh(c*x))**3/(e*x+d)**2,x)
 

Output:

Integral((a + b*atanh(c*x))**3/(d + e*x)**2, x)
 

Maxima [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:

integrate((a+b*arctanh(c*x))^3/(e*x+d)^2,x, algorithm="maxima")
 

Output:

3/2*(c*(log(c*x + 1)/(c*d*e - e^2) - log(c*x - 1)/(c*d*e + e^2) - 2*log(e* 
x + d)/(c^2*d^2 - e^2)) - 2*arctanh(c*x)/(e^2*x + d*e))*a^2*b - a^3/(e^2*x 
 + d*e) - 1/8*(((c^2*d*e - c*e^2)*b^3*x - (c*d*e - e^2)*b^3)*log(-c*x + 1) 
^3 + 3*(2*(c^2*d^2 - e^2)*a*b^2 - ((c^2*d*e + c*e^2)*b^3*x + (c*d*e + e^2) 
*b^3)*log(c*x + 1))*log(-c*x + 1)^2)/(c^2*d^3*e - d*e^3 + (c^2*d^2*e^2 - e 
^4)*x) - integrate(1/8*(((c^2*d*e - c*e^2)*b^3*x - (c*d*e - e^2)*b^3)*log( 
c*x + 1)^3 + 6*((c^2*d*e - c*e^2)*a*b^2*x - (c*d*e - e^2)*a*b^2)*log(c*x + 
 1)^2 + 3*(4*(c^2*d*e - c*e^2)*a*b^2*x + 4*(c^2*d^2 - c*d*e)*a*b^2 - ((c^2 
*d*e - c*e^2)*b^3*x - (c*d*e - e^2)*b^3)*log(c*x + 1)^2 - 2*(b^3*c^2*e^2*x 
^2 + b^3*c*d*e - 2*(c*d*e - e^2)*a*b^2 + (2*(c^2*d*e - c*e^2)*a*b^2 + (c^2 
*d*e + c*e^2)*b^3)*x)*log(c*x + 1))*log(-c*x + 1))/(c*d^3*e - d^2*e^2 - (c 
^2*d*e^3 - c*e^4)*x^3 - (2*c^2*d^2*e^2 - 3*c*d*e^3 + e^4)*x^2 - (c^2*d^3*e 
 - 3*c*d^2*e^2 + 2*d*e^3)*x), x)
 

Giac [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:

integrate((a+b*arctanh(c*x))^3/(e*x+d)^2,x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x) + a)^3/(e*x + d)^2, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3}{{\left (d+e\,x\right )}^2} \,d x \] Input:

int((a + b*atanh(c*x))^3/(d + e*x)^2,x)
 

Output:

int((a + b*atanh(c*x))^3/(d + e*x)^2, x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^2} \, dx=\text {too large to display} \] Input:

int((a+b*atanh(c*x))^3/(e*x+d)^2,x)
 

Output:

(4*atanh(c*x)**3*b**3*c**6*d**6*x - 4*atanh(c*x)**3*b**3*c**4*d**5*e - 4*a 
tanh(c*x)**3*b**3*c**4*d**4*e**2*x + 4*atanh(c*x)**3*b**3*c**2*d**3*e**3 + 
 12*atanh(c*x)**2*a*b**2*c**6*d**6*x - 12*atanh(c*x)**2*a*b**2*c**4*d**5*e 
 - 12*atanh(c*x)**2*a*b**2*c**4*d**4*e**2*x + 12*atanh(c*x)**2*a*b**2*c**2 
*d**3*e**3 - 3*atanh(c*x)**2*b**3*c**5*d**6 + 9*atanh(c*x)**2*b**3*c**5*d* 
*5*e*x - 12*atanh(c*x)**2*b**3*c**3*d**3*e**3*x + 3*atanh(c*x)**2*b**3*c*d 
**2*e**4 + 3*atanh(c*x)**2*b**3*c*d*e**5*x + 12*atanh(c*x)*a**2*b*c**6*d** 
6*x - 12*atanh(c*x)*a**2*b*c**2*d**2*e**4*x + 24*atanh(c*x)*a*b**2*c**5*d* 
*5*e*x - 24*atanh(c*x)*a*b**2*c**3*d**3*e**3*x + 18*atanh(c*x)*b**3*c**4*d 
**4*e**2*x - 24*atanh(c*x)*b**3*c**2*d**2*e**4*x + 6*atanh(c*x)*b**3*e**6* 
x - 6*int(atanh(c*x)/(c**4*d**4*x**2 + 2*c**4*d**3*e*x**3 + c**4*d**2*e**2 
*x**4 - c**2*d**4 - 2*c**2*d**3*e*x + 2*c**2*d*e**3*x**3 + c**2*e**4*x**4 
- d**2*e**2 - 2*d*e**3*x - e**4*x**2),x)*b**3*c**8*d**10 - 6*int(atanh(c*x 
)/(c**4*d**4*x**2 + 2*c**4*d**3*e*x**3 + c**4*d**2*e**2*x**4 - c**2*d**4 - 
 2*c**2*d**3*e*x + 2*c**2*d*e**3*x**3 + c**2*e**4*x**4 - d**2*e**2 - 2*d*e 
**3*x - e**4*x**2),x)*b**3*c**8*d**9*e*x + 12*int(atanh(c*x)/(c**4*d**4*x* 
*2 + 2*c**4*d**3*e*x**3 + c**4*d**2*e**2*x**4 - c**2*d**4 - 2*c**2*d**3*e* 
x + 2*c**2*d*e**3*x**3 + c**2*e**4*x**4 - d**2*e**2 - 2*d*e**3*x - e**4*x* 
*2),x)*b**3*c**6*d**8*e**2 + 12*int(atanh(c*x)/(c**4*d**4*x**2 + 2*c**4*d* 
*3*e*x**3 + c**4*d**2*e**2*x**4 - c**2*d**4 - 2*c**2*d**3*e*x + 2*c**2*...