\(\int (d+e x)^2 (a+b \text {arctanh}(c x))^3 \, dx\) [16]

Optimal result

Integrand size = 18, antiderivative size = 387 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\frac {a b^2 e^2 x}{c^2}+\frac {b^3 e^2 x \text {arctanh}(c x)}{c^2}+\frac {3 b d e (a+b \text {arctanh}(c x))^2}{c^2}-\frac {b e^2 (a+b \text {arctanh}(c x))^2}{2 c^3}+\frac {3 b d e x (a+b \text {arctanh}(c x))^2}{c}+\frac {b e^2 x^2 (a+b \text {arctanh}(c x))^2}{2 c}+\frac {\left (3 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^3}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) (a+b \text {arctanh}(c x))^3}{3 e}+\frac {(d+e x)^3 (a+b \text {arctanh}(c x))^3}{3 e}-\frac {6 b^2 d e (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {b \left (3 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {3 b^3 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^2}-\frac {b^2 \left (3 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \left (3 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^3} \] Output:

a*b^2*e^2*x/c^2+b^3*e^2*x*arctanh(c*x)/c^2+3*b*d*e*(a+b*arctanh(c*x))^2/c^ 
2-1/2*b*e^2*(a+b*arctanh(c*x))^2/c^3+3*b*d*e*x*(a+b*arctanh(c*x))^2/c+1/2* 
b*e^2*x^2*(a+b*arctanh(c*x))^2/c+1/3*(3*c^2*d^2+e^2)*(a+b*arctanh(c*x))^3/ 
c^3-1/3*d*(d^2+3*e^2/c^2)*(a+b*arctanh(c*x))^3/e+1/3*(e*x+d)^3*(a+b*arctan 
h(c*x))^3/e-6*b^2*d*e*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^2-b*(3*c^2*d^2+e 
^2)*(a+b*arctanh(c*x))^2*ln(2/(-c*x+1))/c^3+1/2*b^3*e^2*ln(-c^2*x^2+1)/c^3 
-3*b^3*d*e*polylog(2,1-2/(-c*x+1))/c^2-b^2*(3*c^2*d^2+e^2)*(a+b*arctanh(c* 
x))*polylog(2,1-2/(-c*x+1))/c^3+1/2*b^3*(3*c^2*d^2+e^2)*polylog(3,1-2/(-c* 
x+1))/c^3
 

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.53 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^3 \, dx=\frac {6 a^2 c^2 d (a c d+3 b e) x+3 a^2 c^2 e (2 a c d+b e) x^2+2 a^3 c^3 e^2 x^3+6 a^2 b c^3 x \left (3 d^2+3 d e x+e^2 x^2\right ) \text {arctanh}(c x)+3 a^2 b \left (3 c^2 d^2+3 c d e+e^2\right ) \log (1-c x)+3 a^2 b \left (3 c^2 d^2-3 c d e+e^2\right ) \log (1+c x)+18 a b^2 c d e \left (2 c x \text {arctanh}(c x)+\left (-1+c^2 x^2\right ) \text {arctanh}(c x)^2+\log \left (1-c^2 x^2\right )\right )-6 b^3 c d e \left (\text {arctanh}(c x) \left ((3-3 c x) \text {arctanh}(c x)+\left (1-c^2 x^2\right ) \text {arctanh}(c x)^2+6 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )-3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+6 a b^2 e^2 \left (c x+\left (-1+c^3 x^3\right ) \text {arctanh}(c x)^2+\text {arctanh}(c x) \left (-1+c^2 x^2-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+18 a b^2 c^2 d^2 \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+6 b^3 c^2 d^2 \left (\text {arctanh}(c x)^2 \left ((-1+c x) \text {arctanh}(c x)-3 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+3 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )\right )+b^3 e^2 \left (6 c x \text {arctanh}(c x)-3 \text {arctanh}(c x)^2+3 c^2 x^2 \text {arctanh}(c x)^2-2 \text {arctanh}(c x)^3+2 c^3 x^3 \text {arctanh}(c x)^3-6 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+3 \log \left (1-c^2 x^2\right )+6 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )\right )}{6 c^3} \] Input:

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

Output:

(6*a^2*c^2*d*(a*c*d + 3*b*e)*x + 3*a^2*c^2*e*(2*a*c*d + b*e)*x^2 + 2*a^3*c 
^3*e^2*x^3 + 6*a^2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTanh[c*x] + 3*a^ 
2*b*(3*c^2*d^2 + 3*c*d*e + e^2)*Log[1 - c*x] + 3*a^2*b*(3*c^2*d^2 - 3*c*d* 
e + e^2)*Log[1 + c*x] + 18*a*b^2*c*d*e*(2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2 
)*ArcTanh[c*x]^2 + Log[1 - c^2*x^2]) - 6*b^3*c*d*e*(ArcTanh[c*x]*((3 - 3*c 
*x)*ArcTanh[c*x] + (1 - c^2*x^2)*ArcTanh[c*x]^2 + 6*Log[1 + E^(-2*ArcTanh[ 
c*x])]) - 3*PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 6*a*b^2*e^2*(c*x + (-1 + c 
^3*x^3)*ArcTanh[c*x]^2 + ArcTanh[c*x]*(-1 + c^2*x^2 - 2*Log[1 + E^(-2*ArcT 
anh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 18*a*b^2*c^2*d^2*(ArcTan 
h[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLo 
g[2, -E^(-2*ArcTanh[c*x])]) + 6*b^3*c^2*d^2*(ArcTanh[c*x]^2*((-1 + c*x)*Ar 
cTanh[c*x] - 3*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*ArcTanh[c*x]*PolyLog[2, - 
E^(-2*ArcTanh[c*x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/2) + b^3*e^2*( 
6*c*x*ArcTanh[c*x] - 3*ArcTanh[c*x]^2 + 3*c^2*x^2*ArcTanh[c*x]^2 - 2*ArcTa 
nh[c*x]^3 + 2*c^3*x^3*ArcTanh[c*x]^3 - 6*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcT 
anh[c*x])] + 3*Log[1 - c^2*x^2] + 6*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh 
[c*x])] + 3*PolyLog[3, -E^(-2*ArcTanh[c*x])]))/(6*c^3)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.04, 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.

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

\(\Big \downarrow \) 6480

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^3 (a+b \text {arctanh}(c x))^3}{3 e}-\frac {b c \left (\frac {e^3 (a+b \text {arctanh}(c x))^2}{2 c^4}-\frac {3 d e^2 (a+b \text {arctanh}(c x))^2}{c^3}+\frac {6 b d e^2 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^3}-\frac {3 d e^2 x (a+b \text {arctanh}(c x))^2}{c^2}-\frac {e^3 x^2 (a+b \text {arctanh}(c x))^2}{2 c^2}+\frac {b e \left (3 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^4}-\frac {e \left (3 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^3}{3 b c^4}+\frac {e \left (3 c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{c^4}+\frac {d \left (c^2 d^2+3 e^2\right ) (a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {a b e^3 x}{c^3}-\frac {b^2 e^3 x \text {arctanh}(c x)}{c^3}+\frac {3 b^2 d e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3}-\frac {b^2 e \left (3 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^4}-\frac {b^2 e^3 \log \left (1-c^2 x^2\right )}{2 c^4}\right )}{e}\)

Input:

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

Output:

((d + e*x)^3*(a + b*ArcTanh[c*x])^3)/(3*e) - (b*c*(-((a*b*e^3*x)/c^3) - (b 
^2*e^3*x*ArcTanh[c*x])/c^3 - (3*d*e^2*(a + b*ArcTanh[c*x])^2)/c^3 + (e^3*( 
a + b*ArcTanh[c*x])^2)/(2*c^4) - (3*d*e^2*x*(a + b*ArcTanh[c*x])^2)/c^2 - 
(e^3*x^2*(a + b*ArcTanh[c*x])^2)/(2*c^2) - (e*(3*c^2*d^2 + e^2)*(a + b*Arc 
Tanh[c*x])^3)/(3*b*c^4) + (d*(c^2*d^2 + 3*e^2)*(a + b*ArcTanh[c*x])^3)/(3* 
b*c^3) + (6*b*d*e^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^3 + (e*(3*c^2 
*d^2 + e^2)*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^4 - (b^2*e^3*Log[1 
- c^2*x^2])/(2*c^4) + (3*b^2*d*e^2*PolyLog[2, 1 - 2/(1 - c*x)])/c^3 + (b*e 
*(3*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^4 - 
 (b^2*e*(3*c^2*d^2 + e^2)*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c^4)))/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 = 14.83 (sec) , antiderivative size = 3848, normalized size of antiderivative = 9.94

method	result	size
parts	\(\text {Expression too large to display}\)	\(3848\)
derivativedivides	\(\text {Expression too large to display}\)	\(3873\)
default	\(\text {Expression too large to display}\)	\(3873\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/3*a^3*e^2*x^3 + a^3*d*e*x^2 + 3/2*(2*x^2*arctanh(c*x) + c*(2*x/c^2 - log 
(c*x + 1)/c^3 + log(c*x - 1)/c^3))*a^2*b*d*e + 1/2*(2*x^3*arctanh(c*x) + c 
*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a^2*b*e^2 + a^3*d^2*x + 3/2*(2*c*x*arct 
anh(c*x) + log(-c^2*x^2 + 1))*a^2*b*d^2/c - 1/24*((b^3*c^3*e^2*x^3 + 3*b^3 
*c^3*d*e*x^2 + 3*b^3*c^3*d^2*x - (3*c^2*d^2 + 3*c*d*e + e^2)*b^3)*log(-c*x 
 + 1)^3 - 3*(2*a*b^2*c^3*e^2*x^3 + (6*a*b^2*c^3*d*e + b^3*c^2*e^2)*x^2 + 6 
*(a*b^2*c^3*d^2 + b^3*c^2*d*e)*x + (b^3*c^3*e^2*x^3 + 3*b^3*c^3*d*e*x^2 + 
3*b^3*c^3*d^2*x + (3*c^2*d^2 - 3*c*d*e + e^2)*b^3)*log(c*x + 1))*log(-c*x 
+ 1)^2)/c^3 - integrate(-1/8*((b^3*c^3*e^2*x^3 - b^3*c^2*d^2 + (2*c^3*d*e 
- c^2*e^2)*b^3*x^2 + (c^3*d^2 - 2*c^2*d*e)*b^3*x)*log(c*x + 1)^3 + 6*(a*b^ 
2*c^3*e^2*x^3 - a*b^2*c^2*d^2 + (2*c^3*d*e - c^2*e^2)*a*b^2*x^2 + (c^3*d^2 
 - 2*c^2*d*e)*a*b^2*x)*log(c*x + 1)^2 - (4*a*b^2*c^3*e^2*x^3 + 2*(6*a*b^2* 
c^3*d*e + b^3*c^2*e^2)*x^2 + 3*(b^3*c^3*e^2*x^3 - b^3*c^2*d^2 + (2*c^3*d*e 
 - c^2*e^2)*b^3*x^2 + (c^3*d^2 - 2*c^2*d*e)*b^3*x)*log(c*x + 1)^2 + 12*(a* 
b^2*c^3*d^2 + b^3*c^2*d*e)*x - 2*(6*a*b^2*c^2*d^2 - (3*c^2*d^2 - 3*c*d*e + 
 e^2)*b^3 - (6*a*b^2*c^3*e^2 + b^3*c^3*e^2)*x^3 - 3*(b^3*c^3*d*e + 2*(2*c^ 
3*d*e - c^2*e^2)*a*b^2)*x^2 - 3*(b^3*c^3*d^2 + 2*(c^3*d^2 - 2*c^2*d*e)*a*b 
^2)*x)*log(c*x + 1))*log(-c*x + 1))/(c^3*x - c^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

(6*atanh(c*x)**3*b**3*c**3*d**2*x + 6*atanh(c*x)**3*b**3*c**3*d*e*x**2 + 2 
*atanh(c*x)**3*b**3*c**3*e**2*x**3 - 6*atanh(c*x)**3*b**3*c*d*e + 18*atanh 
(c*x)**2*a*b**2*c**3*d**2*x + 18*atanh(c*x)**2*a*b**2*c**3*d*e*x**2 + 6*at 
anh(c*x)**2*a*b**2*c**3*e**2*x**3 - 18*atanh(c*x)**2*a*b**2*c*d*e + 18*ata 
nh(c*x)**2*b**3*c**2*d*e*x + 3*atanh(c*x)**2*b**3*c**2*e**2*x**2 - 3*atanh 
(c*x)**2*b**3*e**2 + 18*atanh(c*x)*a**2*b*c**3*d**2*x + 18*atanh(c*x)*a**2 
*b*c**3*d*e*x**2 + 6*atanh(c*x)*a**2*b*c**3*e**2*x**3 + 18*atanh(c*x)*a**2 
*b*c**2*d**2 - 18*atanh(c*x)*a**2*b*c*d*e + 6*atanh(c*x)*a**2*b*e**2 + 36* 
atanh(c*x)*a*b**2*c**2*d*e*x + 6*atanh(c*x)*a*b**2*c**2*e**2*x**2 + 36*ata 
nh(c*x)*a*b**2*c*d*e - 6*atanh(c*x)*a*b**2*e**2 + 6*atanh(c*x)*b**3*c*e**2 
*x + 6*atanh(c*x)*b**3*e**2 + 36*int((atanh(c*x)*x)/(c**2*x**2 - 1),x)*a*b 
**2*c**4*d**2 + 12*int((atanh(c*x)*x)/(c**2*x**2 - 1),x)*a*b**2*c**2*e**2 
+ 36*int((atanh(c*x)*x)/(c**2*x**2 - 1),x)*b**3*c**3*d*e + 18*int((atanh(c 
*x)**2*x)/(c**2*x**2 - 1),x)*b**3*c**4*d**2 + 6*int((atanh(c*x)**2*x)/(c** 
2*x**2 - 1),x)*b**3*c**2*e**2 + 18*log(c**2*x - c)*a**2*b*c**2*d**2 + 6*lo 
g(c**2*x - c)*a**2*b*e**2 + 36*log(c**2*x - c)*a*b**2*c*d*e + 6*log(c**2*x 
 - c)*b**3*e**2 + 6*a**3*c**3*d**2*x + 6*a**3*c**3*d*e*x**2 + 2*a**3*c**3* 
e**2*x**3 + 18*a**2*b*c**2*d*e*x + 3*a**2*b*c**2*e**2*x**2 + 6*a*b**2*c*e* 
*2*x)/(6*c**3)