\(\int (d+e x)^3 (a+b \text {arctanh}(c x))^3 \, dx\) [15]

Optimal result

Integrand size = 18, antiderivative size = 614 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\frac {3 a b^2 d e^2 x}{c^2}+\frac {b^3 e^3 x}{4 c^3}-\frac {b^3 e^3 \text {arctanh}(c x)}{4 c^4}+\frac {3 b^3 d e^2 x \text {arctanh}(c x)}{c^2}+\frac {b^2 e^3 x^2 (a+b \text {arctanh}(c x))}{4 c^2}-\frac {3 b d e^2 (a+b \text {arctanh}(c x))^2}{2 c^3}+\frac {b e^3 (a+b \text {arctanh}(c x))^2}{4 c^4}+\frac {3 b e \left (6 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2}{4 c^4}+\frac {3 b e \left (6 c^2 d^2+e^2\right ) x (a+b \text {arctanh}(c x))^2}{4 c^3}+\frac {3 b d e^2 x^2 (a+b \text {arctanh}(c x))^2}{2 c}+\frac {b e^3 x^3 (a+b \text {arctanh}(c x))^2}{4 c}+\frac {d \left (c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^3}{c^3}-\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) (a+b \text {arctanh}(c x))^3}{4 c^4 e}+\frac {(d+e x)^4 (a+b \text {arctanh}(c x))^3}{4 e}-\frac {b^2 e^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{2 c^4}-\frac {3 b^2 e \left (6 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{2 c^4}-\frac {3 b d \left (c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {3 b^3 d e^2 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {b^3 e^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 c^4}-\frac {3 b^3 e \left (6 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 c^4}-\frac {3 b^2 d \left (c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3}+\frac {3 b^3 d \left (c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^3} \] Output:

3*a*b^2*d*e^2*x/c^2+1/4*b^3*e^3*x/c^3-1/4*b^3*e^3*arctanh(c*x)/c^4+3*b^3*d 
*e^2*x*arctanh(c*x)/c^2+1/4*b^2*e^3*x^2*(a+b*arctanh(c*x))/c^2-3/2*b*d*e^2 
*(a+b*arctanh(c*x))^2/c^3+1/4*b*e^3*(a+b*arctanh(c*x))^2/c^4+3/4*b*e*(6*c^ 
2*d^2+e^2)*(a+b*arctanh(c*x))^2/c^4+3/4*b*e*(6*c^2*d^2+e^2)*x*(a+b*arctanh 
(c*x))^2/c^3+3/2*b*d*e^2*x^2*(a+b*arctanh(c*x))^2/c+1/4*b*e^3*x^3*(a+b*arc 
tanh(c*x))^2/c+d*(c^2*d^2+e^2)*(a+b*arctanh(c*x))^3/c^3-1/4*(c^4*d^4+6*c^2 
*d^2*e^2+e^4)*(a+b*arctanh(c*x))^3/c^4/e+1/4*(e*x+d)^4*(a+b*arctanh(c*x))^ 
3/e-1/2*b^2*e^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^4-3/2*b^2*e*(6*c^2*d^2 
+e^2)*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^4-3*b*d*(c^2*d^2+e^2)*(a+b*arcta 
nh(c*x))^2*ln(2/(-c*x+1))/c^3+3/2*b^3*d*e^2*ln(-c^2*x^2+1)/c^3-1/4*b^3*e^3 
*polylog(2,1-2/(-c*x+1))/c^4-3/4*b^3*e*(6*c^2*d^2+e^2)*polylog(2,1-2/(-c*x 
+1))/c^4-3*b^2*d*(c^2*d^2+e^2)*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/ 
c^3+3/2*b^3*d*(c^2*d^2+e^2)*polylog(3,1-2/(-c*x+1))/c^3
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 830, normalized size of antiderivative = 1.35 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(2*a^2*c*(4*a*c^3*d^3 + 3*b*e*(6*c^2*d^2 + e^2))*x + 12*a^2*c^3*d*e*(a*c*d 
 + b*e)*x^2 + 2*a^2*c^3*e^2*(4*a*c*d + b*e)*x^3 + 2*a^3*c^4*e^3*x^4 + 6*a^ 
2*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcTanh[c*x] + 3*a^2 
*b*(4*c^3*d^3 + 6*c^2*d^2*e + 4*c*d*e^2 + e^3)*Log[1 - c*x] + 3*a^2*b*(4*c 
^3*d^3 - 6*c^2*d^2*e + 4*c*d*e^2 - e^3)*Log[1 + c*x] + 36*a*b^2*c^2*d^2*e* 
(2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + Log[1 - c^2*x^2]) + 
2*a*b^2*e^3*(-1 + c^2*x^2 + 2*c*x*(3 + c^2*x^2)*ArcTanh[c*x] + 3*(-1 + c^4 
*x^4)*ArcTanh[c*x]^2 + 4*Log[1 - c^2*x^2]) - 12*b^3*c^2*d^2*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])]) + 24*a*b^2*c*d*e^ 
2*(c*x + (-1 + c^3*x^3)*ArcTanh[c*x]^2 + ArcTanh[c*x]*(-1 + c^2*x^2 - 2*Lo 
g[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 24*a*b^2 
*c^3*d^3*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTanh[ 
c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 2*b^3*e^3*(c*x + (-4 + 3*c*x 
 + c^3*x^3)*ArcTanh[c*x]^2 + (-1 + c^4*x^4)*ArcTanh[c*x]^3 + ArcTanh[c*x]* 
(-1 + c^2*x^2 - 8*Log[1 + E^(-2*ArcTanh[c*x])]) + 4*PolyLog[2, -E^(-2*ArcT 
anh[c*x])]) + 8*b^3*c^3*d^3*(ArcTanh[c*x]^2*((-1 + c*x)*ArcTanh[c*x] - 3*L 
og[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) + 4*b^3*c*d*e^2*(6*c*x*ArcT 
anh[c*x] - 3*ArcTanh[c*x]^2 + 3*c^2*x^2*ArcTanh[c*x]^2 - 2*ArcTanh[c*x]...
 

Rubi [A] (verified)

Time = 1.49 (sec) , antiderivative size = 608, normalized size of antiderivative = 0.99, 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.

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

\(\Big \downarrow \) 6480

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^4 (a+b \text {arctanh}(c x))^3}{4 e}-\frac {3 b c \left (-\frac {e^4 (a+b \text {arctanh}(c x))^2}{3 c^5}+\frac {2 b e^4 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{3 c^5}+\frac {2 d e^3 (a+b \text {arctanh}(c x))^2}{c^4}-\frac {b e^4 x^2 (a+b \text {arctanh}(c x))}{3 c^3}-\frac {2 d e^3 x^2 (a+b \text {arctanh}(c x))^2}{c^2}-\frac {e^4 x^3 (a+b \text {arctanh}(c x))^2}{3 c^2}-\frac {e^2 \left (6 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2}{c^5}+\frac {2 b e^2 \left (6 c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^5}+\frac {4 b d e \left (c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^4}-\frac {e^2 x \left (6 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2}{c^4}-\frac {4 d e \left (c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^3}{3 b c^4}+\frac {4 d e \left (c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{c^4}+\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) (a+b \text {arctanh}(c x))^3}{3 b c^5}-\frac {4 a b d e^3 x}{c^3}+\frac {b^2 e^4 \text {arctanh}(c x)}{3 c^5}-\frac {4 b^2 d e^3 x \text {arctanh}(c x)}{c^3}+\frac {b^2 e^4 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^5}-\frac {b^2 e^4 x}{3 c^4}+\frac {b^2 e^2 \left (6 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^5}-\frac {2 b^2 d e \left (c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{c^4}-\frac {2 b^2 d e^3 \log \left (1-c^2 x^2\right )}{c^4}\right )}{4 e}\)

Input:

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

Output:

((d + e*x)^4*(a + b*ArcTanh[c*x])^3)/(4*e) - (3*b*c*((-4*a*b*d*e^3*x)/c^3 
- (b^2*e^4*x)/(3*c^4) + (b^2*e^4*ArcTanh[c*x])/(3*c^5) - (4*b^2*d*e^3*x*Ar 
cTanh[c*x])/c^3 - (b*e^4*x^2*(a + b*ArcTanh[c*x]))/(3*c^3) + (2*d*e^3*(a + 
 b*ArcTanh[c*x])^2)/c^4 - (e^4*(a + b*ArcTanh[c*x])^2)/(3*c^5) - (e^2*(6*c 
^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^2)/c^5 - (e^2*(6*c^2*d^2 + e^2)*x*(a + 
b*ArcTanh[c*x])^2)/c^4 - (2*d*e^3*x^2*(a + b*ArcTanh[c*x])^2)/c^2 - (e^4*x 
^3*(a + b*ArcTanh[c*x])^2)/(3*c^2) - (4*d*e*(c^2*d^2 + e^2)*(a + b*ArcTanh 
[c*x])^3)/(3*b*c^4) + ((c^4*d^4 + 6*c^2*d^2*e^2 + e^4)*(a + b*ArcTanh[c*x] 
)^3)/(3*b*c^5) + (2*b*e^4*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^5) + 
 (2*b*e^2*(6*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^5 + ( 
4*d*e*(c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^4 - (2*b^ 
2*d*e^3*Log[1 - c^2*x^2])/c^4 + (b^2*e^4*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c 
^5) + (b^2*e^2*(6*c^2*d^2 + e^2)*PolyLog[2, 1 - 2/(1 - c*x)])/c^5 + (4*b*d 
*e*(c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^4 - 
 (2*b^2*d*e*(c^2*d^2 + e^2)*PolyLog[3, 1 - 2/(1 - c*x)])/c^4))/(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 = 28.29 (sec) , antiderivative size = 5093, normalized size of antiderivative = 8.29

method	result	size
parts	\(\text {Expression too large to display}\)	\(5093\)
derivativedivides	\(\text {Expression too large to display}\)	\(5124\)
default	\(\text {Expression too large to display}\)	\(5124\)

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/4*a^3*e^3*x^4 + a^3*d*e^2*x^3 + 3/2*a^3*d^2*e*x^2 + 9/4*(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^2*e + 3/2 
*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a^2*b*d*e^2 + 1 
/8*(6*x^4*arctanh(c*x) + c*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3 
*log(c*x - 1)/c^5))*a^2*b*e^3 + a^3*d^3*x + 3/2*(2*c*x*arctanh(c*x) + log( 
-c^2*x^2 + 1))*a^2*b*d^3/c - 1/32*((b^3*c^4*e^3*x^4 + 4*b^3*c^4*d*e^2*x^3 
+ 6*b^3*c^4*d^2*e*x^2 + 4*b^3*c^4*d^3*x - (4*c^3*d^3 + 6*c^2*d^2*e + 4*c*d 
*e^2 + e^3)*b^3)*log(-c*x + 1)^3 - (6*a*b^2*c^4*e^3*x^4 + 2*(12*a*b^2*c^4* 
d*e^2 + b^3*c^3*e^3)*x^3 + 12*(3*a*b^2*c^4*d^2*e + b^3*c^3*d*e^2)*x^2 + 6* 
(4*a*b^2*c^4*d^3 + (6*c^3*d^2*e + c*e^3)*b^3)*x + 3*(b^3*c^4*e^3*x^4 + 4*b 
^3*c^4*d*e^2*x^3 + 6*b^3*c^4*d^2*e*x^2 + 4*b^3*c^4*d^3*x + (4*c^3*d^3 - 6* 
c^2*d^2*e + 4*c*d*e^2 - e^3)*b^3)*log(c*x + 1))*log(-c*x + 1)^2)/c^4 - int 
egrate(-1/16*(2*(b^3*c^4*e^3*x^4 - b^3*c^3*d^3 + (3*c^4*d*e^2 - c^3*e^3)*b 
^3*x^3 + 3*(c^4*d^2*e - c^3*d*e^2)*b^3*x^2 + (c^4*d^3 - 3*c^3*d^2*e)*b^3*x 
)*log(c*x + 1)^3 + 12*(a*b^2*c^4*e^3*x^4 - a*b^2*c^3*d^3 + (3*c^4*d*e^2 - 
c^3*e^3)*a*b^2*x^3 + 3*(c^4*d^2*e - c^3*d*e^2)*a*b^2*x^2 + (c^4*d^3 - 3*c^ 
3*d^2*e)*a*b^2*x)*log(c*x + 1)^2 - (6*a*b^2*c^4*e^3*x^4 + 2*(12*a*b^2*c^4* 
d*e^2 + b^3*c^3*e^3)*x^3 + 12*(3*a*b^2*c^4*d^2*e + b^3*c^3*d*e^2)*x^2 + 6* 
(b^3*c^4*e^3*x^4 - b^3*c^3*d^3 + (3*c^4*d*e^2 - c^3*e^3)*b^3*x^3 + 3*(c^4* 
d^2*e - c^3*d*e^2)*b^3*x^2 + (c^4*d^3 - 3*c^3*d^2*e)*b^3*x)*log(c*x + 1...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(4*atanh(c*x)**3*b**3*c**4*d**3*x + 6*atanh(c*x)**3*b**3*c**4*d**2*e*x**2 
+ 4*atanh(c*x)**3*b**3*c**4*d*e**2*x**3 + atanh(c*x)**3*b**3*c**4*e**3*x** 
4 - 6*atanh(c*x)**3*b**3*c**2*d**2*e - atanh(c*x)**3*b**3*e**3 + 12*atanh( 
c*x)**2*a*b**2*c**4*d**3*x + 18*atanh(c*x)**2*a*b**2*c**4*d**2*e*x**2 + 12 
*atanh(c*x)**2*a*b**2*c**4*d*e**2*x**3 + 3*atanh(c*x)**2*a*b**2*c**4*e**3* 
x**4 - 18*atanh(c*x)**2*a*b**2*c**2*d**2*e - 3*atanh(c*x)**2*a*b**2*e**3 + 
 18*atanh(c*x)**2*b**3*c**3*d**2*e*x + 6*atanh(c*x)**2*b**3*c**3*d*e**2*x* 
*2 + atanh(c*x)**2*b**3*c**3*e**3*x**3 - 6*atanh(c*x)**2*b**3*c*d*e**2 + 3 
*atanh(c*x)**2*b**3*c*e**3*x + 12*atanh(c*x)*a**2*b*c**4*d**3*x + 18*atanh 
(c*x)*a**2*b*c**4*d**2*e*x**2 + 12*atanh(c*x)*a**2*b*c**4*d*e**2*x**3 + 3* 
atanh(c*x)*a**2*b*c**4*e**3*x**4 + 12*atanh(c*x)*a**2*b*c**3*d**3 - 18*ata 
nh(c*x)*a**2*b*c**2*d**2*e + 12*atanh(c*x)*a**2*b*c*d*e**2 - 3*atanh(c*x)* 
a**2*b*e**3 + 36*atanh(c*x)*a*b**2*c**3*d**2*e*x + 12*atanh(c*x)*a*b**2*c* 
*3*d*e**2*x**2 + 2*atanh(c*x)*a*b**2*c**3*e**3*x**3 + 36*atanh(c*x)*a*b**2 
*c**2*d**2*e - 12*atanh(c*x)*a*b**2*c*d*e**2 + 6*atanh(c*x)*a*b**2*c*e**3* 
x + 8*atanh(c*x)*a*b**2*e**3 + 12*atanh(c*x)*b**3*c**2*d*e**2*x + atanh(c* 
x)*b**3*c**2*e**3*x**2 + 12*atanh(c*x)*b**3*c*d*e**2 - atanh(c*x)*b**3*e** 
3 + 24*int((atanh(c*x)*x)/(c**2*x**2 - 1),x)*a*b**2*c**5*d**3 + 24*int((at 
anh(c*x)*x)/(c**2*x**2 - 1),x)*a*b**2*c**3*d*e**2 + 36*int((atanh(c*x)*x)/ 
(c**2*x**2 - 1),x)*b**3*c**4*d**2*e + 8*int((atanh(c*x)*x)/(c**2*x**2 -...