\(\int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)^2} \, dx\) [110]

Optimal result

Integrand size = 22, antiderivative size = 480 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)^2} \, dx=\frac {b^2 c^2}{2 d^2 (1+c x)}-\frac {b^2 c^2 \text {arctanh}(c x)}{2 d^2}-\frac {b c (a+b \text {arctanh}(c x))}{d^2 x}+\frac {b c^2 (a+b \text {arctanh}(c x))}{d^2 (1+c x)}-\frac {2 c^2 (a+b \text {arctanh}(c x))^2}{d^2}-\frac {(a+b \text {arctanh}(c x))^2}{2 d^2 x^2}+\frac {2 c (a+b \text {arctanh}(c x))^2}{d^2 x}+\frac {c^2 (a+b \text {arctanh}(c x))^2}{d^2 (1+c x)}+\frac {6 c^2 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}+\frac {3 c^2 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}-\frac {4 b c^2 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )}{d^2}-\frac {3 b c^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 b c^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )}{d^2}-\frac {3 b c^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d^2}+\frac {2 b^2 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d^2}+\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 d^2} \] Output:

1/2*b^2*c^2/d^2/(c*x+1)-1/2*b^2*c^2*arctanh(c*x)/d^2-b*c*(a+b*arctanh(c*x) 
)/d^2/x+b*c^2*(a+b*arctanh(c*x))/d^2/(c*x+1)-2*c^2*(a+b*arctanh(c*x))^2/d^ 
2-1/2*(a+b*arctanh(c*x))^2/d^2/x^2+2*c*(a+b*arctanh(c*x))^2/d^2/x+c^2*(a+b 
*arctanh(c*x))^2/d^2/(c*x+1)-6*c^2*(a+b*arctanh(c*x))^2*arctanh(-1+2/(-c*x 
+1))/d^2+b^2*c^2*ln(x)/d^2+3*c^2*(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/d^2-1/ 
2*b^2*c^2*ln(-c^2*x^2+1)/d^2-4*b*c^2*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))/d^ 
2-3*b*c^2*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/d^2+3*b*c^2*(a+b*arct 
anh(c*x))*polylog(2,-1+2/(-c*x+1))/d^2-3*b*c^2*(a+b*arctanh(c*x))*polylog( 
2,1-2/(c*x+1))/d^2+2*b^2*c^2*polylog(2,-1+2/(c*x+1))/d^2+3/2*b^2*c^2*polyl 
og(3,1-2/(-c*x+1))/d^2-3/2*b^2*c^2*polylog(3,-1+2/(-c*x+1))/d^2-3/2*b^2*c^ 
2*polylog(3,1-2/(c*x+1))/d^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.56 (sec) , antiderivative size = 452, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)^2} \, dx=\frac {-\frac {4 a^2}{x^2}+\frac {16 a^2 c}{x}+\frac {8 a^2 c^2}{1+c x}+24 a^2 c^2 \log (x)-24 a^2 c^2 \log (1+c x)+b^2 c^2 \left (i \pi ^3-\frac {8 \text {arctanh}(c x)}{c x}-12 \text {arctanh}(c x)^2-\frac {4 \text {arctanh}(c x)^2}{c^2 x^2}+\frac {16 \text {arctanh}(c x)^2}{c x}-16 \text {arctanh}(c x)^3+2 \cosh (2 \text {arctanh}(c x))+4 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))+4 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))-32 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )+24 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+8 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+16 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+24 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-12 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )-2 \sinh (2 \text {arctanh}(c x))-4 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))-4 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))\right )+\frac {4 a b \left (-6 c^2 x^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+c x \left (-2+c x \cosh (2 \text {arctanh}(c x))-8 c x \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-c x \sinh (2 \text {arctanh}(c x))\right )+2 \text {arctanh}(c x) \left (-1+4 c x+c^2 x^2+c^2 x^2 \cosh (2 \text {arctanh}(c x))+6 c^2 x^2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-c^2 x^2 \sinh (2 \text {arctanh}(c x))\right )\right )}{x^2}}{8 d^2} \] Input:

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

Output:

((-4*a^2)/x^2 + (16*a^2*c)/x + (8*a^2*c^2)/(1 + c*x) + 24*a^2*c^2*Log[x] - 
 24*a^2*c^2*Log[1 + c*x] + b^2*c^2*(I*Pi^3 - (8*ArcTanh[c*x])/(c*x) - 12*A 
rcTanh[c*x]^2 - (4*ArcTanh[c*x]^2)/(c^2*x^2) + (16*ArcTanh[c*x]^2)/(c*x) - 
 16*ArcTanh[c*x]^3 + 2*Cosh[2*ArcTanh[c*x]] + 4*ArcTanh[c*x]*Cosh[2*ArcTan 
h[c*x]] + 4*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - 32*ArcTanh[c*x]*Log[1 - 
E^(-2*ArcTanh[c*x])] + 24*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 8*L 
og[(c*x)/Sqrt[1 - c^2*x^2]] + 16*PolyLog[2, E^(-2*ArcTanh[c*x])] + 24*ArcT 
anh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 12*PolyLog[3, E^(2*ArcTanh[c*x]) 
] - 2*Sinh[2*ArcTanh[c*x]] - 4*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] - 4*ArcTa 
nh[c*x]^2*Sinh[2*ArcTanh[c*x]]) + (4*a*b*(-6*c^2*x^2*PolyLog[2, E^(-2*ArcT 
anh[c*x])] + c*x*(-2 + c*x*Cosh[2*ArcTanh[c*x]] - 8*c*x*Log[(c*x)/Sqrt[1 - 
 c^2*x^2]] - c*x*Sinh[2*ArcTanh[c*x]]) + 2*ArcTanh[c*x]*(-1 + 4*c*x + c^2* 
x^2 + c^2*x^2*Cosh[2*ArcTanh[c*x]] + 6*c^2*x^2*Log[1 - E^(-2*ArcTanh[c*x]) 
] - c^2*x^2*Sinh[2*ArcTanh[c*x]])))/x^2)/(8*d^2)
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6502, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 6502

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

Maple [C] (warning: unable to verify)

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

Time = 6.03 (sec) , antiderivative size = 1572, normalized size of antiderivative = 3.28

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1572\)
default	\(\text {Expression too large to display}\)	\(1572\)
parts	\(\text {Expression too large to display}\)	\(1577\)

Input:

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

Output:

c^2*(a^2/d^2*(-1/2/c^2/x^2+2/c/x+3*ln(c*x)+1/(c*x+1)-3*ln(c*x+1))+b^2/d^2* 
(3*arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+6*arctanh(c*x)*polylog( 
2,(c*x+1)/(-c^2*x^2+1)^(1/2))+3*arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^( 
1/2))+6*arctanh(c*x)*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-3*arctanh(c*x) 
^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)+3/2*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1))) 
*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-( 
c*x+1)^2/(c^2*x^2-1)))*arctanh(c*x)^2-3/2*I*Pi*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*arctanh(c*x)^2*ln(c*x)-3*arctanh(c*x)^ 
2*ln(c*x+1)-4*arctanh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+ln((c*x+1)/(-c 
^2*x^2+1)^(1/2)-1)-1/2*arctanh(c*x)^2/c^2/x^2-2*arctanh(c*x)^3+4*dilog((c* 
x+1)/(-c^2*x^2+1)^(1/2))-4*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-1/4/(c*x+1) 
*(c*x-1)+2*arctanh(c*x)^2-3/2*I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*csgn 
(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2*arctanh(c*x)^2- 
3/2*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)- 
1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2*arctanh(c*x)^2+ln(1+(c*x+1)/(-c^2*x^2+1)^( 
1/2))-6*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))-6*polylog(3,(c*x+1)/(-c^2*x 
^2+1)^(1/2))+2*arctanh(c*x)^2/c/x-3/2*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*c 
sgn(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*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1...
 

Fricas [F]

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

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

Output:

integral((b^2*arctanh(c*x)^2 + 2*a*b*arctanh(c*x) + a^2)/(c^2*d^2*x^5 + 2* 
c*d^2*x^4 + d^2*x^3), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)^2} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx}{d^{2}} \] Input:

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

Output:

(Integral(a**2/(c**2*x**5 + 2*c*x**4 + x**3), x) + Integral(b**2*atanh(c*x 
)**2/(c**2*x**5 + 2*c*x**4 + x**3), x) + Integral(2*a*b*atanh(c*x)/(c**2*x 
**5 + 2*c*x**4 + x**3), x))/d**2
 

Maxima [F]

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

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

Output:

-1/2*a^2*(6*c^2*log(c*x + 1)/d^2 - 6*c^2*log(x)/d^2 - (6*c^2*x^2 + 3*c*x - 
 1)/(c*d^2*x^3 + d^2*x^2)) + 1/8*(6*b^2*c^2*x^2 + 3*b^2*c*x - b^2 - 6*(b^2 
*c^3*x^3 + b^2*c^2*x^2)*log(c*x + 1))*log(-c*x + 1)^2/(c*d^2*x^3 + d^2*x^2 
) + integrate(1/4*((b^2*c*x - b^2)*log(c*x + 1)^2 + 4*(a*b*c*x - a*b)*log( 
c*x + 1) - (6*b^2*c^4*x^4 + 9*b^2*c^3*x^3 + 2*b^2*c^2*x^2 - 4*a*b + (4*a*b 
*c - b^2*c)*x - 2*(3*b^2*c^5*x^5 + 6*b^2*c^4*x^4 + 3*b^2*c^3*x^3 - b^2*c*x 
 + b^2)*log(c*x + 1))*log(-c*x + 1))/(c^3*d^2*x^6 + c^2*d^2*x^5 - c*d^2*x^ 
4 - d^2*x^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*atanh(c*x)**3*b**2*c**3*x**3 - 2*atanh(c*x)**3*b**2*c**2*x**2 - 6*at 
anh(c*x)**2*a*b*c**3*x**3 - 6*atanh(c*x)**2*a*b*c**2*x**2 - 6*atanh(c*x)** 
2*b**2*c**3*x**3 + 6*atanh(c*x)**2*b**2*c*x + 2*atanh(c*x)**2*b**2 - 12*at 
anh(c*x)*a*b*c**3*x**3 + 12*atanh(c*x)*a*b*c*x + 4*atanh(c*x)*a*b - 12*ata 
nh(c*x)*b**2*c**3*x**3 + 6*atanh(c*x)*b**2*c*x - 2*atanh(c*x)*b**2 - 24*in 
t(atanh(c*x)/(c**3*x**6 + c**2*x**5 - c*x**4 - x**3),x)*a*b*c*x**3 - 24*in 
t(atanh(c*x)/(c**3*x**6 + c**2*x**5 - c*x**4 - x**3),x)*a*b*x**2 + 4*int(a 
tanh(c*x)/(c**3*x**6 + c**2*x**5 - c*x**4 - x**3),x)*b**2*c*x**3 + 4*int(a 
tanh(c*x)/(c**3*x**6 + c**2*x**5 - c*x**4 - x**3),x)*b**2*x**2 - 12*int(at 
anh(c*x)**2/(c**3*x**6 + c**2*x**5 - c*x**4 - x**3),x)*b**2*c*x**3 - 12*in 
t(atanh(c*x)**2/(c**3*x**6 + c**2*x**5 - c*x**4 - x**3),x)*b**2*x**2 + log 
(c*x - 1)*a*b*c**3*x**3 + log(c*x - 1)*a*b*c**2*x**2 - 2*log(c*x - 1)*b**2 
*c**3*x**3 - 2*log(c*x - 1)*b**2*c**2*x**2 - 24*log(c*x + 1)*a**2*c**3*x** 
3 - 24*log(c*x + 1)*a**2*c**2*x**2 + 7*log(c*x + 1)*a*b*c**3*x**3 + 7*log( 
c*x + 1)*a*b*c**2*x**2 + 10*log(c*x + 1)*b**2*c**3*x**3 + 10*log(c*x + 1)* 
b**2*c**2*x**2 + 24*log(x)*a**2*c**3*x**3 + 24*log(x)*a**2*c**2*x**2 - 8*l 
og(x)*a*b*c**3*x**3 - 8*log(x)*a*b*c**2*x**2 - 8*log(x)*b**2*c**3*x**3 - 8 
*log(x)*b**2*c**2*x**2 - 24*a**2*c**3*x**3 + 12*a**2*c*x - 4*a**2 - 6*a*b* 
c**3*x**3 + 4*a*b*c*x - 2*b**2*c*x)/(8*d**2*x**2*(c*x + 1))