\(\int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx\) [88]

Optimal result

Integrand size = 22, antiderivative size = 355 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx=3 a b c d^3 x+\frac {1}{3} b^2 c d^3 x-\frac {1}{3} b^2 d^3 \text {arctanh}(c x)+3 b^2 c d^3 x \text {arctanh}(c x)+\frac {1}{3} b c^2 d^3 x^2 (a+b \text {arctanh}(c x))+\frac {11}{6} d^3 (a+b \text {arctanh}(c x))^2+3 c d^3 x (a+b \text {arctanh}(c x))^2+\frac {3}{2} c^2 d^3 x^2 (a+b \text {arctanh}(c x))^2+\frac {1}{3} c^3 d^3 x^3 (a+b \text {arctanh}(c x))^2+2 d^3 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )-\frac {20}{3} b d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )+\frac {3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )-\frac {10}{3} b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-b d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \] Output:

3*a*b*c*d^3*x+1/3*b^2*c*d^3*x-1/3*b^2*d^3*arctanh(c*x)+3*b^2*c*d^3*x*arcta 
nh(c*x)+1/3*b*c^2*d^3*x^2*(a+b*arctanh(c*x))+11/6*d^3*(a+b*arctanh(c*x))^2 
+3*c*d^3*x*(a+b*arctanh(c*x))^2+3/2*c^2*d^3*x^2*(a+b*arctanh(c*x))^2+1/3*c 
^3*d^3*x^3*(a+b*arctanh(c*x))^2-2*d^3*(a+b*arctanh(c*x))^2*arctanh(-1+2/(- 
c*x+1))-20/3*b*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))+3/2*b^2*d^3*ln(-c^2*x 
^2+1)-10/3*b^2*d^3*polylog(2,1-2/(-c*x+1))-b*d^3*(a+b*arctanh(c*x))*polylo 
g(2,1-2/(-c*x+1))+b*d^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(-c*x+1))+1/2*b^ 
2*d^3*polylog(3,1-2/(-c*x+1))-1/2*b^2*d^3*polylog(3,-1+2/(-c*x+1))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.26 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx=\frac {1}{24} d^3 \left (i b^2 \pi ^3+72 a^2 c x+72 a b c x+8 b^2 c x+36 a^2 c^2 x^2+8 a b c^2 x^2+8 a^2 c^3 x^3-8 b^2 \text {arctanh}(c x)+144 a b c x \text {arctanh}(c x)+72 b^2 c x \text {arctanh}(c x)+72 a b c^2 x^2 \text {arctanh}(c x)+8 b^2 c^2 x^2 \text {arctanh}(c x)+16 a b c^3 x^3 \text {arctanh}(c x)-116 b^2 \text {arctanh}(c x)^2+72 b^2 c x \text {arctanh}(c x)^2+36 b^2 c^2 x^2 \text {arctanh}(c x)^2+8 b^2 c^3 x^3 \text {arctanh}(c x)^2-16 b^2 \text {arctanh}(c x)^3-160 b^2 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-24 b^2 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+24 b^2 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+24 a^2 \log (c x)+36 a b \log (1-c x)-36 a b \log (1+c x)+72 a b \log \left (1-c^2 x^2\right )+36 b^2 \log \left (1-c^2 x^2\right )+8 a b \log \left (-1+c^2 x^2\right )+8 b^2 (10+3 \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+24 b^2 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-24 a b \operatorname {PolyLog}(2,-c x)+24 a b \operatorname {PolyLog}(2,c x)+12 b^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-12 b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right ) \] Input:

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

Output:

(d^3*(I*b^2*Pi^3 + 72*a^2*c*x + 72*a*b*c*x + 8*b^2*c*x + 36*a^2*c^2*x^2 + 
8*a*b*c^2*x^2 + 8*a^2*c^3*x^3 - 8*b^2*ArcTanh[c*x] + 144*a*b*c*x*ArcTanh[c 
*x] + 72*b^2*c*x*ArcTanh[c*x] + 72*a*b*c^2*x^2*ArcTanh[c*x] + 8*b^2*c^2*x^ 
2*ArcTanh[c*x] + 16*a*b*c^3*x^3*ArcTanh[c*x] - 116*b^2*ArcTanh[c*x]^2 + 72 
*b^2*c*x*ArcTanh[c*x]^2 + 36*b^2*c^2*x^2*ArcTanh[c*x]^2 + 8*b^2*c^3*x^3*Ar 
cTanh[c*x]^2 - 16*b^2*ArcTanh[c*x]^3 - 160*b^2*ArcTanh[c*x]*Log[1 + E^(-2* 
ArcTanh[c*x])] - 24*b^2*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 24*b 
^2*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*a^2*Log[c*x] + 36*a*b*L 
og[1 - c*x] - 36*a*b*Log[1 + c*x] + 72*a*b*Log[1 - c^2*x^2] + 36*b^2*Log[1 
 - c^2*x^2] + 8*a*b*Log[-1 + c^2*x^2] + 8*b^2*(10 + 3*ArcTanh[c*x])*PolyLo 
g[2, -E^(-2*ArcTanh[c*x])] + 24*b^2*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c 
*x])] - 24*a*b*PolyLog[2, -(c*x)] + 24*a*b*PolyLog[2, c*x] + 12*b^2*PolyLo 
g[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*PolyLog[3, E^(2*ArcTanh[c*x])]))/24
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 355, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} c^3 d^3 x^3 (a+b \text {arctanh}(c x))^2+\frac {3}{2} c^2 d^3 x^2 (a+b \text {arctanh}(c x))^2+\frac {1}{3} b c^2 d^3 x^2 (a+b \text {arctanh}(c x))-b d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+b d^3 \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))+3 c d^3 x (a+b \text {arctanh}(c x))^2+\frac {11}{6} d^3 (a+b \text {arctanh}(c x))^2+2 d^3 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2-\frac {20}{3} b d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+3 a b c d^3 x-\frac {1}{3} b^2 d^3 \text {arctanh}(c x)+3 b^2 c d^3 x \text {arctanh}(c x)+\frac {3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )-\frac {10}{3} b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )+\frac {1}{3} b^2 c d^3 x\)

Input:

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

Output:

3*a*b*c*d^3*x + (b^2*c*d^3*x)/3 - (b^2*d^3*ArcTanh[c*x])/3 + 3*b^2*c*d^3*x 
*ArcTanh[c*x] + (b*c^2*d^3*x^2*(a + b*ArcTanh[c*x]))/3 + (11*d^3*(a + b*Ar 
cTanh[c*x])^2)/6 + 3*c*d^3*x*(a + b*ArcTanh[c*x])^2 + (3*c^2*d^3*x^2*(a + 
b*ArcTanh[c*x])^2)/2 + (c^3*d^3*x^3*(a + b*ArcTanh[c*x])^2)/3 + 2*d^3*(a + 
 b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] - (20*b*d^3*(a + b*ArcTanh[c*x 
])*Log[2/(1 - c*x)])/3 + (3*b^2*d^3*Log[1 - c^2*x^2])/2 - (10*b^2*d^3*Poly 
Log[2, 1 - 2/(1 - c*x)])/3 - b*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/( 
1 - c*x)] + b*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] + (b^2 
*d^3*PolyLog[3, 1 - 2/(1 - c*x)])/2 - (b^2*d^3*PolyLog[3, -1 + 2/(1 - c*x) 
])/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 = 3.38 (sec) , antiderivative size = 959, normalized size of antiderivative = 2.70

method	result	size
parts	\(\text {Expression too large to display}\)	\(959\)
derivativedivides	\(\text {Expression too large to display}\)	\(961\)
default	\(\text {Expression too large to display}\)	\(961\)

Input:

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

Output:

d^3*a^2*(1/3*x^3*c^3+3/2*c^2*x^2+3*c*x+ln(x))+d^3*b^2*(arctanh(c*x)^2*ln(1 
-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+1) 
^(1/2))+arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*pol 
ylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+ 
1)-1)+1/2*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)))*arc 
tanh(c*x)^2+arctanh(c*x)^2*ln(c*x)-1/3-20/3*arctanh(c*x)*ln(1+I*(c*x+1)/(- 
c^2*x^2+1)^(1/2))-20/3*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-20/ 
3*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-20/3*dilog(1-I*(c*x+1)/(-c^2*x^2+1 
)^(1/2))+1/3*c*x+1/3*arctanh(c*x)^2*c^3*x^3+3/2*arctanh(c*x)^2*c^2*x^2+3*a 
rctanh(c*x)^2*c*x+11/6*arctanh(c*x)^2+1/2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1 
))+11/3*(c*x+1)*arctanh(c*x)-3*ln(1+(c*x+1)^2/(-c^2*x^2+1))-arctanh(c*x)*p 
olylog(2,-(c*x+1)^2/(-c^2*x^2+1))-2*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2)) 
-2*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))+1/3*(c*x-3)*(c*x+1)*arctanh(c*x)- 
1/2*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)^2-1/2*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+1/2*I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1) 
^2/(c^2*x^2-1)))^3*arctanh(c*x)^2)+2*d^3*a*b*(1/3*arctanh(c*x)*c^3*x^3+3/2 
*arctanh(c*x)*c^2*x^2+3*arctanh(c*x)*c*x+arctanh(c*x)*ln(c*x)-1/2*dilog...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx=d^{3} \left (\int 3 a^{2} c\, dx + \int \frac {a^{2}}{x}\, dx + \int 3 a^{2} c^{2} x\, dx + \int a^{2} c^{3} x^{2}\, dx + \int 3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int 6 a b c \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{2} x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{2} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \] Input:

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

Output:

d**3*(Integral(3*a**2*c, x) + Integral(a**2/x, x) + Integral(3*a**2*c**2*x 
, x) + Integral(a**2*c**3*x**2, x) + Integral(3*b**2*c*atanh(c*x)**2, x) + 
 Integral(b**2*atanh(c*x)**2/x, x) + Integral(6*a*b*c*atanh(c*x), x) + Int 
egral(2*a*b*atanh(c*x)/x, x) + Integral(3*b**2*c**2*x*atanh(c*x)**2, x) + 
Integral(b**2*c**3*x**2*atanh(c*x)**2, x) + Integral(6*a*b*c**2*x*atanh(c* 
x), x) + Integral(2*a*b*c**3*x**2*atanh(c*x), x))
 

Maxima [F]

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

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

Output:

1/3*a^2*c^3*d^3*x^3 + 3/2*a^2*c^2*d^3*x^2 + 3*a^2*c*d^3*x + 3*(2*c*x*arcta 
nh(c*x) + log(-c^2*x^2 + 1))*a*b*d^3 + a^2*d^3*log(x) + 1/24*(2*b^2*c^3*d^ 
3*x^3 + 9*b^2*c^2*d^3*x^2 + 18*b^2*c*d^3*x)*log(-c*x + 1)^2 - integrate(-1 
/12*(3*(b^2*c^4*d^3*x^4 + 2*b^2*c^3*d^3*x^3 - 2*b^2*c*d^3*x - b^2*d^3)*log 
(c*x + 1)^2 + 12*(a*b*c^4*d^3*x^4 + 2*a*b*c^3*d^3*x^3 - 3*a*b*c^2*d^3*x^2 
+ a*b*c*d^3*x - a*b*d^3)*log(c*x + 1) - (12*a*b*c*d^3*x - 12*a*b*d^3 + 2*( 
6*a*b*c^4*d^3 + b^2*c^4*d^3)*x^4 + 3*(8*a*b*c^3*d^3 + 3*b^2*c^3*d^3)*x^3 - 
 18*(2*a*b*c^2*d^3 - b^2*c^2*d^3)*x^2 + 6*(b^2*c^4*d^3*x^4 + 2*b^2*c^3*d^3 
*x^3 - 2*b^2*c*d^3*x - b^2*d^3)*log(c*x + 1))*log(-c*x + 1))/(c*x^2 - x), 
x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x} \, dx=\frac {d^{3} \left (2 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}+9 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-2 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -9 \mathit {atanh} \left (c x \right )^{2} b^{2}+4 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}+18 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}+36 \mathit {atanh} \left (c x \right ) a b c x +22 \mathit {atanh} \left (c x \right ) a b +2 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}+18 \mathit {atanh} \left (c x \right ) b^{2} c x +16 \mathit {atanh} \left (c x \right ) b^{2}+20 \left (\int \mathit {atanh} \left (c x \right )^{2}d x \right ) b^{2} c +12 \left (\int \frac {\mathit {atanh} \left (c x \right )}{x}d x \right ) a b +6 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2}}{x}d x \right ) b^{2}+40 \,\mathrm {log}\left (c^{2} x -c \right ) a b +18 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2}+6 \,\mathrm {log}\left (x \right ) a^{2}+2 a^{2} c^{3} x^{3}+9 a^{2} c^{2} x^{2}+18 a^{2} c x +2 a b \,c^{2} x^{2}+18 a b c x +2 b^{2} c x \right )}{6} \] Input:

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

Output:

(d**3*(2*atanh(c*x)**2*b**2*c**3*x**3 + 9*atanh(c*x)**2*b**2*c**2*x**2 - 2 
*atanh(c*x)**2*b**2*c*x - 9*atanh(c*x)**2*b**2 + 4*atanh(c*x)*a*b*c**3*x** 
3 + 18*atanh(c*x)*a*b*c**2*x**2 + 36*atanh(c*x)*a*b*c*x + 22*atanh(c*x)*a* 
b + 2*atanh(c*x)*b**2*c**2*x**2 + 18*atanh(c*x)*b**2*c*x + 16*atanh(c*x)*b 
**2 + 20*int(atanh(c*x)**2,x)*b**2*c + 12*int(atanh(c*x)/x,x)*a*b + 6*int( 
atanh(c*x)**2/x,x)*b**2 + 40*log(c**2*x - c)*a*b + 18*log(c**2*x - c)*b**2 
 + 6*log(x)*a**2 + 2*a**2*c**3*x**3 + 9*a**2*c**2*x**2 + 18*a**2*c*x + 2*a 
*b*c**2*x**2 + 18*a*b*c*x + 2*b**2*c*x))/6