\(\int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx\) [90]

Optimal result

Integrand size = 22, antiderivative size = 385 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=-\frac {b c d^3 (a+b \text {arctanh}(c x))}{x}+\frac {9}{2} c^2 d^3 (a+b \text {arctanh}(c x))^2-\frac {d^3 (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{x}+c^3 d^3 x (a+b \text {arctanh}(c x))^2+6 c^2 d^3 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )+b^2 c^2 d^3 \log (x)-2 b c^2 d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^2 d^3 \log \left (1-c^2 x^2\right )+6 b c^2 d^3 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-b^2 c^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-3 b c^2 d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+3 b c^2 d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )-3 b^2 c^2 d^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )+\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \] Output:

-b*c*d^3*(a+b*arctanh(c*x))/x+9/2*c^2*d^3*(a+b*arctanh(c*x))^2-1/2*d^3*(a+ 
b*arctanh(c*x))^2/x^2-3*c*d^3*(a+b*arctanh(c*x))^2/x+c^3*d^3*x*(a+b*arctan 
h(c*x))^2-6*c^2*d^3*(a+b*arctanh(c*x))^2*arctanh(-1+2/(-c*x+1))+b^2*c^2*d^ 
3*ln(x)-2*b*c^2*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))-1/2*b^2*c^2*d^3*ln(- 
c^2*x^2+1)+6*b*c^2*d^3*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))-b^2*c^2*d^3*poly 
log(2,1-2/(-c*x+1))-3*b*c^2*d^3*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1)) 
+3*b*c^2*d^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(-c*x+1))-3*b^2*c^2*d^3*pol 
ylog(2,-1+2/(c*x+1))+3/2*b^2*c^2*d^3*polylog(3,1-2/(-c*x+1))-3/2*b^2*c^2*d 
^3*polylog(3,-1+2/(-c*x+1))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.74 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.20 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\frac {1}{2} d^3 \left (-\frac {a^2}{x^2}-\frac {6 a^2 c}{x}+2 a^2 c^3 x+6 a^2 c^2 \log (x)-\frac {a b (2 \text {arctanh}(c x)+c x (2+c x \log (1-c x)-c x \log (1+c x)))}{x^2}+\frac {b^2 \left (-2 c x \text {arctanh}(c x)+\left (-1+c^2 x^2\right ) \text {arctanh}(c x)^2+2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )\right )}{x^2}+2 a b c^2 \left (2 c x \text {arctanh}(c x)+\log \left (1-c^2 x^2\right )\right )-\frac {6 a b c \left (2 \text {arctanh}(c x)+c x \left (-2 \log (c x)+\log \left (1-c^2 x^2\right )\right )\right )}{x}+2 b^2 c^2 \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+\frac {6 b^2 c \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)+2 c x \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-c x \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{x}-6 a b c^2 (\operatorname {PolyLog}(2,-c x)-\operatorname {PolyLog}(2,c x))+6 b^2 c^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}(c x)^3-\text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )\right ) \] Input:

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

Output:

(d^3*(-(a^2/x^2) - (6*a^2*c)/x + 2*a^2*c^3*x + 6*a^2*c^2*Log[x] - (a*b*(2* 
ArcTanh[c*x] + c*x*(2 + c*x*Log[1 - c*x] - c*x*Log[1 + c*x])))/x^2 + (b^2* 
(-2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + 2*c^2*x^2*Log[(c*x) 
/Sqrt[1 - c^2*x^2]]))/x^2 + 2*a*b*c^2*(2*c*x*ArcTanh[c*x] + Log[1 - c^2*x^ 
2]) - (6*a*b*c*(2*ArcTanh[c*x] + c*x*(-2*Log[c*x] + Log[1 - c^2*x^2])))/x 
+ 2*b^2*c^2*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTa 
nh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + (6*b^2*c*(ArcTanh[c*x]*(( 
-1 + c*x)*ArcTanh[c*x] + 2*c*x*Log[1 - E^(-2*ArcTanh[c*x])]) - c*x*PolyLog 
[2, E^(-2*ArcTanh[c*x])]))/x - 6*a*b*c^2*(PolyLog[2, -(c*x)] - PolyLog[2, 
c*x]) + 6*b^2*c^2*((I/24)*Pi^3 - (2*ArcTanh[c*x]^3)/3 - ArcTanh[c*x]^2*Log 
[1 + E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + A 
rcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^( 
2*ArcTanh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c*x])]/2 - PolyLog[3, E^(2*Ar 
cTanh[c*x])]/2)))/2
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 385, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

\(\displaystyle c^3 d^3 x (a+b \text {arctanh}(c x))^2-3 b c^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+3 b c^2 d^3 \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))+\frac {9}{2} c^2 d^3 (a+b \text {arctanh}(c x))^2+6 c^2 d^3 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2-2 b c^2 d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+6 b c^2 d^3 \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {d^3 (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{x}-\frac {b c d^3 (a+b \text {arctanh}(c x))}{x}-b^2 c^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-3 b^2 c^2 d^3 \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )-\frac {1}{2} b^2 c^2 d^3 \log \left (1-c^2 x^2\right )+b^2 c^2 d^3 \log (x)\)

Input:

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

Output:

-((b*c*d^3*(a + b*ArcTanh[c*x]))/x) + (9*c^2*d^3*(a + b*ArcTanh[c*x])^2)/2 
 - (d^3*(a + b*ArcTanh[c*x])^2)/(2*x^2) - (3*c*d^3*(a + b*ArcTanh[c*x])^2) 
/x + c^3*d^3*x*(a + b*ArcTanh[c*x])^2 + 6*c^2*d^3*(a + b*ArcTanh[c*x])^2*A 
rcTanh[1 - 2/(1 - c*x)] + b^2*c^2*d^3*Log[x] - 2*b*c^2*d^3*(a + b*ArcTanh[ 
c*x])*Log[2/(1 - c*x)] - (b^2*c^2*d^3*Log[1 - c^2*x^2])/2 + 6*b*c^2*d^3*(a 
 + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b^2*c^2*d^3*PolyLog[2, 1 - 2/(1 
- c*x)] - 3*b*c^2*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + 3 
*b*c^2*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - 3*b^2*c^2*d 
^3*PolyLog[2, -1 + 2/(1 + c*x)] + (3*b^2*c^2*d^3*PolyLog[3, 1 - 2/(1 - c*x 
)])/2 - (3*b^2*c^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 = 4.48 (sec) , antiderivative size = 1086, normalized size of antiderivative = 2.82

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1086\)
default	\(\text {Expression too large to display}\)	\(1086\)
parts	\(\text {Expression too large to display}\)	\(1086\)

Input:

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

Output:

c^2*(d^3*a^2*(c*x-1/2/c^2/x^2+3*ln(c*x)-3/c/x)+d^3*b^2*(3*arctanh(c*x)^2*l 
n(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*arctanh(c*x)^2*ln(c*x)+6*arctanh(c*x)*ln(1+(c*x+1)/(- 
c^2*x^2+1)^(1/2))-2*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-2*arct 
anh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-2*dilog(1+I*(c*x+1)/(-c^2*x^2+ 
1)^(1/2))-2*dilog(1-I*(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+arctanh(c*x)^2*c*x-6*dilog((c*x+1)/(-c 
^2*x^2+1)^(1/2))+6*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-3/2*arctanh(c*x)^2+ 
3/2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-3*arctanh(c*x)*polylog(2,-(c*x+1)^2 
/(-c^2*x^2+1))-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*cs 
gn(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*pol 
ylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))-6*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2) 
)-3*arctanh(c*x)^2/c/x-1/2*(c*x-(-c^2*x^2+1)^(1/2)+1)/c/x*arctanh(c*x)-1/2 
*arctanh(c*x)*(c*x+(-c^2*x^2+1)^(1/2)+1)/c/x+3/2*I*Pi*csgn(I*(-(c*x+1)^...
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\frac {d^{3} \left (\mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-6 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -\mathit {atanh} \left (c x \right )^{2} b^{2}+4 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}-6 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}-12 \mathit {atanh} \left (c x \right ) a b c x -2 \mathit {atanh} \left (c x \right ) a b -2 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}-2 \mathit {atanh} \left (c x \right ) b^{2} c x +2 \left (\int \mathit {atanh} \left (c x \right )^{2}d x \right ) b^{2} c^{3} x^{2}-12 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{3}-x}d x \right ) b^{2} c^{2} x^{2}+12 \left (\int \frac {\mathit {atanh} \left (c x \right )}{x}d x \right ) a b \,c^{2} x^{2}+6 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2}}{x}d x \right ) b^{2} c^{2} x^{2}-8 \,\mathrm {log}\left (c^{2} x -c \right ) a b \,c^{2} x^{2}-2 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} c^{2} x^{2}+6 \,\mathrm {log}\left (x \right ) a^{2} c^{2} x^{2}+12 \,\mathrm {log}\left (x \right ) a b \,c^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) b^{2} c^{2} x^{2}+2 a^{2} c^{3} x^{3}-6 a^{2} c x -a^{2}-2 a b c x \right )}{2 x^{2}} \] Input:

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

Output:

(d**3*(atanh(c*x)**2*b**2*c**2*x**2 - 6*atanh(c*x)**2*b**2*c*x - atanh(c*x 
)**2*b**2 + 4*atanh(c*x)*a*b*c**3*x**3 - 6*atanh(c*x)*a*b*c**2*x**2 - 12*a 
tanh(c*x)*a*b*c*x - 2*atanh(c*x)*a*b - 2*atanh(c*x)*b**2*c**2*x**2 - 2*ata 
nh(c*x)*b**2*c*x + 2*int(atanh(c*x)**2,x)*b**2*c**3*x**2 - 12*int(atanh(c* 
x)/(c**2*x**3 - x),x)*b**2*c**2*x**2 + 12*int(atanh(c*x)/x,x)*a*b*c**2*x** 
2 + 6*int(atanh(c*x)**2/x,x)*b**2*c**2*x**2 - 8*log(c**2*x - c)*a*b*c**2*x 
**2 - 2*log(c**2*x - c)*b**2*c**2*x**2 + 6*log(x)*a**2*c**2*x**2 + 12*log( 
x)*a*b*c**2*x**2 + 2*log(x)*b**2*c**2*x**2 + 2*a**2*c**3*x**3 - 6*a**2*c*x 
 - a**2 - 2*a*b*c*x))/(2*x**2)