\(\int \frac {(a+b \text {arctanh}(\frac {c}{x}))^3}{x} \, dx\) [154]

Optimal result

Integrand size = 16, antiderivative size = 208 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x} \, dx=-2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3 \text {arctanh}\left (1-\frac {2}{1-\frac {c}{x}}\right )+\frac {3}{2} b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right )-\frac {3}{2} b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {c}{x}}\right )-\frac {3}{2} b^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-\frac {c}{x}}\right )+\frac {3}{2} b^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-\frac {c}{x}}\right )+\frac {3}{4} b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-\frac {c}{x}}\right )-\frac {3}{4} b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-\frac {c}{x}}\right ) \] Output:

2*(a+b*arccoth(x/c))^3*arctanh(-1+2/(1-c/x))+3/2*b*(a+b*arccoth(x/c))^2*po 
lylog(2,1-2/(1-c/x))-3/2*b*(a+b*arccoth(x/c))^2*polylog(2,-1+2/(1-c/x))-3/ 
2*b^2*(a+b*arccoth(x/c))*polylog(3,1-2/(1-c/x))+3/2*b^2*(a+b*arccoth(x/c)) 
*polylog(3,-1+2/(1-c/x))+3/4*b^3*polylog(4,1-2/(1-c/x))-3/4*b^3*polylog(4, 
-1+2/(1-c/x))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x} \, dx=a^3 \log (x)+\frac {3}{2} a^2 b \left (\operatorname {PolyLog}\left (2,-\frac {c}{x}\right )-\operatorname {PolyLog}\left (2,\frac {c}{x}\right )\right )+3 a b^2 \left (-\frac {i \pi ^3}{24}+\frac {2}{3} \text {arctanh}\left (\frac {c}{x}\right )^3+\text {arctanh}\left (\frac {c}{x}\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-\text {arctanh}\left (\frac {c}{x}\right )^2 \log \left (1-e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-\text {arctanh}\left (\frac {c}{x}\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-\text {arctanh}\left (\frac {c}{x}\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )+\frac {1}{64} b^3 \left (-\pi ^4+32 \text {arctanh}\left (\frac {c}{x}\right )^4+64 \text {arctanh}\left (\frac {c}{x}\right )^3 \log \left (1+e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-64 \text {arctanh}\left (\frac {c}{x}\right )^3 \log \left (1-e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-96 \text {arctanh}\left (\frac {c}{x}\right )^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-96 \text {arctanh}\left (\frac {c}{x}\right )^2 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-96 \text {arctanh}\left (\frac {c}{x}\right ) \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )+96 \text {arctanh}\left (\frac {c}{x}\right ) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-48 \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-48 \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right ) \] Input:

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

Output:

a^3*Log[x] + (3*a^2*b*(PolyLog[2, -(c/x)] - PolyLog[2, c/x]))/2 + 3*a*b^2* 
((-1/24*I)*Pi^3 + (2*ArcTanh[c/x]^3)/3 + ArcTanh[c/x]^2*Log[1 + E^(-2*ArcT 
anh[c/x])] - ArcTanh[c/x]^2*Log[1 - E^(2*ArcTanh[c/x])] - ArcTanh[c/x]*Pol 
yLog[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*ArcTanh[c/x])]/2) 
 + (b^3*(-Pi^4 + 32*ArcTanh[c/x]^4 + 64*ArcTanh[c/x]^3*Log[1 + E^(-2*ArcTa 
nh[c/x])] - 64*ArcTanh[c/x]^3*Log[1 - E^(2*ArcTanh[c/x])] - 96*ArcTanh[c/x 
]^2*PolyLog[2, -E^(-2*ArcTanh[c/x])] - 96*ArcTanh[c/x]^2*PolyLog[2, E^(2*A 
rcTanh[c/x])] - 96*ArcTanh[c/x]*PolyLog[3, -E^(-2*ArcTanh[c/x])] + 96*ArcT 
anh[c/x]*PolyLog[3, E^(2*ArcTanh[c/x])] - 48*PolyLog[4, -E^(-2*ArcTanh[c/x 
])] - 48*PolyLog[4, E^(2*ArcTanh[c/x])]))/64
 

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6450, 6448, 6614, 6620, 6624, 7164}

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.

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

\(\Big \downarrow \) 6450

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

\(\Big \downarrow \) 6448

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

\(\Big \downarrow \) 6614

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 6624

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

-2*ArcTanh[1 - 2/(1 - c/x)]*(a + b*ArcTanh[c/x])^3 + 6*b*c*((((a + b*ArcTa 
nh[c/x])^2*PolyLog[2, 1 - 2/(1 - c/x)])/(2*c) - b*(((a + b*ArcTanh[c/x])*P 
olyLog[3, 1 - 2/(1 - c/x)])/(2*c) - (b*PolyLog[4, 1 - 2/(1 - c/x)])/(4*c)) 
)/2 + (-1/2*((a + b*ArcTanh[c/x])^2*PolyLog[2, -1 + 2/(1 - c/x)])/c + b*(( 
(a + b*ArcTanh[c/x])*PolyLog[3, -1 + 2/(1 - c/x)])/(2*c) - (b*PolyLog[4, - 
1 + 2/(1 - c/x)])/(4*c)))/2)
 

Defintions of rubi rules used

rule 6448

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + 
 b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1 - c*x)], x] - Simp[2*b*c*p   Int[(a + b 
*ArcTanh[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 - c*x)]/(1 - c^2*x^2)), x], x] /; 
FreeQ[{a, b, c}, x] && IGtQ[p, 1]
 

rule 6450

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[ 
1/n   Subst[Int[(a + b*ArcTanh[c*x])^p/x, x], x, x^n], x] /; FreeQ[{a, b, c 
, n}, x] && IGtQ[p, 0]
 

rule 6614

Int[(ArcTanh[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*( 
x_)^2), x_Symbol] :> Simp[1/2   Int[Log[1 + u]*((a + b*ArcTanh[c*x])^p/(d + 
 e*x^2)), x], x] - Simp[1/2   Int[Log[1 - u]*((a + b*ArcTanh[c*x])^p/(d + e 
*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 
 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
 

rule 6620

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 
2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)) 
, x] + Simp[b*(p/2)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/( 
d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d 
 + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
 

rule 6624

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_ 
.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[k + 1, u]/(2* 
c*d)), x] - Simp[b*(p/2)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[k + 1, 
 u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && E 
qQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
 

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

Maple [C] (warning: unable to verify)

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

Time = 15.62 (sec) , antiderivative size = 1452, normalized size of antiderivative = 6.98

method	result	size
parts	\(\text {Expression too large to display}\)	\(1452\)
derivativedivides	\(\text {Expression too large to display}\)	\(1454\)
default	\(\text {Expression too large to display}\)	\(1454\)

Input:

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

Output:

a^3*ln(x)+b^3*(-ln(c/x)*arctanh(c/x)^3+arctanh(c/x)^3*ln((1+c/x)^2/(1-c^2/ 
x^2)-1)-arctanh(c/x)^3*ln(1-(1+c/x)/(1-c^2/x^2)^(1/2))-3*arctanh(c/x)^2*po 
lylog(2,(1+c/x)/(1-c^2/x^2)^(1/2))+6*arctanh(c/x)*polylog(3,(1+c/x)/(1-c^2 
/x^2)^(1/2))-6*polylog(4,(1+c/x)/(1-c^2/x^2)^(1/2))-arctanh(c/x)^3*ln(1+(1 
+c/x)/(1-c^2/x^2)^(1/2))-3*arctanh(c/x)^2*polylog(2,-(1+c/x)/(1-c^2/x^2)^( 
1/2))+6*arctanh(c/x)*polylog(3,-(1+c/x)/(1-c^2/x^2)^(1/2))-6*polylog(4,-(1 
+c/x)/(1-c^2/x^2)^(1/2))-1/2*I*Pi*csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1)/(1-(1+ 
c/x)^2/(c^2/x^2-1)))*(csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1))*csgn(I/(1-(1+c/x) 
^2/(c^2/x^2-1)))-csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1))*csgn(I*(-(1+c/x)^2/(c^ 
2/x^2-1)-1)/(1-(1+c/x)^2/(c^2/x^2-1)))-csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1)/( 
1-(1+c/x)^2/(c^2/x^2-1)))*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))+csgn(I*(-(1+c/ 
x)^2/(c^2/x^2-1)-1)/(1-(1+c/x)^2/(c^2/x^2-1)))^2)*arctanh(c/x)^3+3/2*arcta 
nh(c/x)^2*polylog(2,-(1+c/x)^2/(1-c^2/x^2))-3/2*arctanh(c/x)*polylog(3,-(1 
+c/x)^2/(1-c^2/x^2))+3/4*polylog(4,-(1+c/x)^2/(1-c^2/x^2)))+3*a*b^2*(-ln(c 
/x)*arctanh(c/x)^2+arctanh(c/x)*polylog(2,-(1+c/x)^2/(1-c^2/x^2))-1/2*poly 
log(3,-(1+c/x)^2/(1-c^2/x^2))+arctanh(c/x)^2*ln((1+c/x)^2/(1-c^2/x^2)-1)-a 
rctanh(c/x)^2*ln(1-(1+c/x)/(1-c^2/x^2)^(1/2))-2*arctanh(c/x)*polylog(2,(1+ 
c/x)/(1-c^2/x^2)^(1/2))+2*polylog(3,(1+c/x)/(1-c^2/x^2)^(1/2))-arctanh(c/x 
)^2*ln(1+(1+c/x)/(1-c^2/x^2)^(1/2))-2*arctanh(c/x)*polylog(2,-(1+c/x)/(1-c 
^2/x^2)^(1/2))+2*polylog(3,-(1+c/x)/(1-c^2/x^2)^(1/2))-1/2*I*Pi*csgn(I*...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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