\(\int x^3 (a+b \text {arctanh}(\frac {c}{x}))^3 \, dx\) [150]

Optimal result

Integrand size = 16, antiderivative size = 203 \[ \int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\frac {1}{4} b^3 c^3 x-\frac {1}{4} b^3 c^4 \coth ^{-1}\left (\frac {x}{c}\right )+\frac {1}{4} b^2 c^2 x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-b c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{4} b c^3 x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{4} b c x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {1}{4} c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{4} x^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-2 b^2 c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1+\frac {c}{x}}\right )+b^3 c^4 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+\frac {c}{x}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.41 \[ \int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\frac {1}{8} \left (-2 a b^2 c^4+6 a^2 b c^3 x+2 b^3 c^3 x+2 a b^2 c^2 x^2+2 a^2 b c x^3+2 a^3 x^4+2 b^2 \left (b c \left (-4 c^3+3 c^2 x+x^3\right )+3 a \left (-c^4+x^4\right )\right ) \text {arctanh}\left (\frac {c}{x}\right )^2+2 b^3 \left (-c^4+x^4\right ) \text {arctanh}\left (\frac {c}{x}\right )^3+2 b \text {arctanh}\left (\frac {c}{x}\right ) \left (3 a^2 x^4+2 a b c x \left (3 c^2+x^2\right )+b^2 \left (-c^4+c^2 x^2\right )-8 b^2 c^4 \log \left (1-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )+3 a^2 b c^4 \log \left (1-\frac {c}{x}\right )-16 a b^2 c^4 \log \left (\frac {c}{\sqrt {1-\frac {c^2}{x^2}} x}\right )-3 a^2 b c^4 \log \left (\frac {c+x}{x}\right )+8 b^3 c^4 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right ) \] Input:

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

Output:

(-2*a*b^2*c^4 + 6*a^2*b*c^3*x + 2*b^3*c^3*x + 2*a*b^2*c^2*x^2 + 2*a^2*b*c* 
x^3 + 2*a^3*x^4 + 2*b^2*(b*c*(-4*c^3 + 3*c^2*x + x^3) + 3*a*(-c^4 + x^4))* 
ArcTanh[c/x]^2 + 2*b^3*(-c^4 + x^4)*ArcTanh[c/x]^3 + 2*b*ArcTanh[c/x]*(3*a 
^2*x^4 + 2*a*b*c*x*(3*c^2 + x^2) + b^2*(-c^4 + c^2*x^2) - 8*b^2*c^4*Log[1 
- E^(-2*ArcTanh[c/x])]) + 3*a^2*b*c^4*Log[1 - c/x] - 16*a*b^2*c^4*Log[c/(S 
qrt[1 - c^2/x^2]*x)] - 3*a^2*b*c^4*Log[(c + x)/x] + 8*b^3*c^4*PolyLog[2, E 
^(-2*ArcTanh[c/x])])/8
 

Rubi [A] (verified)

Time = 2.55 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.31, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6454, 6452, 6544, 6452, 6544, 6452, 264, 219, 6510, 6550, 6494, 2897}

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.

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

\(\Big \downarrow \) 6454

\(\displaystyle -\int x^5 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3d\frac {1}{x}\)

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

rule 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -1]
 

rule 6454

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

rule 6494

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

rule 6510

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

rule 6544

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( 
e_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x 
], x] - Simp[e/(d*f^2)   Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x 
^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
 

rule 6550

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

Maple [C] (warning: unable to verify)

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

Time = 20.36 (sec) , antiderivative size = 1267, normalized size of antiderivative = 6.24

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1267\)
default	\(\text {Expression too large to display}\)	\(1267\)
parts	\(\text {Expression too large to display}\)	\(1322\)
risch	\(\text {Expression too large to display}\)	\(44754\)

Input:

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

Output:

-c^4*(-3/16*I*b^3*Pi*csgn(I*(1+c/x)^2/(c^2/x^2-1)/(1-(1+c/x)^2/(c^2/x^2-1) 
))^3*arctanh(c/x)^2+3/8*I*b^3*Pi*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))^3*arcta 
nh(c/x)^2-3/16*I*b^3*Pi*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))^2*csgn(I*(1+c/x) 
^2/(c^2/x^2-1))*arctanh(c/x)^2-3/8*I*b^3*Pi*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/ 
2))*csgn(I*(1+c/x)^2/(c^2/x^2-1))^2*arctanh(c/x)^2-3/16*I*b^3*Pi*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+c/x)^2/(c^2/x^ 
2-1)))^2*arctanh(c/x)^2+3/16*I*b^3*Pi*csgn(I*(1+c/x)^2/(c^2/x^2-1))*csgn(I 
*(1+c/x)^2/(c^2/x^2-1)/(1-(1+c/x)^2/(c^2/x^2-1)))^2*arctanh(c/x)^2+3/16*I* 
b^3*Pi*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))*csgn(I*(1+c/x)^2/(c^2/x^2-1))*csg 
n(I*(1+c/x)^2/(c^2/x^2-1)/(1-(1+c/x)^2/(c^2/x^2-1)))*arctanh(c/x)^2+3/8*I* 
b^3*Pi*arctanh(c/x)^2-1/4*b^3/c^4*x^4*arctanh(c/x)^3-1/4*b^3*arctanh(c/x)^ 
2/c^3*x^3-3/4*b^3*arctanh(c/x)^2/c*x-1/4*b^3*arctanh(c/x)/c^2*x^2-3/16*I*b 
^3*Pi*csgn(I*(1+c/x)^2/(c^2/x^2-1))^3*arctanh(c/x)^2-3/8*I*b^3*Pi*csgn(I/( 
1-(1+c/x)^2/(c^2/x^2-1)))^2*arctanh(c/x)^2-1/4*a^3/c^4*x^4+1/4*b^3*arctanh 
(c/x)^3-b^3*arctanh(c/x)^2+1/4*b^3*arctanh(c/x)+3*a*b^2*(-1/4/c^4*x^4*arct 
anh(c/x)^2-1/6/c^3*x^3*arctanh(c/x)-1/2/c*x*arctanh(c/x)+1/4*arctanh(c/x)* 
ln(1+c/x)-1/4*arctanh(c/x)*ln(c/x-1)-1/16*ln(c/x-1)^2+1/8*ln(c/x-1)*ln(1/2 
*c/x+1/2)+1/8*(ln(1+c/x)-ln(1/2*c/x+1/2))*ln(-1/2*c/x+1/2)-1/16*ln(1+c/x)^ 
2-1/12/c^2*x^2+2/3*ln(c/x)-1/3*ln(1+c/x)-1/3*ln(c/x-1))+3*a^2*b*(-1/4/c^4* 
x^4*arctanh(c/x)+1/8*ln(1+c/x)-1/8*ln(c/x-1)-1/12/c^3*x^3-1/4*x/c)+1/4*...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

3/4*a*b^2*x^4*arctanh(c/x)^2 + 1/4*a^3*x^4 + 1/8*(6*x^4*arctanh(c/x) - (3* 
c^3*log(c + x) - 3*c^3*log(-c + x) - 6*c^2*x - 2*x^3)*c)*a^2*b + 1/16*((3* 
c^2*log(c + x)^2 + 3*c^2*log(-c + x)^2 + 16*c^2*log(c + x) + 4*x^2 - 2*(3* 
c^2*log(c + x) - 8*c^2)*log(-c + x))*c^2 - 4*(3*c^3*log(c + x) - 3*c^3*log 
(-c + x) - 6*c^2*x - 2*x^3)*c*arctanh(c/x))*a*b^2 + 1/32*(16*c^5*integrate 
(-log(c + x)/(c^2 - x^2), x) + 40*c^4*integrate(-x*log(c + x)/(c^2 - x^2), 
 x) - 2*(c*log(c + x) - c*log(-c + x) - 2*x)*c^3 - (c^4 - x^4)*log(c + x)^ 
3 + (c^4 - x^4)*log(-c + x)^3 + 2*(c^2*log(-c^2 + x^2) + x^2)*c^2 + 8*c^2* 
integrate(-x^3*log(c + x)/(c^2 - x^2), x) + 2*(3*c^3*x + c*x^3)*log(c + x) 
^2 - (8*c^4 - 6*c^3*x - 2*c*x^3 + 3*(c^4 - x^4)*log(c + x))*log(-c + x)^2 
- (4*c^2*x^2 - 3*(c^4 - x^4)*log(c + x)^2 + 4*(4*c^4 + 3*c^3*x + c*x^3)*lo 
g(c + x))*log(-c + x))*b^3
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - atanh(c/x)**3*b**3*c**4 + atanh(c/x)**3*b**3*x**4 - 3*atanh(c/x)**2*a* 
b**2*c**4 + 3*atanh(c/x)**2*a*b**2*x**4 + 3*atanh(c/x)**2*b**3*c**3*x + at 
anh(c/x)**2*b**3*c*x**3 - 3*atanh(c/x)*a**2*b*c**4 + 3*atanh(c/x)*a**2*b*x 
**4 - 8*atanh(c/x)*a*b**2*c**4 + 6*atanh(c/x)*a*b**2*c**3*x + 2*atanh(c/x) 
*a*b**2*c*x**3 - atanh(c/x)*b**3*c**4 + atanh(c/x)*b**3*c**2*x**2 - 8*int( 
(atanh(c/x)*x)/(c**2 - x**2),x)*b**3*c**4 + 8*log( - c - x)*a*b**2*c**4 + 
a**3*x**4 + 3*a**2*b*c**3*x + a**2*b*c*x**3 + a*b**2*c**2*x**2 + b**3*c**3 
*x)/4