\(\int x^3 \coth ^{-1}(a x)^3 \, dx\) [25]

Optimal result

Integrand size = 10, antiderivative size = 139 \[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\frac {x}{4 a^3}+\frac {x^2 \coth ^{-1}(a x)}{4 a^2}+\frac {\coth ^{-1}(a x)^2}{a^4}+\frac {3 x \coth ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \coth ^{-1}(a x)^2}{4 a}-\frac {\coth ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {\text {arctanh}(a x)}{4 a^4}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.63 \[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\frac {a x+\left (-4+3 a x+a^3 x^3\right ) \coth ^{-1}(a x)^2+\left (-1+a^4 x^4\right ) \coth ^{-1}(a x)^3+\coth ^{-1}(a x) \left (-1+a^2 x^2-8 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+4 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{4 a^4} \] Input:

Integrate[x^3*ArcCoth[a*x]^3,x]
 

Output:

(a*x + (-4 + 3*a*x + a^3*x^3)*ArcCoth[a*x]^2 + (-1 + a^4*x^4)*ArcCoth[a*x] 
^3 + ArcCoth[a*x]*(-1 + a^2*x^2 - 8*Log[1 - E^(-2*ArcCoth[a*x])]) + 4*Poly 
Log[2, E^(-2*ArcCoth[a*x])])/(4*a^4)
 

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.72, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {6453, 6543, 6453, 6543, 6437, 6453, 262, 219, 6511, 6547, 6471, 2849, 2752}

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.

\(\displaystyle \int x^3 \coth ^{-1}(a x)^3 \, dx\)

\(\Big \downarrow \) 6453

\(\displaystyle \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {3}{4} a \int \frac {x^4 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx\)

\(\Big \downarrow \) 6543

\(\displaystyle \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {3}{4} a \left (\frac {\int \frac {x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\int x^2 \coth ^{-1}(a x)^2dx}{a^2}\right )\)

\(\Big \downarrow \) 6453

\(\displaystyle \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {3}{4} a \left (\frac {\int \frac {x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}\right )\)

\(\Big \downarrow \) 6543

\(\displaystyle \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {3}{4} a \left (\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\int \coth ^{-1}(a x)^2dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x \coth ^{-1}(a x)dx}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6437

\(\displaystyle \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {3}{4} a \left (\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x \coth ^{-1}(a x)dx}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6453

\(\displaystyle \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {3}{4} a \left (\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \int \frac {x^2}{1-a^2 x^2}dx}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {3}{4} a \left (\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\int \frac {1}{1-a^2 x^2}dx}{a^2}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {3}{4} a \left (\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6511

\(\displaystyle \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {3}{4} a \left (\frac {\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6547

\(\displaystyle \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {3}{4} a \left (\frac {\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a x}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\frac {\int \frac {\coth ^{-1}(a x)}{1-a x}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6471

\(\displaystyle \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {3}{4} a \left (\frac {\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \left (\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

Int[x^3*ArcCoth[a*x]^3,x]
 

Output:

(x^4*ArcCoth[a*x]^3)/4 - (3*a*(-(((x^3*ArcCoth[a*x]^2)/3 - (2*a*(-(((x^2*A 
rcCoth[a*x])/2 - (a*(-(x/a^2) + ArcTanh[a*x]/a^3))/2)/a^2) + (-1/2*ArcCoth 
[a*x]^2/a^2 + ((ArcCoth[a*x]*Log[2/(1 - a*x)])/a + PolyLog[2, 1 - 2/(1 - a 
*x)]/(2*a))/a)/a^2))/3)/a^2) + (ArcCoth[a*x]^3/(3*a^3) - (x*ArcCoth[a*x]^2 
 - 2*a*(-1/2*ArcCoth[a*x]^2/a^2 + ((ArcCoth[a*x]*Log[2/(1 - a*x)])/a + Pol 
yLog[2, 1 - 2/(1 - a*x)]/(2*a))/a))/a^2)/a^2))/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 262

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

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 6437

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

rule 6453

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcCoth[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcCoth[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 6471

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcCoth[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[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 6511

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[(a + b*ArcCoth[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 6543

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

rule 6547

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

Maple [C] (warning: unable to verify)

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

Time = 4.69 (sec) , antiderivative size = 871, normalized size of antiderivative = 6.27

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(871\)
default	\(\text {Expression too large to display}\)	\(871\)
parts	\(\text {Expression too large to display}\)	\(871\)

Input:

int(x^3*arccoth(x*a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/a^4*(1/4*x^4*a^4*arccoth(x*a)^3+1/4*x^3*a^3*arccoth(x*a)^2-3/16*I*Pi*csg 
n(I/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/(( 
a*x+1)/(a*x-1)-1))*arccoth(x*a)^2-2*arccoth(x*a)*ln(1+1/((a*x-1)/(a*x+1))^ 
(1/2))-1/4*arccoth(x*a)^3-3/8*arccoth(x*a)^2*ln((a*x-1)/(a*x+1))+1/8*(((a* 
x-1)/(a*x+1))^(1/2)*x*a+((a*x-1)/(a*x+1))^(1/2)+x*a+1)*arccoth(x*a)-1/4/(( 
(a*x-1)/(a*x+1))^(1/2)+1)*((a*x-1)/(a*x+1))^(1/2)-1/4/(((a*x-1)/(a*x+1))^( 
1/2)-1)*((a*x-1)/(a*x+1))^(1/2)+1/8*(((a*x-1)/(a*x+1))^(1/2)*x*a+((a*x-1)/ 
(a*x+1))^(1/2)-x*a)*arccoth(x*a)*(a*x+1)+3/8*arccoth(x*a)^2*ln(a*x-1)-3/8* 
arccoth(x*a)^2*ln(a*x+1)-1/8*(((a*x-1)/(a*x+1))^(1/2)*x*a+((a*x-1)/(a*x+1) 
)^(1/2)-x*a-1)*arccoth(x*a)-3/16*I*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x 
-1)-1))^3*arccoth(x*a)^2-3/16*I*Pi*csgn(I*(a*x+1)/(a*x-1))^3*arccoth(x*a)^ 
2+3/4*arccoth(x*a)^2*x*a+2*dilog(1/((a*x-1)/(a*x+1))^(1/2))-2*dilog(1+1/(( 
a*x-1)/(a*x+1))^(1/2))+arccoth(x*a)^2+3/8*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1 
/2))*csgn(I*(a*x+1)/(a*x-1))^2*arccoth(x*a)^2-3/16*I*Pi*csgn(I/((a*x-1)/(a 
*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth(x*a)^2-1/8*(((a*x-1)/(a*x+ 
1))^(1/2)*x*a+((a*x-1)/(a*x+1))^(1/2)+x*a)*arccoth(x*a)*(a*x+1)-1/4*(((a*x 
-1)/(a*x+1))^(1/2)*x*a-x*a+1)*arccoth(x*a)*(a*x+1)+1/4*(((a*x-1)/(a*x+1))^ 
(1/2)*x*a+x*a-1)*arccoth(x*a)*(a*x+1)+3/16*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1) 
)*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^2*arccoth(x*a)^2+3/16*I*Pi*c 
sgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^2*ar...
 

Fricas [F]

\[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\int { x^{3} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \] Input:

integrate(x^3*arccoth(a*x)^3,x, algorithm="fricas")
 

Output:

integral(x^3*arccoth(a*x)^3, x)
 

Sympy [F]

\[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\int x^{3} \operatorname {acoth}^{3}{\left (a x \right )}\, dx \] Input:

integrate(x**3*acoth(a*x)**3,x)
 

Output:

Integral(x**3*acoth(a*x)**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (122) = 244\).

Time = 0.04 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.88 \[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\frac {1}{4} \, x^{4} \operatorname {arcoth}\left (a x\right )^{3} + \frac {1}{8} \, a {\left (\frac {2 \, {\left (a^{2} x^{3} + 3 \, x\right )}}{a^{4}} - \frac {3 \, \log \left (a x + 1\right )}{a^{5}} + \frac {3 \, \log \left (a x - 1\right )}{a^{5}}\right )} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{32} \, a {\left (\frac {\frac {{\left (3 \, \log \left (a x - 1\right ) - 8\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} + 8 \, a x - {\left (3 \, \log \left (a x - 1\right )^{2} - 16 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 8 \, \log \left (a x - 1\right )^{2} + 4 \, \log \left (a x - 1\right )}{a} - \frac {32 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {4 \, \log \left (a x + 1\right )}{a}}{a^{4}} + \frac {2 \, {\left (4 \, a^{2} x^{2} - 2 \, {\left (3 \, \log \left (a x - 1\right ) - 8\right )} \log \left (a x + 1\right ) + 3 \, \log \left (a x + 1\right )^{2} + 3 \, \log \left (a x - 1\right )^{2} + 16 \, \log \left (a x - 1\right )\right )} \operatorname {arcoth}\left (a x\right )}{a^{5}}\right )} \] Input:

integrate(x^3*arccoth(a*x)^3,x, algorithm="maxima")
 

Output:

1/4*x^4*arccoth(a*x)^3 + 1/8*a*(2*(a^2*x^3 + 3*x)/a^4 - 3*log(a*x + 1)/a^5 
 + 3*log(a*x - 1)/a^5)*arccoth(a*x)^2 + 1/32*a*((((3*log(a*x - 1) - 8)*log 
(a*x + 1)^2 - log(a*x + 1)^3 + log(a*x - 1)^3 + 8*a*x - (3*log(a*x - 1)^2 
- 16*log(a*x - 1))*log(a*x + 1) + 8*log(a*x - 1)^2 + 4*log(a*x - 1))/a - 3 
2*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a - 4*log(a*x 
+ 1)/a)/a^4 + 2*(4*a^2*x^2 - 2*(3*log(a*x - 1) - 8)*log(a*x + 1) + 3*log(a 
*x + 1)^2 + 3*log(a*x - 1)^2 + 16*log(a*x - 1))*arccoth(a*x)/a^5)
 

Giac [F]

\[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\int { x^{3} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \] Input:

integrate(x^3*arccoth(a*x)^3,x, algorithm="giac")
 

Output:

integrate(x^3*arccoth(a*x)^3, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\int x^3\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \] Input:

int(x^3*acoth(a*x)^3,x)
 

Output:

int(x^3*acoth(a*x)^3, x)
 

Reduce [F]

\[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\frac {\mathit {acoth} \left (a x \right )^{3} a^{4} x^{4}-\mathit {acoth} \left (a x \right )^{3}-\mathit {acoth} \left (a x \right )^{2} a^{3} x^{3}-3 \mathit {acoth} \left (a x \right )^{2} a x +\mathit {acoth} \left (a x \right ) a^{2} x^{2}-\mathit {acoth} \left (a x \right )+8 \left (\int \frac {\mathit {acoth} \left (a x \right ) x}{a^{2} x^{2}-1}d x \right ) a^{2}-a x}{4 a^{4}} \] Input:

int(x^3*acoth(a*x)^3,x)
 

Output:

(acoth(a*x)**3*a**4*x**4 - acoth(a*x)**3 - acoth(a*x)**2*a**3*x**3 - 3*aco 
th(a*x)**2*a*x + acoth(a*x)*a**2*x**2 - acoth(a*x) + 8*int((acoth(a*x)*x)/ 
(a**2*x**2 - 1),x)*a**2 - a*x)/(4*a**4)