\(\int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx\) [32]

Optimal result

Integrand size = 10, antiderivative size = 154 \[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=-\frac {a^2 \coth ^{-1}(a x)}{x}+\frac {1}{2} a^3 \coth ^{-1}(a x)^2-\frac {a \coth ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{3 x^3}+a^3 \log (x)-\frac {1}{2} a^3 \log \left (1-a^2 x^2\right )+a^3 \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^3 \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.92 \[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\frac {1}{6} \left (-\frac {6 a^2 \coth ^{-1}(a x)}{x}+3 a^3 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{x^2}+2 a^3 \coth ^{-1}(a x)^3-\frac {2 \coth ^{-1}(a x)^3}{x^3}+6 a^3 \coth ^{-1}(a x)^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )+6 a^3 \log \left (\frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )-6 a^3 \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-3 a^3 \operatorname {PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right )\right ) \] Input:

Integrate[ArcCoth[a*x]^3/x^4,x]
 

Output:

((-6*a^2*ArcCoth[a*x])/x + 3*a^3*ArcCoth[a*x]^2 - (3*a*ArcCoth[a*x]^2)/x^2 
 + 2*a^3*ArcCoth[a*x]^3 - (2*ArcCoth[a*x]^3)/x^3 + 6*a^3*ArcCoth[a*x]^2*Lo 
g[1 + E^(-2*ArcCoth[a*x])] + 6*a^3*Log[1/Sqrt[1 - 1/(a^2*x^2)]] - 6*a^3*Ar 
cCoth[a*x]*PolyLog[2, -E^(-2*ArcCoth[a*x])] - 3*a^3*PolyLog[3, -E^(-2*ArcC 
oth[a*x])])/6
 

Rubi [A] (verified)

Time = 1.58 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {6453, 6545, 6453, 6545, 6453, 243, 47, 14, 16, 6511, 6551, 6495, 6619, 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.

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

\(\Big \downarrow \) 6453

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

\(\Big \downarrow \) 6545

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

\(\Big \downarrow \) 6453

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

\(\Big \downarrow \) 6545

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

\(\Big \downarrow \) 6453

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 6511

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

\(\Big \downarrow \) 6551

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

\(\Big \downarrow \) 6495

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

\(\Big \downarrow \) 6619

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

\(\Big \downarrow \) 7164

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

Input:

Int[ArcCoth[a*x]^3/x^4,x]
 

Output:

-1/3*ArcCoth[a*x]^3/x^3 + a*(-1/2*ArcCoth[a*x]^2/x^2 + a*(-(ArcCoth[a*x]/x 
) + (a*ArcCoth[a*x]^2)/2 + (a*(Log[x^2] - Log[1 - a^2*x^2]))/2) + a^2*(Arc 
Coth[a*x]^3/3 + ArcCoth[a*x]^2*Log[2 - 2/(1 + a*x)] - 2*a*((ArcCoth[a*x]*P 
olyLog[2, -1 + 2/(1 + a*x)])/(2*a) + PolyLog[3, -1 + 2/(1 + a*x)]/(4*a))))
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 6495

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x 
_Symbol] :> Simp[(a + b*ArcCoth[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - 
Simp[b*c*(p/d)   Int[(a + b*ArcCoth[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 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 6545

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

rule 6551

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

rule 6619

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 
2), x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x 
] - Simp[b*(p/2)   Int[(a + b*ArcCoth[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 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 = 6.37 (sec) , antiderivative size = 836, normalized size of antiderivative = 5.43

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(836\)
default	\(\text {Expression too large to display}\)	\(836\)
parts	\(\text {Expression too large to display}\)	\(839\)

Input:

int(arccoth(x*a)^3/x^4,x,method=_RETURNVERBOSE)
 

Output:

a^3*(-1/3/x^3/a^3*arccoth(x*a)^3-1/2*arccoth(x*a)^2*ln(a*x+1)-1/2*arccoth( 
x*a)^2/x^2/a^2+ln(x*a)*arccoth(x*a)^2-1/2*arccoth(x*a)^2*ln(a*x-1)-arccoth 
(x*a)*(a*x+1)/x/a+1/2*arccoth(x*a)^2+ln(2)*arccoth(x*a)^2+1/2*I*Pi*csgn(I/ 
((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-1)))^3*arccoth(x*a)^2+ln(1+(a*x+1)/(a* 
x-1))+1/2*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I*(1+(a*x+1)/(a*x-1)))*csg 
n(I/((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-1)))*arccoth(x*a)^2-1/3*arccoth(x* 
a)^3+1/4*I*Pi*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x 
-1)-1))^2*arccoth(x*a)^2+1/4*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)^2-1/2*polylog(3,-(a*x+1)/(a 
*x-1))-1/2*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I/((a*x+1)/(a*x-1)-1)*(1+ 
(a*x+1)/(a*x-1)))^2*arccoth(x*a)^2-1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*cs 
gn(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)^2-1/2*I*Pi*csgn(I*(1+(a*x+1)/(a*x-1)))*csgn(I/((a*x+1)/(a*x-1)-1)*(1+ 
(a*x+1)/(a*x-1)))^2*arccoth(x*a)^2-1/4*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+arccoth(x*a)*polylog(2,-(a*x+1) 
/(a*x-1))-1/2*arccoth(x*a)^2*ln((a*x-1)/(a*x+1))-1/4*I*Pi*csgn(I*(a*x+1)/( 
a*x-1))^3*arccoth(x*a)^2-1/4*I*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)- 
1))^3*arccoth(x*a)^2+1/2*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)
 

Fricas [F]

\[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{4}} \,d x } \] Input:

integrate(arccoth(a*x)^3/x^4,x, algorithm="fricas")
 

Output:

integral(arccoth(a*x)^3/x^4, x)
 

Sympy [F]

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

integrate(acoth(a*x)**3/x**4,x)
 

Output:

Integral(acoth(a*x)**3/x**4, x)
 

Maxima [F]

\[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{4}} \,d x } \] Input:

integrate(arccoth(a*x)^3/x^4,x, algorithm="maxima")
 

Output:

1/4*a^4*integrate(x^4*log(a*x + 1)*log(a*x - 1)/(a*x^5 + x^4), x) - 1/2*a^ 
4*integrate(x^4*log(a*x + 1)*log(x)/(a*x^5 + x^4), x) + 1/16*(2*a^2*log(a* 
x + 1) - 2*a^2*log(x) - (2*a*x - 1)/x^2)*a*log(a)^3 + 3/8*a*integrate(x*lo 
g(a*x - 1)/(a*x^5 + x^4), x)*log(a)^2 - 3/8*a*integrate(x*log(x)/(a*x^5 + 
x^4), x)*log(a)^2 - 1/48*(6*a^3*log(a*x + 1) - 6*a^3*log(x) - (6*a^2*x^2 - 
 3*a*x + 2)/x^3)*log(a)^3 + 1/4*a^2*integrate(x^2*log(a*x + 1)/(a*x^5 + x^ 
4), x) + 3/4*a*integrate(x*log(a*x - 1)*log(x)/(a*x^5 + x^4), x)*log(a) - 
3/8*a*integrate(x*log(x)^2/(a*x^5 + x^4), x)*log(a) + 3/8*integrate(log(a* 
x - 1)/(a*x^5 + x^4), x)*log(a)^2 - 3/8*integrate(log(x)/(a*x^5 + x^4), x) 
*log(a)^2 + 3/8*a*integrate(x*log(a*x + 1)*log(a*x - 1)^2/(a*x^5 + x^4), x 
) - 3/8*a*integrate(x*log(a*x - 1)^2*log(x)/(a*x^5 + x^4), x) + 3/8*a*inte 
grate(x*log(a*x - 1)*log(x)^2/(a*x^5 + x^4), x) - 1/8*a*integrate(x*log(x) 
^3/(a*x^5 + x^4), x) - 1/4*a*integrate(x*log(a*x + 1)*log(a*x - 1)/(a*x^5 
+ x^4), x) - 3/8*integrate(a*x*log(a*x - 1)^2/(a*x^5 + x^4), x)*log(a) - 3 
/8*integrate(log(a*x - 1)^2/(a*x^5 + x^4), x)*log(a) + 3/4*integrate(log(a 
*x - 1)*log(x)/(a*x^5 + x^4), x)*log(a) - 3/8*integrate(log(x)^2/(a*x^5 + 
x^4), x)*log(a) - 1/48*(6*a^4*log(a*x - 1) - 6*a^4*log(x) + (6*a^3*x^2 + 3 
*a^2*x + 2*a)/x^3)*log(-1/(a*x) + 1)^2/a + 1/864*(6*(18*a^5*x^3*log(a*x - 
1)^2 + 18*a^5*x^3*log(x)^2 - 66*a^5*x^3*log(x) + 66*a^4*x^2 + 15*a^3*x + 4 
*a^2 - 6*(6*a^5*x^3*log(x) - 11*a^5*x^3)*log(a*x - 1))*log(-1/(a*x) + 1...
 

Giac [F]

\[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{4}} \,d x } \] Input:

integrate(arccoth(a*x)^3/x^4,x, algorithm="giac")
 

Output:

integrate(arccoth(a*x)^3/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^4} \,d x \] Input:

int(acoth(a*x)^3/x^4,x)
 

Output:

int(acoth(a*x)^3/x^4, x)
 

Reduce [F]

\[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\frac {-2 \mathit {acoth} \left (a x \right )^{3}-3 \mathit {acoth} \left (a x \right )^{2} a^{3} x^{3}+3 \mathit {acoth} \left (a x \right )^{2} a x -6 \mathit {acoth} \left (a x \right ) a^{3} x^{3}-6 \mathit {acoth} \left (a x \right ) a^{2} x^{2}+6 \left (\int \frac {\mathit {acoth} \left (a x \right )^{2}}{a^{2} x^{3}-x}d x \right ) a^{3} x^{3}+6 \,\mathrm {log}\left (a^{2} x -a \right ) a^{3} x^{3}-6 \,\mathrm {log}\left (x \right ) a^{3} x^{3}}{6 x^{3}} \] Input:

int(acoth(a*x)^3/x^4,x)
                                                                                    
                                                                                    
 

Output:

( - 2*acoth(a*x)**3 - 3*acoth(a*x)**2*a**3*x**3 + 3*acoth(a*x)**2*a*x - 6* 
acoth(a*x)*a**3*x**3 - 6*acoth(a*x)*a**2*x**2 + 6*int(acoth(a*x)**2/(a**2* 
x**3 - x),x)*a**3*x**3 + 6*log(a**2*x - a)*a**3*x**3 - 6*log(x)*a**3*x**3) 
/(6*x**3)