\(\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx\) [31]
Optimal result
Integrand size = 10, antiderivative size = 95 \[
\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\frac {3}{2} a^2 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+3 a^2 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )
\]
[Out]
3/2*a^2*arccoth(a*x)^2-3/2*a*arccoth(a*x)^2/x+1/2*a^2*arccoth(a*x)^3-1/2*arccoth(a*x)^3/x^2+3*a^2*arccoth(a*x)
*ln(2-2/(a*x+1))-3/2*a^2*polylog(2,-1+2/(a*x+1))
Rubi [A] (verified)
Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6038, 6130, 6136, 6080, 2497,
6096} \[
\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=-\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {1}{2} a^2 \coth ^{-1}(a x)^3+\frac {3}{2} a^2 \coth ^{-1}(a x)^2+3 a^2 \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\frac {\coth ^{-1}(a x)^3}{2 x^2}-\frac {3 a \coth ^{-1}(a x)^2}{2 x}
\]
[In]
Int[ArcCoth[a*x]^3/x^3,x]
[Out]
(3*a^2*ArcCoth[a*x]^2)/2 - (3*a*ArcCoth[a*x]^2)/(2*x) + (a^2*ArcCoth[a*x]^3)/2 - ArcCoth[a*x]^3/(2*x^2) + 3*a^
2*ArcCoth[a*x]*Log[2 - 2/(1 + a*x)] - (3*a^2*PolyLog[2, -1 + 2/(1 + a*x)])/2
Rule 2497
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[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]
Rule 6038
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] - Dist[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 6080
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] - Dist[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 6096
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> 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 6130
Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d
, Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x], x] - Dist[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 6136
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] + Dist[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]
Rubi steps \begin{align*}
\text {integral}& = -\frac {\coth ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} (3 a) \int \frac {\coth ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx \\ & = -\frac {\coth ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} (3 a) \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx+\frac {1}{2} \left (3 a^3\right ) \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx \\ & = -\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx \\ & = \frac {3}{2} a^2 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac {\coth ^{-1}(a x)}{x (1+a x)} \, dx \\ & = \frac {3}{2} a^2 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+3 a^2 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\left (3 a^3\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx \\ & = \frac {3}{2} a^2 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+3 a^2 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \\
\end{align*}
Mathematica [A] (verified)
Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83
\[
\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\frac {1}{2} \left (\frac {\coth ^{-1}(a x) \left (3 a x (-1+a x) \coth ^{-1}(a x)+\left (-1+a^2 x^2\right ) \coth ^{-1}(a x)^2+6 a^2 x^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )\right )}{x^2}-3 a^2 \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )\right )
\]
[In]
Integrate[ArcCoth[a*x]^3/x^3,x]
[Out]
((ArcCoth[a*x]*(3*a*x*(-1 + a*x)*ArcCoth[a*x] + (-1 + a^2*x^2)*ArcCoth[a*x]^2 + 6*a^2*x^2*Log[1 + E^(-2*ArcCot
h[a*x])]))/x^2 - 3*a^2*PolyLog[2, -E^(-2*ArcCoth[a*x])])/2
Maple [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.32 (sec) , antiderivative size = 3498, normalized size of antiderivative =
36.82
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| method | result | size |
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| parts |
\(\text {Expression too large to display}\) |
\(3498\) |
| derivativedivides |
\(\text {Expression too large to display}\) |
\(3502\) |
| default |
\(\text {Expression too large to display}\) |
\(3502\) |
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[In]
int(arccoth(a*x)^3/x^3,x,method=_RETURNVERBOSE)
[Out]
-1/2*arccoth(a*x)^3/x^2-3/2*a^2*(-1/2*polylog(2,-(a*x+1)/(a*x-1))+1/a/x*arccoth(a*x)^2-1/4*I*Pi*csgn(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(a*x)^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(a*x)*ln(1-I/((a*x-1)/(a*x+1))^(1
/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(a*x)*ln(1+(a*x+1)/(a*x-1))+1/4
*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth(a*x)*ln(1+(a*x+1)/(a*x-1))-1/4*I*Pi*cs
gn(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(a*x)*ln(1+(a*x+1)/(a*x-1))-1/4
*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1/2))-1
/4*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth(a*x)*ln(1-I/((a*x-1)/(a*x+1))^(1/2))
+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(a*x)*ln(1+I/((a*x-1)/(
a*x+1))^(1/2))+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(a*x)*ln(
1-I/((a*x-1)/(a*x+1))^(1/2))+1/2*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*arccoth(a*x)*l
n(1+I/((a*x-1)/(a*x+1))^(1/2))+1/2*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*arccoth(a*x)
*ln(1-I/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(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))*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(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))*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))+1/8*I*Pi*csgn(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))*polylog(2,-(a*x+1)/(a*
x-1))-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(a*x)*ln(1+(a*x+1)
/(a*x-1))+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(a*x)*ln(1
+I/((a*x-1)/(a*x+1))^(1/2))-1/3*arccoth(a*x)^3-1/4*I*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^3*dilog(1+
I/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^3*dilog(1-I/((a*x-1)/(a*x+1))^
(1/2))+1/8*I*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^3*polylog(2,-(a*x+1)/(a*x-1))+1/8*I*Pi*csgn(I*(a*x
+1)/(a*x-1))^3*polylog(2,-(a*x+1)/(a*x-1))-1/4*I*Pi*csgn(I*(a*x+1)/(a*x-1))^3*dilog(1+I/((a*x-1)/(a*x+1))^(1/2
))-1/4*I*Pi*csgn(I*(a*x+1)/(a*x-1))^3*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(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(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1/2
))-1/4*I*Pi*csgn(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))*ar
ccoth(a*x)*ln(1-I/((a*x-1)/(a*x+1))^(1/2))+1/4*I*Pi*csgn(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(a*x)*ln(1+(a*x+1)/(a*x-1))+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*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))+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*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(I/((a*x
-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*polylog(2,-(a*x+1)/(a*x-1))-dilog(1+I/((a*x-1)/(a*x+1))^(1/2))-d
ilog(1-I/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*dilog(1-I
/((a*x-1)/(a*x+1))^(1/2))+1/2*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*dilog(1-I/((a*x-1
)/(a*x+1))^(1/2))-1/8*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*polylog(2,-(a
*x+1)/(a*x-1))-1/4*I*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^3*arccoth(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1
/2))-1/8*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*polylog(2,-(a*x+1)/(a*
x-1))-1/4*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))+1/
8*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*polylog(2,-(a*x+1)/(a*x-1))+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*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))+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*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))+1/2*I*Pi*
csgn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(I*(
a*x+1)/(a*x-1))^3*arccoth(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1/2))+1/4*I*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1
)-1))^3*arccoth(a*x)*ln(1+(a*x+1)/(a*x-1))-1/4*I*Pi*csgn(I*(a*x+1)/(a*x-1))^3*arccoth(a*x)*ln(1-I/((a*x-1)/(a*
x+1))^(1/2))+1/4*I*Pi*csgn(I*(a*x+1)/(a*x-1))^3*arccoth(a*x)*ln(1+(a*x+1)/(a*x-1))-1/4*I*Pi*csgn(I/(a*x-1)*(a*
x+1)/((a*x+1)/(a*x-1)-1))^3*arccoth(a*x)*ln(1-I/((a*x-1)/(a*x+1))^(1/2))+arccoth(a*x)^2-1/4*I*Pi*csgn(I/(a*x-1
)*(a*x+1)/((a*x+1)/(a*x-1)-1))^3*arccoth(a*x)^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(a*x)^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
(a*x)^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(a*x)^2+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(a*x)^2-1/4*I*Pi*csgn(I*(a*x+1)/(a*x-1))^3*a
rccoth(a*x)^2-arccoth(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1/2))-arccoth(a*x)*ln(1-I/((a*x-1)/(a*x+1))^(1/2))-arccot
h(a*x)*ln(1+(a*x+1)/(a*x-1))-1/2*arccoth(a*x)^2*ln((a*x-1)/(a*x+1))+1/2*arccoth(a*x)^2*ln(a*x-1)-1/2*arccoth(a
*x)^2*ln(a*x+1))
Fricas [F]
\[
\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{3}} \,d x }
\]
[In]
integrate(arccoth(a*x)^3/x^3,x, algorithm="fricas")
[Out]
integral(arccoth(a*x)^3/x^3, x)
Sympy [F]
\[
\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x^{3}}\, dx
\]
[In]
integrate(acoth(a*x)**3/x**3,x)
[Out]
Integral(acoth(a*x)**3/x**3, x)
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (84) = 168\).
Time = 0.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.65
\[
\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\frac {3}{4} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} a \operatorname {arcoth}\left (a x\right )^{2} - \frac {1}{16} \, {\left (a^{2} {\left (\frac {3 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} - 3 \, {\left (\log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 6 \, \log \left (a x - 1\right )^{2}}{a} - \frac {24 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} + \frac {24 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} - \frac {24 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} - 6 \, {\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right ) + 8 \, \log \left (x\right )\right )} a \operatorname {arcoth}\left (a x\right )\right )} a - \frac {\operatorname {arcoth}\left (a x\right )^{3}}{2 \, x^{2}}
\]
[In]
integrate(arccoth(a*x)^3/x^3,x, algorithm="maxima")
[Out]
3/4*(a*log(a*x + 1) - a*log(a*x - 1) - 2/x)*a*arccoth(a*x)^2 - 1/16*(a^2*((3*(log(a*x - 1) - 2)*log(a*x + 1)^2
- log(a*x + 1)^3 + log(a*x - 1)^3 - 3*(log(a*x - 1)^2 - 4*log(a*x - 1))*log(a*x + 1) + 6*log(a*x - 1)^2)/a -
24*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a + 24*(log(a*x + 1)*log(x) + dilog(-a*x))/a - 24
*(log(-a*x + 1)*log(x) + dilog(a*x))/a) - 6*(2*(log(a*x - 1) - 2)*log(a*x + 1) - log(a*x + 1)^2 - log(a*x - 1)
^2 - 4*log(a*x - 1) + 8*log(x))*a*arccoth(a*x))*a - 1/2*arccoth(a*x)^3/x^2
Giac [F]
\[
\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{3}} \,d x }
\]
[In]
integrate(arccoth(a*x)^3/x^3,x, algorithm="giac")
[Out]
integrate(arccoth(a*x)^3/x^3, x)
Mupad [F(-1)]
Timed out. \[
\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^3} \,d x
\]
[In]
int(acoth(a*x)^3/x^3,x)
[Out]
int(acoth(a*x)^3/x^3, x)