\(\int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx\) [30]

Optimal result

Integrand size = 10, antiderivative size = 79 \[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.91 \[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=\frac {(-1+a x) \coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )-3 a \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right ) \] Input:

Integrate[ArcCoth[a*x]^3/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6453, 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.

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

\(\Big \downarrow \) 6453

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

\(\Big \downarrow \) 6551

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

\(\Big \downarrow \) 6495

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

\(\Big \downarrow \) 6619

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

\(\Big \downarrow \) 7164

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

Input:

Int[ArcCoth[a*x]^3/x^2,x]
 

Output:

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

Defintions of rubi rules used

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 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 = 5.00 (sec) , antiderivative size = 718, normalized size of antiderivative = 9.09

method	result	size
parts	\(-\frac {\operatorname {arccoth}\left (x a \right )^{3}}{x}-3 a \left (\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a +1\right )}{2}-\ln \left (x a \right ) \operatorname {arccoth}\left (x a \right )^{2}+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a -1\right )}{2}+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a -1}{x a +1}\right )}{2}+\frac {\operatorname {arccoth}\left (x a \right )^{3}}{3}-\frac {\left (-i \pi \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )+2 i \pi \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2}-2 i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{3}-2 i \pi \,\operatorname {csgn}\left (i \left (1+\frac {x a +1}{x a -1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{2}+2 i \pi \,\operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{2}+2 i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {x a +1}{x a -1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )+4 \ln \left (2\right )\right ) \operatorname {arccoth}\left (x a \right )^{2}}{4}-\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {x a +1}{x a -1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {x a +1}{x a -1}\right )}{2}\right )\)	\(718\)
derivativedivides	\(a \left (-\frac {\operatorname {arccoth}\left (x a \right )^{3}}{x a}-\frac {3 \operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a +1\right )}{2}+3 \ln \left (x a \right ) \operatorname {arccoth}\left (x a \right )^{2}-\frac {3 \operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a -1\right )}{2}-\frac {3 \operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a -1}{x a +1}\right )}{2}-\operatorname {arccoth}\left (x a \right )^{3}+\frac {3 \left (-i \pi \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )+2 i \pi \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2}-2 i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{3}-2 i \pi \,\operatorname {csgn}\left (i \left (1+\frac {x a +1}{x a -1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{2}+2 i \pi \,\operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{2}+2 i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {x a +1}{x a -1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )+4 \ln \left (2\right )\right ) \operatorname {arccoth}\left (x a \right )^{2}}{4}+3 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {x a +1}{x a -1}\right )-\frac {3 \operatorname {polylog}\left (3, -\frac {x a +1}{x a -1}\right )}{2}\right )\)	\(719\)
default	\(a \left (-\frac {\operatorname {arccoth}\left (x a \right )^{3}}{x a}-\frac {3 \operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a +1\right )}{2}+3 \ln \left (x a \right ) \operatorname {arccoth}\left (x a \right )^{2}-\frac {3 \operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a -1\right )}{2}-\frac {3 \operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a -1}{x a +1}\right )}{2}-\operatorname {arccoth}\left (x a \right )^{3}+\frac {3 \left (-i \pi \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )+2 i \pi \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2}-2 i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{3}-2 i \pi \,\operatorname {csgn}\left (i \left (1+\frac {x a +1}{x a -1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{2}+2 i \pi \,\operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{2}+2 i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {x a +1}{x a -1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )+4 \ln \left (2\right )\right ) \operatorname {arccoth}\left (x a \right )^{2}}{4}+3 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {x a +1}{x a -1}\right )-\frac {3 \operatorname {polylog}\left (3, -\frac {x a +1}{x a -1}\right )}{2}\right )\)	\(719\)

Input:

int(arccoth(x*a)^3/x^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{2}} \,d x } \] Input:

integrate(arccoth(a*x)^3/x^2,x, algorithm="fricas")
 

Output:

integral(arccoth(a*x)^3/x^2, x)
                                                                                    
                                                                                    
 

Sympy [F]

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

integrate(acoth(a*x)**3/x**2,x)
 

Output:

Integral(acoth(a*x)**3/x**2, x)
 

Maxima [F]

\[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{2}} \,d x } \] Input:

integrate(arccoth(a*x)^3/x^2,x, algorithm="maxima")
 

Output:

1/8*a*(log(a*x + 1) - log(x))*log(a)^3 + 3/8*a*integrate(x*log(a*x - 1)/(a 
*x^3 + x^2), x)*log(a)^2 - 3/8*a*integrate(x*log(x)/(a*x^3 + x^2), x)*log( 
a)^2 - 1/8*(a*log(a*x + 1) - a*log(x) - 1/x)*log(a)^3 + 3/4*a^2*integrate( 
x^2*log(a*x + 1)*log(a*x - 1)/(a*x^3 + x^2), x) - 3/2*a^2*integrate(x^2*lo 
g(a*x + 1)*log(x)/(a*x^3 + x^2), x) + 3/4*a*integrate(x*log(a*x - 1)*log(x 
)/(a*x^3 + x^2), x)*log(a) - 3/8*a*integrate(x*log(x)^2/(a*x^3 + x^2), x)* 
log(a) + 3/8*integrate(log(a*x - 1)/(a*x^3 + x^2), x)*log(a)^2 - 3/8*integ 
rate(log(x)/(a*x^3 + x^2), x)*log(a)^2 + 3/8*a*integrate(x*log(a*x + 1)*lo 
g(a*x - 1)^2/(a*x^3 + x^2), x) - 3/8*a*integrate(x*log(a*x - 1)^2*log(x)/( 
a*x^3 + x^2), x) + 3/8*a*integrate(x*log(a*x - 1)*log(x)^2/(a*x^3 + x^2), 
x) - 1/8*a*integrate(x*log(x)^3/(a*x^3 + x^2), x) - 3/4*a*integrate(x*log( 
a*x + 1)*log(a*x - 1)/(a*x^3 + x^2), x) - 3/8*integrate(a*x*log(a*x - 1)^2 
/(a*x^3 + x^2), x)*log(a) - 3/8*integrate(log(a*x - 1)^2/(a*x^3 + x^2), x) 
*log(a) + 3/4*integrate(log(a*x - 1)*log(x)/(a*x^3 + x^2), x)*log(a) - 3/8 
*integrate(log(x)^2/(a*x^3 + x^2), x)*log(a) - 3/8*(a^2*log(a*x - 1) - a^2 
*log(x) + a/x)*log(-1/(a*x) + 1)^2/a + 1/8*log(-1/(a*x) + 1)^3/x - 1/8*((a 
*x + 1)*log(a*x + 1)^3 - 3*(2*a*x*log(x) - (a*x - 1)*log(a*x - 1))*log(a*x 
 + 1)^2)/x + 1/8*(3*(a^3*x*log(a*x - 1)^2 + a^3*x*log(x)^2 - 2*a^3*x*log(x 
) + 2*a^2 - 2*(a^3*x*log(x) - a^3*x)*log(a*x - 1))*log(-1/(a*x) + 1)/(a*x) 
 - (a^4*x*log(a*x - 1)^3 - a^4*x*log(x)^3 + 3*a^4*x*log(x)^2 - 6*a^4*x*...
 

Giac [F]

\[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{2}} \,d x } \] Input:

integrate(arccoth(a*x)^3/x^2,x, algorithm="giac")
 

Output:

integrate(arccoth(a*x)^3/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^2} \,d x \] Input:

int(acoth(a*x)^3/x^2,x)
 

Output:

int(acoth(a*x)^3/x^2, x)
 

Reduce [F]

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

int(acoth(a*x)^3/x^2,x)
 

Output:

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