\(\int \frac {\coth ^{-1}(a x)^2}{x} \, dx\) [18]

Optimal result

Integrand size = 10, antiderivative size = 97 \[ \int \frac {\coth ^{-1}(a x)^2}{x} \, dx=2 \coth ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )+\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+a x}\right )-\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2 a x}{1+a x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{1+a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 a x}{1+a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.18 \[ \int \frac {\coth ^{-1}(a x)^2}{x} \, dx=\frac {2}{3} \coth ^{-1}(a x)^3+\coth ^{-1}(a x)^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )-\coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.32, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6449, 6615, 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.

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

\(\Big \downarrow \) 6449

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

\(\Big \downarrow \) 6615

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

\(\Big \downarrow \) 6619

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

2*ArcCoth[a*x]^2*ArcCoth[1 - 2/(1 - a*x)] - 4*a*((-1/2*(ArcCoth[a*x]*PolyL 
og[2, 1 - 2/(1 + a*x)])/a - PolyLog[3, 1 - 2/(1 + a*x)]/(4*a))/2 + ((ArcCo 
th[a*x]*PolyLog[2, 1 - (2*a*x)/(1 + a*x)])/(2*a) + PolyLog[3, 1 - (2*a*x)/ 
(1 + a*x)]/(4*a))/2)
 

Defintions of rubi rules used

rule 6449

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

rule 6615

Int[(ArcCoth[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*( 
x_)^2), x_Symbol] :> Simp[1/2   Int[Log[SimplifyIntegrand[1 + 1/u, x]]*((a 
+ b*ArcCoth[c*x])^p/(d + e*x^2)), x], x] - Simp[1/2   Int[Log[SimplifyInteg 
rand[1 - 1/u, x]]*((a + b*ArcCoth[c*x])^p/(d + e*x^2)), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 
- c*x))^2, 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 = 3.54 (sec) , antiderivative size = 459, normalized size of antiderivative = 4.73

method	result	size
derivativedivides	\(\ln \left (x a \right ) \operatorname {arccoth}\left (x a \right )^{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right ) \left (\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {x a +1}{x a -1}\right )\right )-\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {x a +1}{x a -1}\right )\right )+\operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (x a \right )^{2}}{2}+\operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a +1}{x a -1}-1\right )-\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-2 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-2 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {x a +1}{x a -1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {x a +1}{x a -1}\right )}{2}\)	\(459\)
default	\(\ln \left (x a \right ) \operatorname {arccoth}\left (x a \right )^{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right ) \left (\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {x a +1}{x a -1}\right )\right )-\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {x a +1}{x a -1}\right )\right )+\operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (x a \right )^{2}}{2}+\operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a +1}{x a -1}-1\right )-\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-2 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-2 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {x a +1}{x a -1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {x a +1}{x a -1}\right )}{2}\)	\(459\)
parts	\(\ln \left (x \right ) \operatorname {arccoth}\left (x a \right )^{2}+2 a \left (\frac {\left (i \pi \,\operatorname {csgn}\left (\frac {i}{a}\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 ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{a \left (\frac {x a +1}{x a -1}-1\right )}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{a}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{a \left (\frac {x a +1}{x a -1}-1\right )}\right )^{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 (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right )^{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}+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 (1+\frac {x a +1}{x a -1}\right )}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{a \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (1+\frac {x a +1}{x a -1}\right )}{a \left (\frac {x a +1}{x a -1}-1\right )}\right )^{3}+2 \ln \left (a \right )\right ) \operatorname {arccoth}\left (x a \right )^{2}}{4 a}+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a +1}{x a -1}-1\right )}{2 a}-\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )}{2 a}-\frac {\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )}{a}+\frac {\operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )}{a}-\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )}{2 a}-\frac {\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )}{a}+\frac {\operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )}{a}+\frac {\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {x a +1}{x a -1}\right )}{2 a}-\frac {\operatorname {polylog}\left (3, -\frac {x a +1}{x a -1}\right )}{4 a}\right )\)	\(772\)

Input:

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

Output:

ln(x*a)*arccoth(x*a)^2+1/2*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x 
-1)))*(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))*csgn(I/((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-1)))-csgn(I/(( 
a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-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)))^2)*arccoth(x*a)^2+arccoth(x*a)^2* 
ln((a*x+1)/(a*x-1)-1)-arccoth(x*a)^2*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-2*arc 
coth(x*a)*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))+2*polylog(3,1/((a*x-1)/(a*x 
+1))^(1/2))-arccoth(x*a)^2*ln(1+1/((a*x-1)/(a*x+1))^(1/2))-2*arccoth(x*a)* 
polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))+2*polylog(3,-1/((a*x-1)/(a*x+1))^(1/ 
2))+arccoth(x*a)*polylog(2,-(a*x+1)/(a*x-1))-1/2*polylog(3,-(a*x+1)/(a*x-1 
))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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