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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6034, 6200, 6096, 6204, 6745} \[ \int \frac {\coth ^{-1}(a x)^2}{x} \, dx=\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 a x}{a x+1}\right )+\operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+1}\right ) \coth ^{-1}(a x)+2 \coth ^{-1}\left (1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2 \]

[In]

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

[Out]

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

Rule 6034

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] - Dist[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 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 6200

Int[(ArcCoth[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[
Log[SimplifyIntegrand[1 + 1/u, x]]*((a + b*ArcCoth[c*x])^p/(d + e*x^2)), x], x] - Dist[1/2, Int[Log[SimplifyIn
tegrand[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 6204

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCot
h[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] - Dist[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 6745

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]

Rubi steps \begin{align*} \text {integral}& = 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 \\ & = 2 \coth ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )+(2 a) \int \frac {\coth ^{-1}(a x) \log \left (\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx-(2 a) \int \frac {\coth ^{-1}(a x) \log \left (\frac {2 a x}{1+a x}\right )}{1-a^2 x^2} \, 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 )-a \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx+a \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2 a x}{1+a x}\right )}{1-a^2 x^2} \, 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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.18 \[ \int \frac {\coth ^{-1}(a x)^2}{x} \, dx=\frac {2}{3} \coth ^{-1}(a x)^3+\coth ^{-1}(a x)^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )-\coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right ) \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 3.37 (sec) , antiderivative size = 459, normalized size of antiderivative = 4.73

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(arccoth(a*x)^2/x, x)

Sympy [F]

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

[In]

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

[Out]

Integral(acoth(a*x)**2/x, x)

Maxima [F]

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

[In]

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

[Out]

integrate(arccoth(a*x)^2/x, x)

Giac [F]

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

[In]

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

[Out]

integrate(arccoth(a*x)^2/x, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(acoth(a*x)^2/x, x)