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

Optimal. Leaf size=97 \[ \frac{1}{2} \text{PolyLog}\left (3,1-\frac{2}{a x+1}\right )-\frac{1}{2} \text{PolyLog}\left (3,1-\frac{2 a x}{a x+1}\right )+\coth ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{a x+1}\right )-\coth ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2 a x}{a x+1}\right )+2 \coth ^{-1}\left (1-\frac{2}{1-a x}\right ) \coth ^{-1}(a x)^2 \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.232009, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5915, 6053, 5949, 6057, 6610} \[ \frac{1}{2} \text{PolyLog}\left (3,1-\frac{2}{a x+1}\right )-\frac{1}{2} \text{PolyLog}\left (3,1-\frac{2 a x}{a x+1}\right )+\coth ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{a x+1}\right )-\coth ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2 a x}{a x+1}\right )+2 \coth ^{-1}\left (1-\frac{2}{1-a x}\right ) \coth ^{-1}(a x)^2 \]

Antiderivative was successfully verified.

[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 5915

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 6053

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[SimplifyI
ntegrand[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 5949

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 6057

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcCo
th[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 6610

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*} \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 )-(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) \text{Li}_2\left (1-\frac{2}{1+a x}\right )-\coth ^{-1}(a x) \text{Li}_2\left (1-\frac{2 a x}{1+a x}\right )-a \int \frac{\text{Li}_2\left (1-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx+a \int \frac{\text{Li}_2\left (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) \text{Li}_2\left (1-\frac{2}{1+a x}\right )-\coth ^{-1}(a x) \text{Li}_2\left (1-\frac{2 a x}{1+a x}\right )+\frac{1}{2} \text{Li}_3\left (1-\frac{2}{1+a x}\right )-\frac{1}{2} \text{Li}_3\left (1-\frac{2 a x}{1+a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0634424, size = 114, normalized size = 1.18 \[ -\coth ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-\coth ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right )+\frac{2}{3} \coth ^{-1}(a x)^3+\coth ^{-1}(a x)^2 \log \left (e^{-2 \coth ^{-1}(a x)}+1\right )-\coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right ) \]

Warning: Unable to verify antiderivative.

[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]  time = 0.381, size = 487, normalized size = 5. \begin{align*} \ln \left ( ax \right ) \left ({\rm arccoth} \left (ax\right ) \right ) ^{2}+{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ({\frac{ax+1}{ax-1}}+1 \right ) \right ){\it csgn} \left ({i \left ({\frac{ax+1}{ax-1}}-1 \right ) ^{-1}} \right ){\it csgn} \left ({i \left ({\frac{ax+1}{ax-1}}+1 \right ) \left ({\frac{ax+1}{ax-1}}-1 \right ) ^{-1}} \right ) \left ({\rm arccoth} \left (ax\right ) \right ) ^{2}-{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ({\frac{ax+1}{ax-1}}+1 \right ) \right ) \left ({\it csgn} \left ({i \left ({\frac{ax+1}{ax-1}}+1 \right ) \left ({\frac{ax+1}{ax-1}}-1 \right ) ^{-1}} \right ) \right ) ^{2} \left ({\rm arccoth} \left (ax\right ) \right ) ^{2}-{\frac{i}{2}}\pi \,{\it csgn} \left ({i \left ({\frac{ax+1}{ax-1}}-1 \right ) ^{-1}} \right ) \left ({\it csgn} \left ({i \left ({\frac{ax+1}{ax-1}}+1 \right ) \left ({\frac{ax+1}{ax-1}}-1 \right ) ^{-1}} \right ) \right ) ^{2} \left ({\rm arccoth} \left (ax\right ) \right ) ^{2}+{\frac{i}{2}}\pi \, \left ({\it csgn} \left ({i \left ({\frac{ax+1}{ax-1}}+1 \right ) \left ({\frac{ax+1}{ax-1}}-1 \right ) ^{-1}} \right ) \right ) ^{3} \left ({\rm arccoth} \left (ax\right ) \right ) ^{2}+ \left ({\rm arccoth} \left (ax\right ) \right ) ^{2}\ln \left ({\frac{ax+1}{ax-1}}-1 \right ) - \left ({\rm arccoth} \left (ax\right ) \right ) ^{2}\ln \left ( 1-{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}} \right ) -2\,{\rm arccoth} \left (ax\right ){\it polylog} \left ( 2,{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}} \right ) +2\,{\it polylog} \left ( 3,{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}} \right ) - \left ({\rm arccoth} \left (ax\right ) \right ) ^{2}\ln \left ( 1+{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}} \right ) -2\,{\rm arccoth} \left (ax\right ){\it polylog} \left ( 2,-{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}} \right ) +2\,{\it polylog} \left ( 3,-{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}} \right ) +{\rm arccoth} \left (ax\right ){\it polylog} \left ( 2,-{\frac{ax+1}{ax-1}} \right ) -{\frac{1}{2}{\it polylog} \left ( 3,-{\frac{ax+1}{ax-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (a x\right )^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccoth(a*x)^2/x, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcoth}\left (a x\right )^{2}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccoth(a*x)^2/x, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acoth}^{2}{\left (a x \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(a*x)**2/x, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (a x\right )^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccoth(a*x)^2/x, x)