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

Optimal. Leaf size=150 \[ \frac{3}{4} \text{PolyLog}\left (4,1-\frac{2}{a x+1}\right )-\frac{3}{4} \text{PolyLog}\left (4,1-\frac{2 a x}{a x+1}\right )+\frac{3}{2} \coth ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{a x+1}\right )-\frac{3}{2} \coth ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2 a x}{a x+1}\right )+\frac{3}{2} \coth ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{a x+1}\right )-\frac{3}{2} \coth ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2 a x}{a x+1}\right )+2 \coth ^{-1}\left (1-\frac{2}{1-a x}\right ) \coth ^{-1}(a x)^3 \]

[Out]

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

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Rubi [A]  time = 0.350385, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5915, 6053, 5949, 6057, 6061, 6610} \[ \frac{3}{4} \text{PolyLog}\left (4,1-\frac{2}{a x+1}\right )-\frac{3}{4} \text{PolyLog}\left (4,1-\frac{2 a x}{a x+1}\right )+\frac{3}{2} \coth ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{a x+1}\right )-\frac{3}{2} \coth ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2 a x}{a x+1}\right )+\frac{3}{2} \coth ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{a x+1}\right )-\frac{3}{2} \coth ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2 a x}{a x+1}\right )+2 \coth ^{-1}\left (1-\frac{2}{1-a x}\right ) \coth ^{-1}(a x)^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 6061

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[((a
+ b*ArcCoth[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] + Dist[(b*p)/2, Int[((a + b*ArcCoth[c*x])^(p - 1)*PolyLog[k
+ 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[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)^3}{x} \, dx &=2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1-a x}\right )-(6 a) \int \frac{\coth ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1-a x}\right )+(3 a) \int \frac{\coth ^{-1}(a x)^2 \log \left (\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx-(3 a) \int \frac{\coth ^{-1}(a x)^2 \log \left (\frac{2 a x}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1-a x}\right )+\frac{3}{2} \coth ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+a x}\right )-\frac{3}{2} \coth ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2 a x}{1+a x}\right )-(3 a) \int \frac{\coth ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx+(3 a) \int \frac{\coth ^{-1}(a x) \text{Li}_2\left (1-\frac{2 a x}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1-a x}\right )+\frac{3}{2} \coth ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+a x}\right )-\frac{3}{2} \coth ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2 a x}{1+a x}\right )+\frac{3}{2} \coth ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+a x}\right )-\frac{3}{2} \coth ^{-1}(a x) \text{Li}_3\left (1-\frac{2 a x}{1+a x}\right )-\frac{1}{2} (3 a) \int \frac{\text{Li}_3\left (1-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx+\frac{1}{2} (3 a) \int \frac{\text{Li}_3\left (1-\frac{2 a x}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1-a x}\right )+\frac{3}{2} \coth ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+a x}\right )-\frac{3}{2} \coth ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2 a x}{1+a x}\right )+\frac{3}{2} \coth ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+a x}\right )-\frac{3}{2} \coth ^{-1}(a x) \text{Li}_3\left (1-\frac{2 a x}{1+a x}\right )+\frac{3}{4} \text{Li}_4\left (1-\frac{2}{1+a x}\right )-\frac{3}{4} \text{Li}_4\left (1-\frac{2 a x}{1+a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0758937, size = 156, normalized size = 1.04 \[ \frac{1}{64} \left (-96 \coth ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-96 \coth ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )-96 \coth ^{-1}(a x) \text{PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right )+96 \coth ^{-1}(a x) \text{PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right )-48 \text{PolyLog}\left (4,-e^{-2 \coth ^{-1}(a x)}\right )-48 \text{PolyLog}\left (4,e^{2 \coth ^{-1}(a x)}\right )+32 \coth ^{-1}(a x)^4+64 \coth ^{-1}(a x)^3 \log \left (e^{-2 \coth ^{-1}(a x)}+1\right )-64 \coth ^{-1}(a x)^3 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-\pi ^4\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-Pi^4 + 32*ArcCoth[a*x]^4 + 64*ArcCoth[a*x]^3*Log[1 + E^(-2*ArcCoth[a*x])] - 64*ArcCoth[a*x]^3*Log[1 - E^(2*A
rcCoth[a*x])] - 96*ArcCoth[a*x]^2*PolyLog[2, -E^(-2*ArcCoth[a*x])] - 96*ArcCoth[a*x]^2*PolyLog[2, E^(2*ArcCoth
[a*x])] - 96*ArcCoth[a*x]*PolyLog[3, -E^(-2*ArcCoth[a*x])] + 96*ArcCoth[a*x]*PolyLog[3, E^(2*ArcCoth[a*x])] -
48*PolyLog[4, -E^(-2*ArcCoth[a*x])] - 48*PolyLog[4, E^(2*ArcCoth[a*x])])/64

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Maple [C]  time = 0.253, size = 564, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x)^3/x,x)

[Out]

ln(a*x)*arccoth(a*x)^3+arccoth(a*x)^3*ln((a*x+1)/(a*x-1)-1)+3/2*arccoth(a*x)^2*polylog(2,-(a*x+1)/(a*x-1))-3/2
*arccoth(a*x)*polylog(3,-(a*x+1)/(a*x-1))+3/4*polylog(4,-(a*x+1)/(a*x-1))+1/2*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1)*
((a*x+1)/(a*x-1)+1))^3*arccoth(a*x)^3+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)^3-1/2*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)^3-1/2*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)^3-arccoth(a*x)^3*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-3*arccoth(a*x)^2*polylo
g(2,1/((a*x-1)/(a*x+1))^(1/2))+6*arccoth(a*x)*polylog(3,1/((a*x-1)/(a*x+1))^(1/2))-6*polylog(4,1/((a*x-1)/(a*x
+1))^(1/2))-arccoth(a*x)^3*ln(1+1/((a*x-1)/(a*x+1))^(1/2))-3*arccoth(a*x)^2*polylog(2,-1/((a*x-1)/(a*x+1))^(1/
2))+6*arccoth(a*x)*polylog(3,-1/((a*x-1)/(a*x+1))^(1/2))-6*polylog(4,-1/((a*x-1)/(a*x+1))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (a x\right )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccoth(a*x)^3/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcoth}\left (a x\right )^{3}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccoth(a*x)^3/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acoth}^{3}{\left (a x \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(a*x)**3/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (a x\right )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccoth(a*x)^3/x, x)