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

Optimal. Leaf size=97 \[ \frac {1}{2} \text {Li}_3\left (1-\frac {2}{a x+1}\right )-\frac {1}{2} \text {Li}_3\left (1-\frac {2 a x}{a x+1}\right )+\text {Li}_2\left (1-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\text {Li}_2\left (1-\frac {2 a x}{a x+1}\right ) \coth ^{-1}(a x)+2 \coth ^{-1}\left (1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2 \]

[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))

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Rubi [A]  time = 0.23, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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 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 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*}

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Mathematica [A]  time = 0.06, size = 114, normalized size = 1.18 \[ -\coth ^{-1}(a x) \text {Li}_2\left (-e^{-2 \coth ^{-1}(a x)}\right )-\coth ^{-1}(a x) \text {Li}_2\left (e^{2 \coth ^{-1}(a x)}\right )-\frac {1}{2} \text {Li}_3\left (-e^{-2 \coth ^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_3\left (e^{2 \coth ^{-1}(a x)}\right )+\frac {2}{3} \coth ^{-1}(a x)^3+\coth ^{-1}(a x)^2 \log \left (e^{-2 \coth ^{-1}(a x)}+1\right )-\coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right ) \]

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

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fricas [F]  time = 0.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (a x\right )^{2}}{x}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (a x\right )^{2}}{x}\,{d x} \]

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)

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maple [C]  time = 0.68, size = 487, normalized size = 5.02 \[ \ln \left (a x \right ) \mathrm {arccoth}\left (a x \right )^{2}+\frac {i \pi \,\mathrm {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {arccoth}\left (a x \right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2} \mathrm {arccoth}\left (a x \right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2} \mathrm {arccoth}\left (a x \right )^{2}}{2}+\frac {i \pi \mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{3} \mathrm {arccoth}\left (a x \right )^{2}}{2}+\mathrm {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x +1}{a x -1}-1\right )-\mathrm {arccoth}\left (a x \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \polylog \left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\mathrm {arccoth}\left (a x \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \polylog \left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+\mathrm {arccoth}\left (a x \right ) \polylog \left (2, -\frac {a x +1}{a x -1}\right )-\frac {\polylog \left (3, -\frac {a x +1}{a x -1}\right )}{2} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (a x\right )^{2}}{x}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acoth}\left (a\,x\right )}^2}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acoth(a*x)^2/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{x}\, dx \]

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)

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