3.1.29 \(\int \frac {\coth ^{-1}(a x)^3}{x} \, dx\) [29]

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

[Out]

2*arccoth(a*x)^3*arccoth(1-2/(-a*x+1))+3/2*arccoth(a*x)^2*polylog(2,1-2/(a*x+1))-3/2*arccoth(a*x)^2*polylog(2,
1-2*a*x/(a*x+1))+3/2*arccoth(a*x)*polylog(3,1-2/(a*x+1))-3/2*arccoth(a*x)*polylog(3,1-2*a*x/(a*x+1))+3/4*polyl
og(4,1-2/(a*x+1))-3/4*polylog(4,1-2*a*x/(a*x+1))

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Rubi [A]
time = 0.24, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6034, 6200, 6096, 6204, 6208, 6745} \begin {gather*} \frac {3}{4} \text {Li}_4\left (1-\frac {2}{a x+1}\right )-\frac {3}{4} \text {Li}_4\left (1-\frac {2 a x}{a x+1}\right )+\frac {3}{2} \text {Li}_2\left (1-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2-\frac {3}{2} \text {Li}_2\left (1-\frac {2 a x}{a x+1}\right ) \coth ^{-1}(a x)^2+\frac {3}{2} \text {Li}_3\left (1-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\frac {3}{2} \text {Li}_3\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)^3 \end {gather*}

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 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 6208

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 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*} \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*}

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Mathematica [A]
time = 0.05, size = 156, normalized size = 1.04 \begin {gather*} \frac {1}{64} \left (-\pi ^4+32 \coth ^{-1}(a x)^4+64 \coth ^{-1}(a x)^3 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )-64 \coth ^{-1}(a x)^3 \log \left (1-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)^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 )\right ) \end {gather*}

Antiderivative was successfully verified.

[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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.21, size = 564, normalized size = 3.76

method	result	size
derivativedivides	\(\ln \left (a x \right ) \mathrm {arccoth}\left (a x \right )^{3}+\mathrm {arccoth}\left (a x \right )^{3} \ln \left (\frac {a x +1}{a x -1}-1\right )+\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{a x -1}\right )}{2}-\frac {3 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (3, -\frac {a x +1}{a x -1}\right )}{2}+\frac {3 \polylog \left (4, -\frac {a x +1}{a x -1}\right )}{4}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \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 ) \mathrm {arccoth}\left (a x \right )^{3}}{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 )^{3}}{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 )^{3}}{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 )^{3}}{2}-\mathrm {arccoth}\left (a x \right )^{3} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \mathrm {arccoth}\left (a x \right )^{2} \polylog \left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \polylog \left (4, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\mathrm {arccoth}\left (a x \right )^{3} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \mathrm {arccoth}\left (a x \right )^{2} \polylog \left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \polylog \left (4, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )\)	\(564\)
default	\(\ln \left (a x \right ) \mathrm {arccoth}\left (a x \right )^{3}+\mathrm {arccoth}\left (a x \right )^{3} \ln \left (\frac {a x +1}{a x -1}-1\right )+\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{a x -1}\right )}{2}-\frac {3 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (3, -\frac {a x +1}{a x -1}\right )}{2}+\frac {3 \polylog \left (4, -\frac {a x +1}{a x -1}\right )}{4}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \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 ) \mathrm {arccoth}\left (a x \right )^{3}}{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 )^{3}}{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 )^{3}}{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 )^{3}}{2}-\mathrm {arccoth}\left (a x \right )^{3} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \mathrm {arccoth}\left (a x \right )^{2} \polylog \left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \polylog \left (4, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\mathrm {arccoth}\left (a x \right )^{3} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \mathrm {arccoth}\left (a x \right )^{2} \polylog \left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \polylog \left (4, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )\)	\(564\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x)^3/x,x,method=_RETURNVERBOSE)

[Out]

ln(a*x)*arccoth(a*x)^3+arccoth(a*x)^3*ln(1/(a*x-1)*(a*x+1)-1)+3/2*arccoth(a*x)^2*polylog(2,-1/(a*x-1)*(a*x+1))
-3/2*arccoth(a*x)*polylog(3,-1/(a*x-1)*(a*x+1))+3/4*polylog(4,-1/(a*x-1)*(a*x+1))+1/2*I*Pi*csgn(I/(1/(a*x-1)*(
a*x+1)-1))*csgn(I*(1+1/(a*x-1)*(a*x+1)))*csgn(I/(1/(a*x-1)*(a*x+1)-1)*(1+1/(a*x-1)*(a*x+1)))*arccoth(a*x)^3-1/
2*I*Pi*csgn(I/(1/(a*x-1)*(a*x+1)-1))*csgn(I/(1/(a*x-1)*(a*x+1)-1)*(1+1/(a*x-1)*(a*x+1)))^2*arccoth(a*x)^3-1/2*
I*Pi*csgn(I*(1+1/(a*x-1)*(a*x+1)))*csgn(I/(1/(a*x-1)*(a*x+1)-1)*(1+1/(a*x-1)*(a*x+1)))^2*arccoth(a*x)^3+1/2*I*
Pi*csgn(I/(1/(a*x-1)*(a*x+1)-1)*(1+1/(a*x-1)*(a*x+1)))^3*arccoth(a*x)^3-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))-arccoth(a*x)^3*ln(1+1/((a*x-1)/(a*x+1))^(1/2))-3*arccoth(a*x)^2*po
lylog(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acoth(a*x)^3/x, x)

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