3.1.28 \(\int \coth ^{-1}(a x)^3 \, dx\) [28]

Optimal. Leaf size=85 \[ \frac {\coth ^{-1}(a x)^3}{a}+x \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-\frac {3 \coth ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a}+\frac {3 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6022, 6132, 6056, 6096, 6206, 6745} \begin {gather*} \frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a}-\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}+x \coth ^{-1}(a x)^3+\frac {\coth ^{-1}(a x)^3}{a}-\frac {3 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6022

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x^n])^p, x] - Dist[b
*c*n*p, Int[x^n*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p
, 0] && (EqQ[n, 1] || EqQ[p, 1])

Rule 6056

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^
2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

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 6132

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 6206

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcC
oth[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*} \int \coth ^{-1}(a x)^3 \, dx &=x \coth ^{-1}(a x)^3-(3 a) \int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=\frac {\coth ^{-1}(a x)^3}{a}+x \coth ^{-1}(a x)^3-3 \int \frac {\coth ^{-1}(a x)^2}{1-a x} \, dx\\ &=\frac {\coth ^{-1}(a x)^3}{a}+x \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}+6 \int \frac {\coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {\coth ^{-1}(a x)^3}{a}+x \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-\frac {3 \coth ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a}+3 \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {\coth ^{-1}(a x)^3}{a}+x \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-\frac {3 \coth ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a}+\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 79, normalized size = 0.93 \begin {gather*} \frac {\coth ^{-1}(a x)^3}{a}+x \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )}{a}-\frac {3 \coth ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )}{a}+\frac {3 \text {PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right )}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(83)=166\).
time = 0.33, size = 168, normalized size = 1.98

method	result	size
derivativedivides	\(\frac {\mathrm {arccoth}\left (a x \right )^{3} \left (a x -1\right )+2 \mathrm {arccoth}\left (a x \right )^{3}-3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \polylog \left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \polylog \left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}\)	\(168\)
default	\(\frac {\mathrm {arccoth}\left (a x \right )^{3} \left (a x -1\right )+2 \mathrm {arccoth}\left (a x \right )^{3}-3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \polylog \left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \polylog \left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(arccoth(a*x)^3*(a*x-1)+2*arccoth(a*x)^3-3*arccoth(a*x)^2*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-6*arccoth(a*x)*p
olylog(2,1/((a*x-1)/(a*x+1))^(1/2))+6*polylog(3,1/((a*x-1)/(a*x+1))^(1/2))-3*arccoth(a*x)^2*ln(1+1/((a*x-1)/(a
*x+1))^(1/2))-6*arccoth(a*x)*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))+6*polylog(3,-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, algorithm="maxima")

[Out]

1/8*((a*x + 1)*log(a*x + 1)^3 - 3*(a*x - 1)*log(a*x + 1)^2*log(a*x - 1))/a + 1/8*integrate(-((a*x + 1)*log(a*x
- 1)^3 - 3*((a*x + 1)*log(a*x - 1)^2 + 2*(a*x - 1)*log(a*x - 1))*log(a*x + 1))/(a*x + 1), 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, algorithm="fricas")

[Out]

integral(arccoth(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(a*x)**3, 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, algorithm="giac")

[Out]

integrate(arccoth(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acoth(a*x)^3, x)

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