3.1.26 \(\int x^2 \coth ^{-1}(a x)^3 \, dx\) [26]

Optimal. Leaf size=149 \[ \frac {x \coth ^{-1}(a x)}{a^2}-\frac {\coth ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \coth ^{-1}(a x)^2}{2 a}+\frac {\coth ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {\log \left (1-a^2 x^2\right )}{2 a^3}-\frac {\coth ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3}+\frac {\text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^3} \]

[Out]

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

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Rubi [A]
time = 0.23, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6038, 6128, 6022, 266, 6096, 6132, 6056, 6206, 6745} \begin {gather*} \frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^3}-\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a^3}+\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a^3}+\frac {x \coth ^{-1}(a x)}{a^2}+\frac {\log \left (1-a^2 x^2\right )}{2 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3+\frac {x^2 \coth ^{-1}(a x)^2}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCoth[a*x]^3,x]

[Out]

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 6038

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCoth[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -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 6128

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcCoth[c*x])
^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 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 x^2 \coth ^{-1}(a x)^3 \, dx &=\frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \int \frac {x^3 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \coth ^{-1}(a x)^3+\frac {\int x \coth ^{-1}(a x)^2 \, dx}{a}-\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{a}\\ &=\frac {x^2 \coth ^{-1}(a x)^2}{2 a}+\frac {\coth ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3-\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a x} \, dx}{a^2}-\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {x^2 \coth ^{-1}(a x)^2}{2 a}+\frac {\coth ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {\int \coth ^{-1}(a x) \, dx}{a^2}-\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{a^2}+\frac {2 \int \frac {\coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2}\\ &=\frac {x \coth ^{-1}(a x)}{a^2}-\frac {\coth ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \coth ^{-1}(a x)^2}{2 a}+\frac {\coth ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}-\frac {\coth ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3}+\frac {\int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2}-\frac {\int \frac {x}{1-a^2 x^2} \, dx}{a}\\ &=\frac {x \coth ^{-1}(a x)}{a^2}-\frac {\coth ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \coth ^{-1}(a x)^2}{2 a}+\frac {\coth ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {\log \left (1-a^2 x^2\right )}{2 a^3}-\frac {\coth ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3}+\frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^3}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.26, size = 140, normalized size = 0.94 \begin {gather*} \frac {-i \pi ^3+24 a x \coth ^{-1}(a x)-12 \coth ^{-1}(a x)^2+12 a^2 x^2 \coth ^{-1}(a x)^2+8 \coth ^{-1}(a x)^3+8 a^3 x^3 \coth ^{-1}(a x)^3-24 \coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-24 \log \left (\frac {1}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\right )-24 \coth ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )+12 \text {PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right )}{24 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcCoth[a*x]^3,x]

[Out]

((-I)*Pi^3 + 24*a*x*ArcCoth[a*x] - 12*ArcCoth[a*x]^2 + 12*a^2*x^2*ArcCoth[a*x]^2 + 8*ArcCoth[a*x]^3 + 8*a^3*x^
3*ArcCoth[a*x]^3 - 24*ArcCoth[a*x]^2*Log[1 - E^(2*ArcCoth[a*x])] - 24*Log[1/(a*Sqrt[1 - 1/(a^2*x^2)]*x)] - 24*
ArcCoth[a*x]*PolyLog[2, E^(2*ArcCoth[a*x])] + 12*PolyLog[3, E^(2*ArcCoth[a*x])])/(24*a^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.45, size = 683, normalized size = 4.58 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccoth(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^3*(1/3*a^3*x^3*arccoth(a*x)^3+1/2*a^2*x^2*arccoth(a*x)^2+1/2*arccoth(a*x)^2*ln(a*x-1)+1/2*arccoth(a*x)^2*l
n(a*x+1)+1/2*arccoth(a*x)^2*ln((a*x-1)/(a*x+1))+arccoth(a*x)^2*ln(1/(a*x-1)*(a*x+1)-1)+1/12*arccoth(a*x)*(-3*I
*arccoth(a*x)*csgn(I/(1/(a*x-1)*(a*x+1)-1))*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^2*Pi+3*I*arccoth(a*x
)*csgn(I/(1/(a*x-1)*(a*x+1)-1))*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))*csgn(I/(a*x-1)*(a*x+1))*Pi+3*I*a
rccoth(a*x)*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^3*Pi-3*I*arccoth(a*x)*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x
-1)*(a*x+1)-1))^2*csgn(I/(a*x-1)*(a*x+1))*Pi+3*I*arccoth(a*x)*csgn(I/(a*x-1)*(a*x+1))^3*Pi-6*I*arccoth(a*x)*cs
gn(I/(a*x-1)*(a*x+1))^2*csgn(I/((a*x-1)/(a*x+1))^(1/2))*Pi+3*I*arccoth(a*x)*csgn(I/(a*x-1)*(a*x+1))*csgn(I/((a
*x-1)/(a*x+1))^(1/2))^2*Pi+4*arccoth(a*x)^2-12*arccoth(a*x)*ln(2)-6*arccoth(a*x)+12*a*x+12)-ln(1/((a*x-1)/(a*x
+1))^(1/2)-1)-ln(1+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)*po
lylog(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)))

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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(x^2*arccoth(a*x)^3,x, algorithm="maxima")

[Out]

1/24*((a^3*x^3 + 1)*log(a*x + 1)^3 + 3*(a^2*x^2 - (a^3*x^3 - 1)*log(a*x - 1))*log(a*x + 1)^2)/a^3 + 1/8*integr
ate(-((a^3*x^3 + a^2*x^2)*log(a*x - 1)^3 + (2*a^2*x^2 - 3*(a^3*x^3 + a^2*x^2)*log(a*x - 1)^2 - 2*(a^3*x^3 - 1)
*log(a*x - 1))*log(a*x + 1))/(a^3*x + a^2), 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(x^2*arccoth(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^2*arccoth(a*x)^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \operatorname {acoth}^{3}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*acoth(a*x)**3,x)

[Out]

Integral(x**2*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(x^2*arccoth(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^2*arccoth(a*x)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*acoth(a*x)^3,x)

[Out]

int(x^2*acoth(a*x)^3, x)

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