3.279 \(\int \frac{\tanh ^{-1}(a x)^3}{x^3 (1-a^2 x^2)^2} \, dx\)

Optimal. Leaf size=302 \[ -\frac{3}{2} a^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{3}{2} a^2 \text{PolyLog}\left (4,\frac{2}{a x+1}-1\right )-3 a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-\frac{3 a^3 x}{8 \left (1-a^2 x^2\right )}+\frac{a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac{3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac{1}{2} a^2 \tanh ^{-1}(a x)^4+\frac{1}{4} a^2 \tanh ^{-1}(a x)^3+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2-\frac{3}{8} a^2 \tanh ^{-1}(a x)+2 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3+3 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^3}{2 x^2}-\frac{3 a \tanh ^{-1}(a x)^2}{2 x} \]

[Out]

(-3*a^3*x)/(8*(1 - a^2*x^2)) - (3*a^2*ArcTanh[a*x])/8 + (3*a^2*ArcTanh[a*x])/(4*(1 - a^2*x^2)) + (3*a^2*ArcTan
h[a*x]^2)/2 - (3*a*ArcTanh[a*x]^2)/(2*x) - (3*a^3*x*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)) + (a^2*ArcTanh[a*x]^3)/4
- ArcTanh[a*x]^3/(2*x^2) + (a^2*ArcTanh[a*x]^3)/(2*(1 - a^2*x^2)) + (a^2*ArcTanh[a*x]^4)/2 + 3*a^2*ArcTanh[a*
x]*Log[2 - 2/(1 + a*x)] + 2*a^2*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - (3*a^2*PolyLog[2, -1 + 2/(1 + a*x)])/2 -
3*a^2*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)] - 3*a^2*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)] - (3*a^2*
PolyLog[4, -1 + 2/(1 + a*x)])/2

________________________________________________________________________________________

Rubi [A]  time = 0.955259, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 14, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {6030, 5982, 5916, 5988, 5932, 2447, 5948, 6056, 6060, 6610, 5994, 5956, 199, 206} \[ -\frac{3}{2} a^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{3}{2} a^2 \text{PolyLog}\left (4,\frac{2}{a x+1}-1\right )-3 a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-\frac{3 a^3 x}{8 \left (1-a^2 x^2\right )}+\frac{a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac{3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac{1}{2} a^2 \tanh ^{-1}(a x)^4+\frac{1}{4} a^2 \tanh ^{-1}(a x)^3+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2-\frac{3}{8} a^2 \tanh ^{-1}(a x)+2 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3+3 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^3}{2 x^2}-\frac{3 a \tanh ^{-1}(a x)^2}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)^2),x]

[Out]

(-3*a^3*x)/(8*(1 - a^2*x^2)) - (3*a^2*ArcTanh[a*x])/8 + (3*a^2*ArcTanh[a*x])/(4*(1 - a^2*x^2)) + (3*a^2*ArcTan
h[a*x]^2)/2 - (3*a*ArcTanh[a*x]^2)/(2*x) - (3*a^3*x*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)) + (a^2*ArcTanh[a*x]^3)/4
- ArcTanh[a*x]^3/(2*x^2) + (a^2*ArcTanh[a*x]^3)/(2*(1 - a^2*x^2)) + (a^2*ArcTanh[a*x]^4)/2 + 3*a^2*ArcTanh[a*
x]*Log[2 - 2/(1 + a*x)] + 2*a^2*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - (3*a^2*PolyLog[2, -1 + 2/(1 + a*x)])/2 -
3*a^2*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)] - 3*a^2*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)] - (3*a^2*
PolyLog[4, -1 + 2/(1 + a*x)])/2

Rule 6030

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int
[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh
[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[
m, 0] && NeQ[p, -1]

Rule 5982

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d
, Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/(d*f^2), Int[((f*x)^(m + 2)*(a + b*ArcTanh[c*x])^p)/(d +
e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rule 5916

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

Rule 5988

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

Rule 5932

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTanh[c*
x])^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[2 - 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 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 5948

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[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 6056

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcTa
nh[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p)/2, Int[((a + b*ArcTanh[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 6060

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[((a
+ b*ArcTanh[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] + Dist[(b*p)/2, Int[((a + b*ArcTanh[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]

Rule 5994

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

Rule 5956

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{x^3 \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )^2} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=2 \left (a^2 \int \frac{\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx\right )+a^4 \int \frac{x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x^3} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{1}{2} (3 a) \int \frac{\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx+2 \left (\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \int \frac{\tanh ^{-1}(a x)^3}{x (1+a x)} \, dx\right )-\frac{1}{2} \left (3 a^3\right ) \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac{1}{4} a^2 \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{1}{2} (3 a) \int \frac{\tanh ^{-1}(a x)^2}{x^2} \, dx+\frac{1}{2} \left (3 a^3\right ) \int \frac{\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx+2 \left (\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\left (3 a^3\right ) \int \frac{\tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\right )+\frac{1}{2} \left (3 a^4\right ) \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac{3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac{3 a \tanh ^{-1}(a x)^2}{2 x}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac{1}{4} a^2 \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\frac{1}{4} \left (3 a^3\right ) \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx+2 \left (\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+\left (3 a^3\right ) \int \frac{\tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\right )\\ &=-\frac{3 a^3 x}{8 \left (1-a^2 x^2\right )}+\frac{3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{2 x}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac{1}{4} a^2 \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\frac{1}{8} \left (3 a^3\right ) \int \frac{1}{1-a^2 x^2} \, dx+2 \left (\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )+\frac{1}{2} \left (3 a^3\right ) \int \frac{\text{Li}_3\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\right )\\ &=-\frac{3 a^3 x}{8 \left (1-a^2 x^2\right )}-\frac{3}{8} a^2 \tanh ^{-1}(a x)+\frac{3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{2 x}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac{1}{4} a^2 \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )+2 \left (\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )-\frac{3}{4} a^2 \text{Li}_4\left (-1+\frac{2}{1+a x}\right )\right )-\left (3 a^3\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{3 a^3 x}{8 \left (1-a^2 x^2\right )}-\frac{3}{8} a^2 \tanh ^{-1}(a x)+\frac{3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{2 x}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac{1}{4} a^2 \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+2 \left (\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )-\frac{3}{4} a^2 \text{Li}_4\left (-1+\frac{2}{1+a x}\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.74957, size = 215, normalized size = 0.71 \[ \frac{1}{32} a^2 \left (96 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )-96 \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )-48 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )+48 \text{PolyLog}\left (4,e^{2 \tanh ^{-1}(a x)}\right )-\frac{16 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}{a^2 x^2}-16 \tanh ^{-1}(a x)^4-\frac{48 \tanh ^{-1}(a x)^2}{a x}+48 \tanh ^{-1}(a x)^2+64 \tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+96 \tanh ^{-1}(a x) \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )-12 \tanh ^{-1}(a x)^2 \sinh \left (2 \tanh ^{-1}(a x)\right )-6 \sinh \left (2 \tanh ^{-1}(a x)\right )+8 \tanh ^{-1}(a x)^3 \cosh \left (2 \tanh ^{-1}(a x)\right )+12 \tanh ^{-1}(a x) \cosh \left (2 \tanh ^{-1}(a x)\right )+\pi ^4\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)^2),x]

[Out]

(a^2*(Pi^4 + 48*ArcTanh[a*x]^2 - (48*ArcTanh[a*x]^2)/(a*x) - (16*(1 - a^2*x^2)*ArcTanh[a*x]^3)/(a^2*x^2) - 16*
ArcTanh[a*x]^4 + 12*ArcTanh[a*x]*Cosh[2*ArcTanh[a*x]] + 8*ArcTanh[a*x]^3*Cosh[2*ArcTanh[a*x]] + 96*ArcTanh[a*x
]*Log[1 - E^(-2*ArcTanh[a*x])] + 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] - 48*PolyLog[2, E^(-2*ArcTanh[a
*x])] + 96*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] - 96*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] + 48
*PolyLog[4, E^(2*ArcTanh[a*x])] - 6*Sinh[2*ArcTanh[a*x]] - 12*ArcTanh[a*x]^2*Sinh[2*ArcTanh[a*x]]))/32

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Maple [B]  time = 1.29, size = 672, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)^3/x^3/(-a^2*x^2+1)^2,x)

[Out]

12*a^2*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))+12*a^2*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))-3/2*a^2*arctanh(a*x
)^2-3/2*a*arctanh(a*x)^2/x+3/16*a^2*arctanh(a*x)/(a*x+1)-3/16*a^2*arctanh(a*x)/(a*x-1)-12*a^2*arctanh(a*x)*pol
ylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a^2*arctanh(a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a^2*arctanh(a
*x)^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-12*a^2*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-3/16*a^
3/(a*x+1)*arctanh(a*x)*x-1/8*a^3/(a*x-1)*arctanh(a*x)^3*x+3/16*a^3/(a*x-1)*arctanh(a*x)^2*x-3/16*a^3/(a*x-1)*a
rctanh(a*x)*x-1/8*a^3/(a*x+1)*arctanh(a*x)^3*x-3/16*a^3/(a*x+1)*arctanh(a*x)^2*x+3/32*a^3*x/(a*x-1)-3/32*a^3/(
a*x+1)*x-1/8*a^2/(a*x-1)*arctanh(a*x)^3+1/8*a^2/(a*x+1)*arctanh(a*x)^3+2*a^2*arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2
*x^2+1)^(1/2))+2*a^2*arctanh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2
+1)^(1/2))+3*a^2*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3/32*a^2/(a*x-1)+3/32*a^2/(a*x+1)+1/2*a^2*arcta
nh(a*x)^3-1/2*arctanh(a*x)^3/x^2-1/2*a^2*arctanh(a*x)^4+3*a^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*pol
ylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+3/16*a^2*arctanh(a*x)^2/(a*x-1)+3/16*a^2*arctanh(a*x)^2/(a*x+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (a^{4} x^{4} - a^{2} x^{2}\right )} \log \left (-a x + 1\right )^{4} + 2 \,{\left (2 \, a^{2} x^{2} + 2 \,{\left (a^{4} x^{4} - a^{2} x^{2}\right )} \log \left (a x + 1\right ) - 1\right )} \log \left (-a x + 1\right )^{3}}{32 \,{\left (a^{2} x^{4} - x^{2}\right )}} - \frac{1}{8} \, \int -\frac{2 \, \log \left (a x + 1\right )^{3} - 6 \, \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) - 3 \,{\left (2 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - a^{2} x^{2} - a x + 2 \,{\left (a^{6} x^{6} + a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} - 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{2 \,{\left (a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^3/x^3/(-a^2*x^2+1)^2,x, algorithm="maxima")

[Out]

1/32*((a^4*x^4 - a^2*x^2)*log(-a*x + 1)^4 + 2*(2*a^2*x^2 + 2*(a^4*x^4 - a^2*x^2)*log(a*x + 1) - 1)*log(-a*x +
1)^3)/(a^2*x^4 - x^2) - 1/8*integrate(-1/2*(2*log(a*x + 1)^3 - 6*log(a*x + 1)^2*log(-a*x + 1) - 3*(2*a^4*x^4 +
2*a^3*x^3 - a^2*x^2 - a*x + 2*(a^6*x^6 + a^5*x^5 - a^4*x^4 - a^3*x^3 - 1)*log(a*x + 1))*log(-a*x + 1)^2)/(a^4
*x^7 - 2*a^2*x^5 + x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{artanh}\left (a x\right )^{3}}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arctanh(a*x)^3/(a^4*x^7 - 2*a^2*x^5 + x^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}^{3}{\left (a x \right )}}{x^{3} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**3/x**3/(-a**2*x**2+1)**2,x)

[Out]

Integral(atanh(a*x)**3/(x**3*(a*x - 1)**2*(a*x + 1)**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{2} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^3/x^3/(-a^2*x^2+1)^2,x, algorithm="giac")

[Out]

integrate(arctanh(a*x)^3/((a^2*x^2 - 1)^2*x^3), x)