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

Optimal. Leaf size=281 \[ -\frac{3}{2} a \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-3 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{93 a}{128 \left (1-a^2 x^2\right )}-\frac{3 a}{128 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac{a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac{21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}-\frac{3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac{93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{15}{32} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{93}{128} a \tanh ^{-1}(a x)^2+3 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]

[Out]

(-3*a)/(128*(1 - a^2*x^2)^2) - (93*a)/(128*(1 - a^2*x^2)) + (3*a^2*x*ArcTanh[a*x])/(32*(1 - a^2*x^2)^2) + (93*
a^2*x*ArcTanh[a*x])/(64*(1 - a^2*x^2)) + (93*a*ArcTanh[a*x]^2)/128 - (3*a*ArcTanh[a*x]^2)/(16*(1 - a^2*x^2)^2)
- (21*a*ArcTanh[a*x]^2)/(16*(1 - a^2*x^2)) + a*ArcTanh[a*x]^3 - ArcTanh[a*x]^3/x + (a^2*x*ArcTanh[a*x]^3)/(4*
(1 - a^2*x^2)^2) + (7*a^2*x*ArcTanh[a*x]^3)/(8*(1 - a^2*x^2)) + (15*a*ArcTanh[a*x]^4)/32 + 3*a*ArcTanh[a*x]^2*
Log[2 - 2/(1 + a*x)] - 3*a*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (3*a*PolyLog[3, -1 + 2/(1 + a*x)])/2

________________________________________________________________________________________

Rubi [A]  time = 0.690237, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {6030, 5982, 5916, 5988, 5932, 5948, 6056, 6610, 5956, 5994, 261, 5964, 5960} \[ -\frac{3}{2} a \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-3 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{93 a}{128 \left (1-a^2 x^2\right )}-\frac{3 a}{128 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac{a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac{21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}-\frac{3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac{93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{15}{32} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{93}{128} a \tanh ^{-1}(a x)^2+3 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a)/(128*(1 - a^2*x^2)^2) - (93*a)/(128*(1 - a^2*x^2)) + (3*a^2*x*ArcTanh[a*x])/(32*(1 - a^2*x^2)^2) + (93*
a^2*x*ArcTanh[a*x])/(64*(1 - a^2*x^2)) + (93*a*ArcTanh[a*x]^2)/128 - (3*a*ArcTanh[a*x]^2)/(16*(1 - a^2*x^2)^2)
- (21*a*ArcTanh[a*x]^2)/(16*(1 - a^2*x^2)) + a*ArcTanh[a*x]^3 - ArcTanh[a*x]^3/x + (a^2*x*ArcTanh[a*x]^3)/(4*
(1 - a^2*x^2)^2) + (7*a^2*x*ArcTanh[a*x]^3)/(8*(1 - a^2*x^2)) + (15*a*ArcTanh[a*x]^4)/32 + 3*a*ArcTanh[a*x]^2*
Log[2 - 2/(1 + a*x)] - 3*a*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (3*a*PolyLog[3, -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 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 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 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 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 261

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

Rule 5964

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*p*(d + e*x^2)^(
q + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q
+ 1)*(a + b*ArcTanh[c*x])^p, x], x] + Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTanh[c*
x])^(p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c
, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

Rule 5960

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*(d + e*x^2)^(q + 1))
/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x] -
Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*
d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )^3} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac{a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{1}{8} \left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+\frac{1}{4} \left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx+a^2 \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{3 a}{128 \left (1-a^2 x^2\right )^2}+\frac{3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac{3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac{a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac{7}{32} a \tanh ^{-1}(a x)^4+\frac{1}{32} \left (9 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+a^2 \int \frac{\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx-\frac{1}{8} \left (9 a^3\right ) \int \frac{x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx-\frac{1}{2} \left (3 a^3\right ) \int \frac{x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x^2} \, dx\\ &=-\frac{3 a}{128 \left (1-a^2 x^2\right )^2}+\frac{3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{9 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac{9}{128} a \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac{21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac{15}{32} a \tanh ^{-1}(a x)^4+(3 a) \int \frac{\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx+\frac{1}{8} \left (9 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\frac{1}{2} \left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac{1}{64} \left (9 a^3\right ) \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{3 a}{128 \left (1-a^2 x^2\right )^2}-\frac{9 a}{128 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac{93}{128} a \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac{21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac{15}{32} a \tanh ^{-1}(a x)^4+(3 a) \int \frac{\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx-\frac{1}{16} \left (9 a^3\right ) \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx-\frac{1}{4} \left (3 a^3\right ) \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{3 a}{128 \left (1-a^2 x^2\right )^2}-\frac{93 a}{128 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac{93}{128} a \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac{21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac{15}{32} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-\left (6 a^2\right ) \int \frac{\tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{3 a}{128 \left (1-a^2 x^2\right )^2}-\frac{93 a}{128 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac{93}{128} a \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac{21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac{15}{32} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+\left (3 a^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{3 a}{128 \left (1-a^2 x^2\right )^2}-\frac{93 a}{128 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac{93}{128} a \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac{21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac{15}{32} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} a \text{Li}_3\left (-1+\frac{2}{1+a x}\right )\\ \end{align*}

Mathematica [C]  time = 0.690842, size = 218, normalized size = 0.78 \[ -a \left (-3 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )-\frac{a x \tanh ^{-1}(a x)^3}{1-a^2 x^2}-\frac{15}{32} \tanh ^{-1}(a x)^4+\frac{\tanh ^{-1}(a x)^3}{a x}+\tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-\frac{1}{32} \tanh ^{-1}(a x)^3 \sinh \left (4 \tanh ^{-1}(a x)\right )-\frac{3}{4} \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )-\frac{3}{256} \tanh ^{-1}(a x) \sinh \left (4 \tanh ^{-1}(a x)\right )+\frac{3}{4} \tanh ^{-1}(a x)^2 \cosh \left (2 \tanh ^{-1}(a x)\right )+\frac{3}{128} \tanh ^{-1}(a x)^2 \cosh \left (4 \tanh ^{-1}(a x)\right )+\frac{3}{8} \cosh \left (2 \tanh ^{-1}(a x)\right )+\frac{3 \cosh \left (4 \tanh ^{-1}(a x)\right )}{1024}-\frac{i \pi ^3}{8}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a*((-I/8)*Pi^3 + ArcTanh[a*x]^3 + ArcTanh[a*x]^3/(a*x) - (a*x*ArcTanh[a*x]^3)/(1 - a^2*x^2) - (15*ArcTanh[a*
x]^4)/32 + (3*Cosh[2*ArcTanh[a*x]])/8 + (3*ArcTanh[a*x]^2*Cosh[2*ArcTanh[a*x]])/4 + (3*Cosh[4*ArcTanh[a*x]])/1
024 + (3*ArcTanh[a*x]^2*Cosh[4*ArcTanh[a*x]])/128 - 3*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - 3*ArcTanh[a
*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + (3*PolyLog[3, E^(2*ArcTanh[a*x])])/2 - (3*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x
]])/4 - (3*ArcTanh[a*x]*Sinh[4*ArcTanh[a*x]])/256 - (ArcTanh[a*x]^3*Sinh[4*ArcTanh[a*x]])/32))

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Maple [B]  time = 0.714, size = 842, normalized size = 3. \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^2/(-a^2*x^2+1)^3,x)

[Out]

-arctanh(a*x)^3/x-6*a*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*a*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a*arc
tanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+3/512*
a*arctanh(a*x)/(a*x-1)^2-3/512*a*arctanh(a*x)/(a*x+1)^2-1/64*a/(a*x+1)^2*arctanh(a*x)^3+1/64*a/(a*x-1)^2*arcta
nh(a*x)^3+3/16/(a*x+1)*a^2*x+3/16*a^2*x/(a*x-1)+3/8*a*arctanh(a*x)^2/(a*x-1)-3/8*a*arctanh(a*x)^2/(a*x+1)-3/8*
arctanh(a*x)/(a*x-1)*a^2*x+3/8*arctanh(a*x)/(a*x+1)*a^2*x-3/2048*a/(a*x-1)^2-3/2048*a/(a*x+1)^2+3*a*arctanh(a*
x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a*arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*a/(a*x-1)*arctan
h(a*x)^3-1/64/(a*x+1)^2*arctanh(a*x)^3*x^2*a^3-3/256/(a*x+1)^2*arctanh(a*x)^2*x^2*a^3+1/32/(a*x+1)^2*arctanh(a
*x)^3*x*a^2+3/128/(a*x+1)^2*arctanh(a*x)^2*x*a^2+1/64/(a*x-1)^2*arctanh(a*x)^3*x^2*a^3-3/256/(a*x-1)^2*arctanh
(a*x)^2*x^2*a^3+1/32/(a*x-1)^2*arctanh(a*x)^3*x*a^2-3/128/(a*x-1)^2*arctanh(a*x)^2*x*a^2+3/16*a/(a*x-1)-3/16*a
/(a*x+1)-a*arctanh(a*x)^3+15/32*a*arctanh(a*x)^4-1/4*a/(a*x+1)*arctanh(a*x)^3+3/512*arctanh(a*x)/(a*x-1)^2*a^3
*x^2+3/256*arctanh(a*x)/(a*x-1)^2*a^2*x-3/512*arctanh(a*x)/(a*x+1)^2*a^3*x^2+3/256*arctanh(a*x)/(a*x+1)^2*a^2*
x-3/2048/(a*x+1)^2*a^3*x^2+3/1024/(a*x+1)^2*a^2*x-3/2048/(a*x-1)^2*a^3*x^2-3/1024/(a*x-1)^2*a^2*x-3/256*a*arct
anh(a*x)^2/(a*x-1)^2-3/256*a*arctanh(a*x)^2/(a*x+1)^2-1/4/(a*x-1)*arctanh(a*x)^3*x*a^2+3/8/(a*x-1)*arctanh(a*x
)^2*x*a^2+1/4/(a*x+1)*arctanh(a*x)^3*x*a^2+3/8/(a*x+1)*arctanh(a*x)^2*x*a^2-3/8*a*arctanh(a*x)/(a*x-1)-3/8*a*a
rctanh(a*x)/(a*x+1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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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^{6} x^{8} - 3 \, a^{4} x^{6} + 3 \, a^{2} x^{4} - x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-arctanh(a*x)^3/(a^6*x^8 - 3*a^4*x^6 + 3*a^2*x^4 - x^2), 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 )}}{a^{6} x^{8} - 3 a^{4} x^{6} + 3 a^{2} x^{4} - x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(atanh(a*x)**3/(a**6*x**8 - 3*a**4*x**6 + 3*a**2*x**4 - x**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 )}^{3} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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