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

Optimal. Leaf size=277 \[ -\frac{3}{4} \text{PolyLog}\left (4,\frac{2}{a x+1}-1\right )-\frac{3}{2} \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{3}{2} \tanh ^{-1}(a x) \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-\frac{141 a x}{256 \left (1-a^2 x^2\right )}-\frac{3 a x}{128 \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac{33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac{3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac{33 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{1}{4} \tanh ^{-1}(a x)^4-\frac{11}{32} \tanh ^{-1}(a x)^3-\frac{141}{256} \tanh ^{-1}(a x)+\log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3 \]

[Out]

(-3*a*x)/(128*(1 - a^2*x^2)^2) - (141*a*x)/(256*(1 - a^2*x^2)) - (141*ArcTanh[a*x])/256 + (3*ArcTanh[a*x])/(32
*(1 - a^2*x^2)^2) + (33*ArcTanh[a*x])/(32*(1 - a^2*x^2)) - (3*a*x*ArcTanh[a*x]^2)/(16*(1 - a^2*x^2)^2) - (33*a
*x*ArcTanh[a*x]^2)/(32*(1 - a^2*x^2)) - (11*ArcTanh[a*x]^3)/32 + ArcTanh[a*x]^3/(4*(1 - a^2*x^2)^2) + ArcTanh[
a*x]^3/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^4/4 + ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - (3*ArcTanh[a*x]^2*PolyLog[
2, -1 + 2/(1 + a*x)])/2 - (3*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/2 - (3*PolyLog[4, -1 + 2/(1 + a*x)])/4

________________________________________________________________________________________

Rubi [A]  time = 0.626081, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {6030, 5988, 5932, 5948, 6056, 6060, 6610, 5994, 5956, 199, 206, 5964} \[ -\frac{3}{4} \text{PolyLog}\left (4,\frac{2}{a x+1}-1\right )-\frac{3}{2} \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{3}{2} \tanh ^{-1}(a x) \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-\frac{141 a x}{256 \left (1-a^2 x^2\right )}-\frac{3 a x}{128 \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac{33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac{3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac{33 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{1}{4} \tanh ^{-1}(a x)^4-\frac{11}{32} \tanh ^{-1}(a x)^3-\frac{141}{256} \tanh ^{-1}(a x)+\log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*x)/(128*(1 - a^2*x^2)^2) - (141*a*x)/(256*(1 - a^2*x^2)) - (141*ArcTanh[a*x])/256 + (3*ArcTanh[a*x])/(32
*(1 - a^2*x^2)^2) + (33*ArcTanh[a*x])/(32*(1 - a^2*x^2)) - (3*a*x*ArcTanh[a*x]^2)/(16*(1 - a^2*x^2)^2) - (33*a
*x*ArcTanh[a*x]^2)/(32*(1 - a^2*x^2)) - (11*ArcTanh[a*x]^3)/32 + ArcTanh[a*x]^3/(4*(1 - a^2*x^2)^2) + ArcTanh[
a*x]^3/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^4/4 + ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - (3*ArcTanh[a*x]^2*PolyLog[
2, -1 + 2/(1 + a*x)])/2 - (3*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/2 - (3*PolyLog[4, -1 + 2/(1 + a*x)])/4

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 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 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])

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]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )^3} \, dx &=a^2 \int \frac{x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )^2} \, dx\\ &=\frac{\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac{1}{4} (3 a) \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx+a^2 \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 \left (1-a^2 x^2\right )} \, dx\\ &=\frac{3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac{3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{1}{4} \tanh ^{-1}(a x)^4-\frac{1}{32} (3 a) \int \frac{1}{\left (1-a^2 x^2\right )^3} \, dx-\frac{1}{16} (9 a) \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx-\frac{1}{2} (3 a) \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x (1+a x)} \, dx\\ &=-\frac{3 a x}{128 \left (1-a^2 x^2\right )^2}+\frac{3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac{3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac{33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac{11}{32} \tanh ^{-1}(a x)^3+\frac{\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{1}{128} (9 a) \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx-(3 a) \int \frac{\tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx+\frac{1}{16} \left (9 a^2\right ) \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\frac{1}{2} \left (3 a^2\right ) \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{3 a x}{128 \left (1-a^2 x^2\right )^2}-\frac{9 a x}{256 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{33 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )}-\frac{3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac{33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac{11}{32} \tanh ^{-1}(a x)^3+\frac{\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{1}{256} (9 a) \int \frac{1}{1-a^2 x^2} \, dx-\frac{1}{32} (9 a) \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx-\frac{1}{4} (3 a) \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx+(3 a) \int \frac{\tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{3 a x}{128 \left (1-a^2 x^2\right )^2}-\frac{141 a x}{256 \left (1-a^2 x^2\right )}-\frac{9}{256} \tanh ^{-1}(a x)+\frac{3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{33 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )}-\frac{3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac{33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac{11}{32} \tanh ^{-1}(a x)^3+\frac{\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )-\frac{1}{64} (9 a) \int \frac{1}{1-a^2 x^2} \, dx-\frac{1}{8} (3 a) \int \frac{1}{1-a^2 x^2} \, dx+\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{3 a x}{128 \left (1-a^2 x^2\right )^2}-\frac{141 a x}{256 \left (1-a^2 x^2\right )}-\frac{141}{256} \tanh ^{-1}(a x)+\frac{3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{33 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )}-\frac{3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac{33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac{11}{32} \tanh ^{-1}(a x)^3+\frac{\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )-\frac{3}{4} \text{Li}_4\left (-1+\frac{2}{1+a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.28276, size = 189, normalized size = 0.68 \[ \frac{1536 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )-1536 \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )+768 \text{PolyLog}\left (4,e^{2 \tanh ^{-1}(a x)}\right )-256 \tanh ^{-1}(a x)^4+1024 \tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-576 \tanh ^{-1}(a x)^2 \sinh \left (2 \tanh ^{-1}(a x)\right )-24 \tanh ^{-1}(a x)^2 \sinh \left (4 \tanh ^{-1}(a x)\right )-288 \sinh \left (2 \tanh ^{-1}(a x)\right )-3 \sinh \left (4 \tanh ^{-1}(a x)\right )+384 \tanh ^{-1}(a x)^3 \cosh \left (2 \tanh ^{-1}(a x)\right )+32 \tanh ^{-1}(a x)^3 \cosh \left (4 \tanh ^{-1}(a x)\right )+576 \tanh ^{-1}(a x) \cosh \left (2 \tanh ^{-1}(a x)\right )+12 \tanh ^{-1}(a x) \cosh \left (4 \tanh ^{-1}(a x)\right )+16 \pi ^4}{1024} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(16*Pi^4 - 256*ArcTanh[a*x]^4 + 576*ArcTanh[a*x]*Cosh[2*ArcTanh[a*x]] + 384*ArcTanh[a*x]^3*Cosh[2*ArcTanh[a*x]
] + 12*ArcTanh[a*x]*Cosh[4*ArcTanh[a*x]] + 32*ArcTanh[a*x]^3*Cosh[4*ArcTanh[a*x]] + 1024*ArcTanh[a*x]^3*Log[1
- E^(2*ArcTanh[a*x])] + 1536*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] - 1536*ArcTanh[a*x]*PolyLog[3, E^(2
*ArcTanh[a*x])] + 768*PolyLog[4, E^(2*ArcTanh[a*x])] - 288*Sinh[2*ArcTanh[a*x]] - 576*ArcTanh[a*x]^2*Sinh[2*Ar
cTanh[a*x]] - 3*Sinh[4*ArcTanh[a*x]] - 24*ArcTanh[a*x]^2*Sinh[4*ArcTanh[a*x]])/1024

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Maple [C]  time = 0.477, size = 1533, normalized size = 5.5 \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/(-a^2*x^2+1)^3,x)

[Out]

3*arctanh(a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))
-6*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*arctanh(a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+ar
ctanh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^3*l
n((a*x+1)^2/(-a^2*x^2+1)-1)+ln(2)*arctanh(a*x)^3+arctanh(a*x)^3*ln(a*x)-9/32*(a*x+1)*arctanh(a*x)/(a*x-1)-9/32
*(a*x-1)*arctanh(a*x)/(a*x+1)+9/32*(a*x+1)*arctanh(a*x)^2/(a*x-1)+1/2*I*Pi*arctanh(a*x)^3+3/512*arctanh(a*x)*(
a*x+1)^2/(a*x-1)^2-9/64*(a*x-1)/(a*x+1)+9/64*(a*x+1)/(a*x-1)+3/2048*(a*x-1)^2/(a*x+1)^2-3/256*arctanh(a*x)^2*(
a*x+1)^2/(a*x-1)^2+3/256*arctanh(a*x)^2*(a*x-1)^2/(a*x+1)^2-3/2048*(a*x+1)^2/(a*x-1)^2-1/4*arctanh(a*x)^4-11/3
2*arctanh(a*x)^3+1/2*I*Pi*arctanh(a*x)^3*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3-1/2*I
*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^3+1/2*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(
a*x)^3+1/4*I*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3-9/32*arctanh(a*x)^2*(a*x-1)/(a*x+1)+6*polylog(4
,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)/(
-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))-1/2*I*arctanh(a*x)^3*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*
csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2-1/4*I*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*
x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2+1/2*I*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)/(-a^
2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2+1/4*I*Pi*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csg
n(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2+1/16*arctanh(a*x)^3/(a*x-1)^2+1/16*arctanh(a*x)^3/(a*x
+1)^2+3/512*(a*x-1)^2*arctanh(a*x)/(a*x+1)^2-5/16*arctanh(a*x)^3/(a*x-1)-1/2*arctanh(a*x)^3*ln(a*x-1)+5/16*arc
tanh(a*x)^3/(a*x+1)-1/2*arctanh(a*x)^3*ln(a*x+1)+arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*Pi*arctan
h(a*x)^3*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2+1/
4*I*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3-1/4*I*Pi*arctanh(a*x)^3*csgn(
I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+
1)+1))+1/2*I*Pi*arctanh(a*x)^3*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*((
a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-a x + 1\right )^{4} + 2 \,{\left (2 \, a^{2} x^{2} + 2 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) - 3\right )} \log \left (-a x + 1\right )^{3}}{64 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )}} - \frac{1}{8} \, \int \frac{4 \, \log \left (a x + 1\right )^{3} - 12 \, \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} - 3 \, a^{2} x^{2} - 3 \, a x + 2 \,{\left (a^{6} x^{6} + a^{5} x^{5} - 2 \, a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2} + a x + 2\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{4 \,{\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*((a^4*x^4 - 2*a^2*x^2 + 1)*log(-a*x + 1)^4 + 2*(2*a^2*x^2 + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1) - 3)
*log(-a*x + 1)^3)/(a^4*x^4 - 2*a^2*x^2 + 1) - 1/8*integrate(1/4*(4*log(a*x + 1)^3 - 12*log(a*x + 1)^2*log(-a*x
+ 1) + 3*(2*a^4*x^4 + 2*a^3*x^3 - 3*a^2*x^2 - 3*a*x + 2*(a^6*x^6 + a^5*x^5 - 2*a^4*x^4 - 2*a^3*x^3 + a^2*x^2
+ a*x + 2)*log(a*x + 1))*log(-a*x + 1)^2)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), 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^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-arctanh(a*x)^3/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), 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^{7} - 3 a^{4} x^{5} + 3 a^{2} x^{3} - x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(atanh(a*x)**3/(a**6*x**7 - 3*a**4*x**5 + 3*a**2*x**3 - x), 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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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