3.3.8 \(\int \frac {(1-a^2 x^2)^2 \tanh ^{-1}(a x)^2}{x} \, dx\) [208]

Optimal. Leaf size=186 \[ \frac {a^2 x^2}{12}-\frac {3}{2} a x \tanh ^{-1}(a x)+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac {3}{4} \tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\frac {2}{3} \log \left (1-a^2 x^2\right )-\tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )+\tanh ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )+\frac {1}{2} \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )-\frac {1}{2} \text {PolyLog}\left (3,-1+\frac {2}{1-a x}\right ) \]

[Out]

1/12*a^2*x^2-3/2*a*x*arctanh(a*x)+1/6*a^3*x^3*arctanh(a*x)+3/4*arctanh(a*x)^2-a^2*x^2*arctanh(a*x)^2+1/4*a^4*x
^4*arctanh(a*x)^2-2*arctanh(a*x)^2*arctanh(-1+2/(-a*x+1))-2/3*ln(-a^2*x^2+1)-arctanh(a*x)*polylog(2,1-2/(-a*x+
1))+arctanh(a*x)*polylog(2,-1+2/(-a*x+1))+1/2*polylog(3,1-2/(-a*x+1))-1/2*polylog(3,-1+2/(-a*x+1))

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Rubi [A]
time = 0.36, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {6159, 6033, 6199, 6095, 6205, 6745, 6037, 6127, 6021, 266, 272, 45} \begin {gather*} \frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac {a^2 x^2}{12}-\frac {2}{3} \log \left (1-a^2 x^2\right )-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{2} \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \text {Li}_3\left (\frac {2}{1-a x}-1\right )-\text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)+\text {Li}_2\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)-\frac {3}{2} a x \tanh ^{-1}(a x)+\frac {3}{4} \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^2)/12 - (3*a*x*ArcTanh[a*x])/2 + (a^3*x^3*ArcTanh[a*x])/6 + (3*ArcTanh[a*x]^2)/4 - a^2*x^2*ArcTanh[a*x]
^2 + (a^4*x^4*ArcTanh[a*x]^2)/4 + 2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] - (2*Log[1 - a^2*x^2])/3 - ArcTanh
[a*x]*PolyLog[2, 1 - 2/(1 - a*x)] + ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 - a*x)] + PolyLog[3, 1 - 2/(1 - a*x)]/2
- PolyLog[3, -1 + 2/(1 - a*x)]/2

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 272

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

Rule 6021

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Dist[b
*c*n*p, Int[x^n*((a + b*ArcTanh[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 6033

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

Rule 6037

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTanh[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTanh[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 6095

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 6127

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

Rule 6159

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

Rule 6199

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

Rule 6205

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcT
anh[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 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 \frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x} \, dx &=\int \left (\frac {\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+a^4 x^3 \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x \tanh ^{-1}(a x)^2 \, dx\right )+a^4 \int x^3 \tanh ^{-1}(a x)^2 \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x} \, dx\\ &=-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-(4 a) \int \frac {\tanh ^{-1}(a x) \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx+\left (2 a^3\right ) \int \frac {x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac {1}{2} a^5 \int \frac {x^4 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-(2 a) \int \tanh ^{-1}(a x) \, dx+(2 a) \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+(2 a) \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx-(2 a) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx+\frac {1}{2} a^3 \int x^2 \tanh ^{-1}(a x) \, dx-\frac {1}{2} a^3 \int \frac {x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-2 a x \tanh ^{-1}(a x)+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)+\tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} a \int \tanh ^{-1}(a x) \, dx-\frac {1}{2} a \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+a \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx-a \int \frac {\text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx+\left (2 a^2\right ) \int \frac {x}{1-a^2 x^2} \, dx-\frac {1}{6} a^4 \int \frac {x^3}{1-a^2 x^2} \, dx\\ &=-\frac {3}{2} a x \tanh ^{-1}(a x)+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac {3}{4} \tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\log \left (1-a^2 x^2\right )-\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \text {Li}_3\left (-1+\frac {2}{1-a x}\right )-\frac {1}{2} a^2 \int \frac {x}{1-a^2 x^2} \, dx-\frac {1}{12} a^4 \text {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {3}{2} a x \tanh ^{-1}(a x)+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac {3}{4} \tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\frac {3}{4} \log \left (1-a^2 x^2\right )-\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \text {Li}_3\left (-1+\frac {2}{1-a x}\right )-\frac {1}{12} a^4 \text {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 x^2}{12}-\frac {3}{2} a x \tanh ^{-1}(a x)+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac {3}{4} \tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\frac {2}{3} \log \left (1-a^2 x^2\right )-\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \text {Li}_3\left (-1+\frac {2}{1-a x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 191, normalized size = 1.03 \begin {gather*} \frac {a^2 x^2}{12}-2 a x \tanh ^{-1}(a x)+\frac {1}{6} a x \left (3+a^2 x^2\right ) \tanh ^{-1}(a x)-\left (-1+a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {1}{4} \left (-1+a^4 x^4\right ) \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\frac {2}{3} \log \left (1-a^2 x^2\right )+\tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {-1-a x}{-1+a x}\right )-\tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {1+a x}{-1+a x}\right )-\frac {1}{2} \text {PolyLog}\left (3,\frac {-1-a x}{-1+a x}\right )+\frac {1}{2} \text {PolyLog}\left (3,\frac {1+a x}{-1+a x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^2)/12 - 2*a*x*ArcTanh[a*x] + (a*x*(3 + a^2*x^2)*ArcTanh[a*x])/6 - (-1 + a^2*x^2)*ArcTanh[a*x]^2 + ((-1
+ a^4*x^4)*ArcTanh[a*x]^2)/4 + 2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] - (2*Log[1 - a^2*x^2])/3 + ArcTanh[a*
x]*PolyLog[2, (-1 - a*x)/(-1 + a*x)] - ArcTanh[a*x]*PolyLog[2, (1 + a*x)/(-1 + a*x)] - PolyLog[3, (-1 - a*x)/(
-1 + a*x)]/2 + PolyLog[3, (1 + a*x)/(-1 + a*x)]/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 21.44, size = 733, normalized size = 3.94 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*x^2+1)^2*arctanh(a*x)^2/x,x,method=_RETURNVERBOSE)

[Out]

1/4*a^4*x^4*arctanh(a*x)^2-a^2*x^2*arctanh(a*x)^2+arctanh(a*x)^2*ln(a*x)-arctanh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2
+1)-1)+arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-2*
polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2
,-(a*x+1)/(-a^2*x^2+1)^(1/2))-2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2
*x^2+1))+1/2*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))-1/2*I*arctanh(a*x)^2*Pi*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*csg
n(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2+1/2*(a*x-3)*(a*x+1)*arctanh(a*x)+1/6*(a^2*x^2-4*a
*x+7)*(a*x+1)*arctanh(a*x)+1/2*I*arctanh(a*x)^2*Pi*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1
))^3+1/6*a*x-1/6+4/3*ln((a*x+1)^2/(-a^2*x^2+1)+1)+3/4*arctanh(a*x)^2+1/12*(a*x-1)^2+1/2*I*arctanh(a*x)^2*Pi*cs
gn(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))-(a*x+1)*arctanh(a*x)-1/2*I*arctanh(a*x)^2*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

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

[Out]

1/16*(a^4*x^4 - 4*a^2*x^2)*log(-a*x + 1)^2 - 1/4*integrate(-1/2*(2*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2
+ a*x - 1)*log(a*x + 1)^2 - (a^5*x^5 - 4*a^3*x^3 + 4*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*log
(a*x + 1))*log(-a*x + 1))/(a*x^2 - x), 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((-a^2*x^2+1)^2*arctanh(a*x)^2/x,x, algorithm="fricas")

[Out]

integral((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)^2/x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x,x)

[Out]

int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x, x)

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