3.211 \(\int \frac{(1-a^2 x^2)^2 \tanh ^{-1}(a x)^2}{x^4} \, dx\)

Optimal. Leaf size=167 \[ -a^3 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )+\frac{5}{3} a^3 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{a^2}{3 x}+a^4 x \tanh ^{-1}(a x)^2-\frac{2}{3} a^3 \tanh ^{-1}(a x)^2+\frac{1}{3} a^3 \tanh ^{-1}(a x)+\frac{2 a^2 \tanh ^{-1}(a x)^2}{x}-2 a^3 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)-\frac{10}{3} a^3 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{3 x^2}-\frac{\tanh ^{-1}(a x)^2}{3 x^3} \]

[Out]

-a^2/(3*x) + (a^3*ArcTanh[a*x])/3 - (a*ArcTanh[a*x])/(3*x^2) - (2*a^3*ArcTanh[a*x]^2)/3 - ArcTanh[a*x]^2/(3*x^
3) + (2*a^2*ArcTanh[a*x]^2)/x + a^4*x*ArcTanh[a*x]^2 - 2*a^3*ArcTanh[a*x]*Log[2/(1 - a*x)] - (10*a^3*ArcTanh[a
*x]*Log[2 - 2/(1 + a*x)])/3 - a^3*PolyLog[2, 1 - 2/(1 - a*x)] + (5*a^3*PolyLog[2, -1 + 2/(1 + a*x)])/3

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Rubi [A]  time = 0.430452, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {6012, 5910, 5984, 5918, 2402, 2315, 5916, 5982, 325, 206, 5988, 5932, 2447} \[ -a^3 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )+\frac{5}{3} a^3 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{a^2}{3 x}+a^4 x \tanh ^{-1}(a x)^2-\frac{2}{3} a^3 \tanh ^{-1}(a x)^2+\frac{1}{3} a^3 \tanh ^{-1}(a x)+\frac{2 a^2 \tanh ^{-1}(a x)^2}{x}-2 a^3 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)-\frac{10}{3} a^3 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{3 x^2}-\frac{\tanh ^{-1}(a x)^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^4,x]

[Out]

-a^2/(3*x) + (a^3*ArcTanh[a*x])/3 - (a*ArcTanh[a*x])/(3*x^2) - (2*a^3*ArcTanh[a*x]^2)/3 - ArcTanh[a*x]^2/(3*x^
3) + (2*a^2*ArcTanh[a*x]^2)/x + a^4*x*ArcTanh[a*x]^2 - 2*a^3*ArcTanh[a*x]*Log[2/(1 - a*x)] - (10*a^3*ArcTanh[a
*x]*Log[2 - 2/(1 + a*x)])/3 - a^3*PolyLog[2, 1 - 2/(1 - a*x)] + (5*a^3*PolyLog[2, -1 + 2/(1 + a*x)])/3

Rule 6012

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 5910

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcTanh[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 5984

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[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 5918

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^p*
Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTanh[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 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

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 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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 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]]

Rubi steps

\begin{align*} \int \frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x^4} \, dx &=\int \left (a^4 \tanh ^{-1}(a x)^2+\frac{\tanh ^{-1}(a x)^2}{x^4}-\frac{2 a^2 \tanh ^{-1}(a x)^2}{x^2}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac{\tanh ^{-1}(a x)^2}{x^2} \, dx\right )+a^4 \int \tanh ^{-1}(a x)^2 \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^4} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{3 x^3}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2+\frac{1}{3} (2 a) \int \frac{\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx-\left (4 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\left (2 a^5\right ) \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-a^3 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{3 x^3}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2+\frac{1}{3} (2 a) \int \frac{\tanh ^{-1}(a x)}{x^3} \, dx+\frac{1}{3} \left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\left (4 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\left (2 a^4\right ) \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{3 x^2}-\frac{2}{3} a^3 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{3 x^3}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2-2 a^3 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )-4 a^3 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )+\frac{1}{3} a^2 \int \frac{1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac{1}{3} \left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\left (2 a^4\right ) \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx+\left (4 a^4\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a^2}{3 x}-\frac{a \tanh ^{-1}(a x)}{3 x^2}-\frac{2}{3} a^3 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{3 x^3}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2-2 a^3 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )-\frac{10}{3} a^3 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )+2 a^3 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )+\frac{1}{3} a^4 \int \frac{1}{1-a^2 x^2} \, dx-\frac{1}{3} \left (2 a^4\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a^2}{3 x}+\frac{1}{3} a^3 \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{3 x^2}-\frac{2}{3} a^3 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{3 x^3}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2-2 a^3 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )-\frac{10}{3} a^3 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )-a^3 \text{Li}_2\left (1-\frac{2}{1-a x}\right )+\frac{5}{3} a^3 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0688928, size = 153, normalized size = 0.92 \[ \frac{1}{3} \left (3 a^3 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+5 a^3 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )-\frac{a^2}{x}+3 a^4 x \tanh ^{-1}(a x)^2-8 a^3 \tanh ^{-1}(a x)^2+a^3 \tanh ^{-1}(a x)+\frac{6 a^2 \tanh ^{-1}(a x)^2}{x}-10 a^3 \tanh ^{-1}(a x) \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )-6 a^3 \tanh ^{-1}(a x) \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-\frac{a \tanh ^{-1}(a x)}{x^2}-\frac{\tanh ^{-1}(a x)^2}{x^3}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^4,x]

[Out]

(-(a^2/x) + a^3*ArcTanh[a*x] - (a*ArcTanh[a*x])/x^2 - 8*a^3*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/x^3 + (6*a^2*ArcTa
nh[a*x]^2)/x + 3*a^4*x*ArcTanh[a*x]^2 - 10*a^3*ArcTanh[a*x]*Log[1 - E^(-2*ArcTanh[a*x])] - 6*a^3*ArcTanh[a*x]*
Log[1 + E^(-2*ArcTanh[a*x])] + 3*a^3*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 5*a^3*PolyLog[2, E^(-2*ArcTanh[a*x])])
/3

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Maple [A]  time = 0.061, size = 249, normalized size = 1.5 \begin{align*}{a}^{4}x \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+2\,{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{x}}-{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{3\,{x}^{3}}}+{\frac{8\,{a}^{3}{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{3}}-{\frac{a{\it Artanh} \left ( ax \right ) }{3\,{x}^{2}}}-{\frac{10\,{a}^{3}{\it Artanh} \left ( ax \right ) \ln \left ( ax \right ) }{3}}+{\frac{8\,{a}^{3}{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{3}}-{\frac{{a}^{2}}{3\,x}}-{\frac{{a}^{3}\ln \left ( ax-1 \right ) }{6}}+{\frac{{a}^{3}\ln \left ( ax+1 \right ) }{6}}+{\frac{5\,{a}^{3}{\it dilog} \left ( ax \right ) }{3}}+{\frac{5\,{a}^{3}{\it dilog} \left ( ax+1 \right ) }{3}}+{\frac{5\,{a}^{3}\ln \left ( ax \right ) \ln \left ( ax+1 \right ) }{3}}+{\frac{2\,{a}^{3} \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{3}}-{\frac{8\,{a}^{3}}{3}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4\,{a}^{3}\ln \left ( ax-1 \right ) }{3}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{2\,{a}^{3} \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{3}}-{\frac{4\,{a}^{3}}{3}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{4\,{a}^{3}\ln \left ( ax+1 \right ) }{3}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^4*x*arctanh(a*x)^2+2*a^2*arctanh(a*x)^2/x-1/3*arctanh(a*x)^2/x^3+8/3*a^3*arctanh(a*x)*ln(a*x-1)-1/3*a*arctan
h(a*x)/x^2-10/3*a^3*arctanh(a*x)*ln(a*x)+8/3*a^3*arctanh(a*x)*ln(a*x+1)-1/3*a^2/x-1/6*a^3*ln(a*x-1)+1/6*a^3*ln
(a*x+1)+5/3*a^3*dilog(a*x)+5/3*a^3*dilog(a*x+1)+5/3*a^3*ln(a*x)*ln(a*x+1)+2/3*a^3*ln(a*x-1)^2-8/3*a^3*dilog(1/
2+1/2*a*x)-4/3*a^3*ln(a*x-1)*ln(1/2+1/2*a*x)-2/3*a^3*ln(a*x+1)^2-4/3*a^3*ln(-1/2*a*x+1/2)*ln(1/2+1/2*a*x)+4/3*
a^3*ln(-1/2*a*x+1/2)*ln(a*x+1)

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Maxima [A]  time = 0.989153, size = 274, normalized size = 1.64 \begin{align*} -\frac{1}{6} \,{\left (16 \,{\left (\log \left (a x - 1\right ) \log \left (\frac{1}{2} \, a x + \frac{1}{2}\right ) +{\rm Li}_2\left (-\frac{1}{2} \, a x + \frac{1}{2}\right )\right )} a - 10 \,{\left (\log \left (a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x\right )\right )} a + 10 \,{\left (\log \left (-a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x\right )\right )} a - a \log \left (a x + 1\right ) + a \log \left (a x - 1\right ) + \frac{2 \,{\left (2 \, a x \log \left (a x + 1\right )^{2} - 4 \, a x \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 2 \, a x \log \left (a x - 1\right )^{2} + 1\right )}}{x}\right )} a^{2} + \frac{1}{3} \,{\left (8 \, a^{2} \log \left (a x + 1\right ) + 8 \, a^{2} \log \left (a x - 1\right ) - 10 \, a^{2} \log \left (x\right ) - \frac{1}{x^{2}}\right )} a \operatorname{artanh}\left (a x\right ) + \frac{1}{3} \,{\left (3 \, a^{4} x + \frac{6 \, a^{2} x^{2} - 1}{x^{3}}\right )} \operatorname{artanh}\left (a x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^4,x, algorithm="maxima")

[Out]

-1/6*(16*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a - 10*(log(a*x + 1)*log(x) + dilog(-a*x))*
a + 10*(log(-a*x + 1)*log(x) + dilog(a*x))*a - a*log(a*x + 1) + a*log(a*x - 1) + 2*(2*a*x*log(a*x + 1)^2 - 4*a
*x*log(a*x + 1)*log(a*x - 1) - 2*a*x*log(a*x - 1)^2 + 1)/x)*a^2 + 1/3*(8*a^2*log(a*x + 1) + 8*a^2*log(a*x - 1)
- 10*a^2*log(x) - 1/x^2)*a*arctanh(a*x) + 1/3*(3*a^4*x + (6*a^2*x^2 - 1)/x^3)*arctanh(a*x)^2

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname{artanh}\left (a x\right )^{2}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname{atanh}^{2}{\left (a x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*x**2+1)**2*atanh(a*x)**2/x**4,x)

[Out]

Integral((a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**2/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname{artanh}\left (a x\right )^{2}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^4,x, algorithm="giac")

[Out]

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