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

Optimal. Leaf size=156 \[ \frac{5}{3} a \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )-a \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac{1}{3} a^3 x^2 \tanh ^{-1}(a x)+\frac{a^2 x}{3}-2 a^2 x \tanh ^{-1}(a x)^2-\frac{2}{3} a \tanh ^{-1}(a x)^2-\frac{1}{3} a \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{10}{3} a \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)+2 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.418459, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 20, 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, 5988, 5932, 2447, 5980, 321, 206} \[ \frac{5}{3} a \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )-a \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac{1}{3} a^3 x^2 \tanh ^{-1}(a x)+\frac{a^2 x}{3}-2 a^2 x \tanh ^{-1}(a x)^2-\frac{2}{3} a \tanh ^{-1}(a x)^2-\frac{1}{3} a \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{10}{3} a \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)+2 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 5980

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 321

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

Rubi steps

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

Mathematica [A]  time = 0.056467, size = 182, normalized size = 1.17 \[ \frac{1}{3} a \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )-\left (1-a^2 x^2\right ) \left (a x \tanh ^{-1}(a x)+1\right ) \tanh ^{-1}(a x)+a x+a x \tanh ^{-1}(a x)^2-\tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x) \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )\right )-2 a \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+a \left (\tanh ^{-1}(a x) \left (-\frac{\tanh ^{-1}(a x)}{a x}+\tanh ^{-1}(a x)+2 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )-\text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )\right )-2 a \tanh ^{-1}(a x) \left (a x \tanh ^{-1}(a x)-\tanh ^{-1}(a x)-2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-2*a*ArcTanh[a*x]*(-ArcTanh[a*x] + a*x*ArcTanh[a*x] - 2*Log[1 + E^(-2*ArcTanh[a*x])]) - 2*a*PolyLog[2, -E^(-2*
ArcTanh[a*x])] + (a*(a*x - ArcTanh[a*x]^2 + a*x*ArcTanh[a*x]^2 - (1 - a^2*x^2)*ArcTanh[a*x]*(1 + a*x*ArcTanh[a
*x]) - 2*ArcTanh[a*x]*Log[1 + E^(-2*ArcTanh[a*x])] + PolyLog[2, -E^(-2*ArcTanh[a*x])]))/3 + a*(ArcTanh[a*x]*(A
rcTanh[a*x] - ArcTanh[a*x]/(a*x) + 2*Log[1 - E^(-2*ArcTanh[a*x])]) - PolyLog[2, E^(-2*ArcTanh[a*x])])

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Maple [A]  time = 0.056, size = 222, normalized size = 1.4 \begin{align*}{\frac{{a}^{4}{x}^{3} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{3}}-2\,{a}^{2}x \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{x}}+{\frac{{a}^{3}{x}^{2}{\it Artanh} \left ( ax \right ) }{3}}-{\frac{8\,a{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{3}}+2\,a{\it Artanh} \left ( ax \right ) \ln \left ( ax \right ) -{\frac{8\,a{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{3}}+{\frac{{a}^{2}x}{3}}+{\frac{a\ln \left ( ax-1 \right ) }{6}}-{\frac{a\ln \left ( ax+1 \right ) }{6}}-a{\it dilog} \left ( ax \right ) -a{\it dilog} \left ( ax+1 \right ) -a\ln \left ( ax \right ) \ln \left ( ax+1 \right ) -{\frac{2\,a \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{3}}+{\frac{8\,a}{3}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{4\,a\ln \left ( ax-1 \right ) }{3}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{4\,a}{3}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4\,a\ln \left ( ax+1 \right ) }{3}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }+{\frac{2\,a \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{3}} \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^2,x)

[Out]

1/3*a^4*x^3*arctanh(a*x)^2-2*a^2*x*arctanh(a*x)^2-arctanh(a*x)^2/x+1/3*a^3*x^2*arctanh(a*x)-8/3*a*arctanh(a*x)
*ln(a*x-1)+2*a*arctanh(a*x)*ln(a*x)-8/3*a*arctanh(a*x)*ln(a*x+1)+1/3*a^2*x+1/6*a*ln(a*x-1)-1/6*a*ln(a*x+1)-a*d
ilog(a*x)-a*dilog(a*x+1)-a*ln(a*x)*ln(a*x+1)-2/3*a*ln(a*x-1)^2+8/3*a*dilog(1/2+1/2*a*x)+4/3*a*ln(a*x-1)*ln(1/2
+1/2*a*x)+4/3*a*ln(-1/2*a*x+1/2)*ln(1/2+1/2*a*x)-4/3*a*ln(-1/2*a*x+1/2)*ln(a*x+1)+2/3*a*ln(a*x+1)^2

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Maxima [A]  time = 0.976506, size = 270, normalized size = 1.73 \begin{align*} \frac{1}{6} \, a^{2}{\left (\frac{2 \,{\left (a x + 2 \, \log \left (a x + 1\right )^{2} - 4 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 2 \, \log \left (a x - 1\right )^{2}\right )}}{a} + \frac{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} - \frac{6 \,{\left (\log \left (a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x\right )\right )}}{a} + \frac{6 \,{\left (\log \left (-a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x\right )\right )}}{a} - \frac{\log \left (a x + 1\right )}{a} + \frac{\log \left (a x - 1\right )}{a}\right )} + \frac{1}{3} \,{\left (a^{2} x^{2} - 8 \, \log \left (a x + 1\right ) - 8 \, \log \left (a x - 1\right ) + 6 \, \log \left (x\right )\right )} a \operatorname{artanh}\left (a x\right ) + \frac{1}{3} \,{\left (a^{4} x^{3} - 6 \, a^{2} x - \frac{3}{x}\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^2,x, algorithm="maxima")

[Out]

1/6*a^2*(2*(a*x + 2*log(a*x + 1)^2 - 4*log(a*x + 1)*log(a*x - 1) - 2*log(a*x - 1)^2)/a + 16*(log(a*x - 1)*log(
1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a - 6*(log(a*x + 1)*log(x) + dilog(-a*x))/a + 6*(log(-a*x + 1)*log(x)
+ dilog(a*x))/a - log(a*x + 1)/a + log(a*x - 1)/a) + 1/3*(a^2*x^2 - 8*log(a*x + 1) - 8*log(a*x - 1) + 6*log(x)
)*a*arctanh(a*x) + 1/3*(a^4*x^3 - 6*a^2*x - 3/x)*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^{2}}, 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^2,x, algorithm="fricas")

[Out]

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

[Out]

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

[Out]

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