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

Optimal. Leaf size=66 \[ -a \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{1}{3} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+2 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.214768, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {5982, 5916, 5988, 5932, 2447, 5948} \[ -a \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{1}{3} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+2 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^2} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{1}{3} a \tanh ^{-1}(a x)^3+(2 a) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{1}{3} a \tanh ^{-1}(a x)^3+(2 a) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx\\ &=a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{1}{3} a \tanh ^{-1}(a x)^3+2 a \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )-\left (2 a^2\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{1}{3} a \tanh ^{-1}(a x)^3+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 )\\ \end{align*}

Mathematica [A]  time = 0.222912, size = 61, normalized size = 0.92 \[ -a \left (\text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )-\frac{1}{3} \tanh ^{-1}(a x) \left (\left (\tanh ^{-1}(a x)+3\right ) \tanh ^{-1}(a x)-\frac{3 \tanh ^{-1}(a x)}{a x}+6 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.374, size = 4449, normalized size = 67.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*arctanh(a*x)^2+1/2*I*a*Pi*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*a*Pi*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1
/2))-1/2*I*a*Pi*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*a*Pi*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I*a*Pi*
arctanh(a*x)^2-1/4*I*a*Pi*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))*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^2/x+1/2*a*arc
tanh(a*x)^2*ln(a*x+1)-a*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+2*a*arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+
1)^(1/2))+1/3*a*arctanh(a*x)^3+1/4*I*a*Pi*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*arctanh(a*x)^2-1/4*I*a*Pi*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*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn
(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^2-1/2*I*a*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*arctanh(a*x)^2-1/4*I*a*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a
*x+1)^2/(a^2*x^2-1))*arctanh(a*x)^2+1/4*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)
/((a*x+1)^2/(-a^2*x^2+1)+1))^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I*a*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*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*a*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*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*csgn(I*(a*x+1)/(-a^
2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*csgn(I*(a*x+1)^2/
(a^2*x^2-1))^3*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*a
rctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*a*Pi*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*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^
2+1)+1))^3*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*
(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*csgn(I/((a*
x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*polylog(2,(a*x+1)/(-a^2*x^2
+1)^(1/2))+1/2*I*a*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*polylog(2,-(a*x+1)/(-
a^2*x^2+1)^(1/2))+1/4*I*a*Pi*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*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*a*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(
a^2*x^2-1))*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)
^2/(a^2*x^2-1))*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*a*Pi*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*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*csgn(I*(a*x+1)^
2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*csgn(I*(a
*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*csg
n(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*dilog((a*x+1)/(-a^2
*x^2+1)^(1/2))+1/2*I*a*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*dilog((a*x+1)/(-a
^2*x^2+1)^(1/2))-a*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+a*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+a*polylog(2,-(a*x+1
)/(-a^2*x^2+1)^(1/2))+a*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+a*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-
1/2*a*arctanh(a*x)^2*ln(a*x-1)+1/2*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^
(1/2))+1/2*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*a*Pi*csgn(I/(
(a*x+1)^2/(-a^2*x^2+1)+1))^2*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))
^3*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)^2+1/2*I
*a*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*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1
)^(1/2))+1/4*I*a*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arctanh(a*x)*ln(1-(a*x+
1)/(-a^2*x^2+1)^(1/2))-1/4*I*a*Pi*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))*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*a*Pi*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*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(
1/2))+1/4*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1
/2))-1/4*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^2+1/2*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2
*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*polyl
og(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*a*Pi*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*csgn(I*(a
*x+1)^2/(a^2*x^2-1))^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*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))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))*arctanh(a*x)^2+1/4*I*a*Pi*
csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)*ln(
1-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*a*Pi*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))*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*a*Pi*csgn(I/(
(a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^2-1/4*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1)
)^3*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*
dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*dilog((a*x+1)/(-a^2*x^2+1)^(
1/2))+1/4*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*a*Pi*csgn(I*(a*x+1)^2
/(a^2*x^2-1))^3*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*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))*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))
-1/4*I*a*Pi*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))*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*poly
log(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*polylog(2,-(a*x+1)/(-a^2*x^
2+1)^(1/2))-1/2*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*a*Pi*c
sgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)^2

________________________________________________________________________________________

Maxima [B]  time = 0.993476, size = 320, normalized size = 4.85 \begin{align*} -\frac{1}{24} \, a^{2}{\left (\frac{3 \,{\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} - 3 \,{\left (\log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 6 \, \log \left (a x - 1\right )^{2}}{a} - \frac{24 \,{\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{24 \,{\left (\log \left (a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x\right )\right )}}{a} - \frac{24 \,{\left (\log \left (-a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x\right )\right )}}{a}\right )} + \frac{1}{4} \,{\left (2 \,{\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right ) + 8 \, \log \left (x\right )\right )} a \operatorname{artanh}\left (a x\right ) + \frac{1}{2} \,{\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac{2}{x}\right )} \operatorname{artanh}\left (a x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*a^2*((3*(log(a*x - 1) - 2)*log(a*x + 1)^2 - log(a*x + 1)^3 + log(a*x - 1)^3 - 3*(log(a*x - 1)^2 - 4*log(
a*x - 1))*log(a*x + 1) + 6*log(a*x - 1)^2)/a - 24*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a
+ 24*(log(a*x + 1)*log(x) + dilog(-a*x))/a - 24*(log(-a*x + 1)*log(x) + dilog(a*x))/a) + 1/4*(2*(log(a*x - 1)
- 2)*log(a*x + 1) - log(a*x + 1)^2 - log(a*x - 1)^2 - 4*log(a*x - 1) + 8*log(x))*a*arctanh(a*x) + 1/2*(a*log(a
*x + 1) - a*log(a*x - 1) - 2/x)*arctanh(a*x)^2

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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