3.3.39 \(\int \frac {\tanh ^{-1}(a x)^2}{x^2 (1-a^2 x^2)} \, dx\) [239]
Optimal. Leaf size=66 \[ a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}+\frac {1}{3} a \tanh ^{-1}(a x)^3+2 a \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
[Out]
a*arctanh(a*x)^2-arctanh(a*x)^2/x+1/3*a*arctanh(a*x)^3+2*a*arctanh(a*x)*ln(2-2/(a*x+1))-a*polylog(2,-1+2/(a*x+
1))
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Rubi [A]
time = 0.14, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 = {6129, 6037,
6135, 6079, 2497, 6095} \begin {gather*} -a \text {Li}_2\left (\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) \end {gather*}
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 2497
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 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 6079
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 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 6129
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 6135
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]
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*}
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Mathematica [A]
time = 0.16, size = 61, normalized size = 0.92 \begin {gather*} -a \left (-\frac {1}{3} \tanh ^{-1}(a x) \left (-\frac {3 \tanh ^{-1}(a x)}{a x}+\tanh ^{-1}(a x) \left (3+\tanh ^{-1}(a x)\right )+6 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )+\text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)),x]
[Out]
-(a*(-1/3*(ArcTanh[a*x]*((-3*ArcTanh[a*x])/(a*x) + ArcTanh[a*x]*(3 + ArcTanh[a*x]) + 6*Log[1 - E^(-2*ArcTanh[a
*x])])) + PolyLog[2, E^(-2*ArcTanh[a*x])]))
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 35.34, size = 4380, normalized size = 66.36
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| method |
result |
size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(4380\) |
| default |
\(\text {Expression too large to display}\) |
\(4380\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(arctanh(a*x)^2/x^2/(-a^2*x^2+1),x,method=_RETURNVERBOSE)
[Out]
a*(-1/4*I*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))+polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+polylog(
2,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/3*arctanh(a*x)^3-arctanh(a*x)^2+1/4*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*cs
gn(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*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*a
rctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*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
*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*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)^2+1/4*I*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))+1/2*I*Pi*csg
n(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*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))+arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*ln(1+(a*x+
1)/(-a^2*x^2+1)^(1/2))+1/2*arctanh(a*x)^2*ln(a*x+1)-1/2*arctanh(a*x)^2*ln(a*x-1)-1/2*I*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))+1/2*I*arctanh(a*x)^2*Pi*csg
n(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3-1/2*I*arctanh(a*x)^2*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2-1/4*I*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))+1/2*I*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*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*Pi*csgn(I/((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))-1/2*I*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*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))+1/2*I*Pi*cs
gn(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))-1/4*I*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)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^2-1/2
*I*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-dilog((a*x+1)/(-a^2*x^
2+1)^(1/2))+dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*Pi*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*Pi*polylo
g(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^2/a/x-1/2*I*Pi*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I*Pi*dilog(
1+(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*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))+1/4*I*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*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*d
ilog((a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*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*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((a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*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*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*cs
gn(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*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*Pi*cs
gn(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*
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*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))+1/4*I*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))-1/4*I*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*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*a
rctanh(a*x)^2-1/4*I*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...
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 237 vs.
\(2 (63) = 126\).
time = 0.27, size = 237, normalized size = 3.59 \begin {gather*} -\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 {gather*}
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
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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(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)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{a^{2} x^{4} - x^{2}}\, dx \end {gather*}
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)
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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(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)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x^2\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-atanh(a*x)^2/(x^2*(a^2*x^2 - 1)),x)
[Out]
-int(atanh(a*x)^2/(x^2*(a^2*x^2 - 1)), x)
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