3.271 \(\int \frac{\tanh ^{-1}(a x)^2}{x^2 (1-a^2 x^2)^2} \, dx\)
Optimal. Leaf size=142 \[ -a \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{a^2 x}{4 \left (1-a^2 x^2\right )}+\frac{a^2 x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}-\frac{a \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac{1}{2} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{1}{4} a \tanh ^{-1}(a x)+2 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x) \]
[Out]
(a^2*x)/(4*(1 - a^2*x^2)) + (a*ArcTanh[a*x])/4 - (a*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + a*ArcTanh[a*x]^2 - ArcTa
nh[a*x]^2/x + (a^2*x*ArcTanh[a*x]^2)/(2*(1 - a^2*x^2)) + (a*ArcTanh[a*x]^3)/2 + 2*a*ArcTanh[a*x]*Log[2 - 2/(1
+ a*x)] - a*PolyLog[2, -1 + 2/(1 + a*x)]
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Rubi [A] time = 0.315069, antiderivative size = 142, normalized size of antiderivative =
1., number of steps used = 11, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) =
0.5, Rules used = {6030, 5982, 5916, 5988, 5932, 2447, 5948, 5956, 5994, 199, 206}
\[ -a \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{a^2 x}{4 \left (1-a^2 x^2\right )}+\frac{a^2 x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}-\frac{a \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac{1}{2} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{1}{4} a \tanh ^{-1}(a x)+2 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
Int[ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)^2),x]
[Out]
(a^2*x)/(4*(1 - a^2*x^2)) + (a*ArcTanh[a*x])/4 - (a*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + a*ArcTanh[a*x]^2 - ArcTa
nh[a*x]^2/x + (a^2*x*ArcTanh[a*x]^2)/(2*(1 - a^2*x^2)) + (a*ArcTanh[a*x]^3)/2 + 2*a*ArcTanh[a*x]*Log[2 - 2/(1
+ a*x)] - a*PolyLog[2, -1 + 2/(1 + a*x)]
Rule 6030
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int
[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh
[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[
m, 0] && NeQ[p, -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 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]
Rule 5956
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(x*(a + b*ArcTanh[c*x
])^p)/(2*d*(d + e*x^2)), x] + (-Dist[(b*c*p)/2, Int[(x*(a + b*ArcTanh[c*x])^(p - 1))/(d + e*x^2)^2, x], x] + S
imp[(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] &&
GtQ[p, 0]
Rule 5994
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[((d + e*x^2)
^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*e*(q + 1)), x] + Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan
h[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
Rule 199
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)
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{\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=\frac{a^2 x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac{1}{6} a \tanh ^{-1}(a x)^3+a^2 \int \frac{\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx-a^3 \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^2} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{a^2 x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac{1}{2} a \tanh ^{-1}(a x)^3+(2 a) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx+\frac{1}{2} a^2 \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac{a^2 x}{4 \left (1-a^2 x^2\right )}-\frac{a \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{a^2 x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac{1}{2} a \tanh ^{-1}(a x)^3+(2 a) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\frac{1}{4} a^2 \int \frac{1}{1-a^2 x^2} \, dx\\ &=\frac{a^2 x}{4 \left (1-a^2 x^2\right )}+\frac{1}{4} a \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{a^2 x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac{1}{2} 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\\ &=\frac{a^2 x}{4 \left (1-a^2 x^2\right )}+\frac{1}{4} a \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{a^2 x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac{1}{2} 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 = 0.34847, size = 97, normalized size = 0.68 \[ \frac{-8 a x \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )+4 a x \tanh ^{-1}(a x)^3+2 \tanh ^{-1}(a x)^2 \left (4 a x+a x \sinh \left (2 \tanh ^{-1}(a x)\right )-4\right )+a x \sinh \left (2 \tanh ^{-1}(a x)\right )-2 a x \tanh ^{-1}(a x) \left (\cosh \left (2 \tanh ^{-1}(a x)\right )-8 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )}{8 x} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)^2),x]
[Out]
(4*a*x*ArcTanh[a*x]^3 - 2*a*x*ArcTanh[a*x]*(Cosh[2*ArcTanh[a*x]] - 8*Log[1 - E^(-2*ArcTanh[a*x])]) - 8*a*x*Pol
yLog[2, E^(-2*ArcTanh[a*x])] + a*x*Sinh[2*ArcTanh[a*x]] + 2*ArcTanh[a*x]^2*(-4 + 4*a*x + a*x*Sinh[2*ArcTanh[a*
x]]))/(8*x)
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Maple [C] time = 0.55, size = 4589, normalized size = 32.3 \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)^2,x)
[Out]
3/8*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+3/8*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))+3/8*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))+3/4*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))+3/8*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))+3/8*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))-3/8*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))+3/4*I*a*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))+3/8*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)^2-3/8*I*a*Pi*c
sgn(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))-a*arctanh(a*x)^2-a*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+a*polyl
og(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*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2)
)+3/8*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*pol
ylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3/8*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)^2-3/4*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))+3/8*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))*dil
og((a*x+1)/(-a^2*x^2+1)^(1/2))-3/8*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))*d
ilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+3/8*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((a*x+1)/(-a^2*x^2+1)^(1/2))+3/8*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))-3/8*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))-3/8*I*a*P
i*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))-3/8*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))+3/4*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))-3/4*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))+3/4*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))-3/4*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)^2-3/8*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+3/8*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))*p
olylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-3/8*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))+3/4
*a*arctanh(a*x)^2*ln(a*x+1)-3/2*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))+a*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/16/(a*x+1)*a^2*x-1/16*a^2*x/(a*x-1)-1/4*
a*arctanh(a*x)^2/(a*x-1)-3/4*a*arctanh(a*x)^2*ln(a*x-1)-1/4*a*arctanh(a*x)^2/(a*x+1)+1/8*arctanh(a*x)/(a*x-1)*
a^2*x+1/8*arctanh(a*x)/(a*x+1)*a^2*x-3/4*I*a*Pi*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-3/4*I*a*Pi*polylog(2,(a
*x+1)/(-a^2*x^2+1)^(1/2))+3/4*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))-
3/4*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))+3/4*I*a*Pi*csgn(I/((a*x
+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)^2+3/4*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))+3/4*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))-3/4*I*a
*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^2-3/4*I*a*Pi*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2
))+3/4*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))-3/4*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))-3/4*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*pol
ylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+3/8*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))+3/8*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))-3/8*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))+3/8*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))+3/8*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))+3/8*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))-3/8*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))+3/8*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))-3/8*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^2-3/8*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)^2-3/4*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))-arctanh(a*x)^2/x+3/4*I*a*Pi*arctanh(a*x)^2+3/4*I*a*Pi*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-3/4*
I*a*Pi*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-1/16*a/(a*x-1)-1/16*a/(a*x+1)+1/2*a*arctanh(a*x)^3-3/8*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))-3/8*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))+3/4*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))+3/8*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))+3/8
*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))-3/8*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))+3/8*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))+1/8*a*arctanh(a*x)/(a*x-1)-1/8*a*arctanh(a*x)/(a*x+1)
________________________________________________________________________________________
Maxima [B] time = 1.01456, size = 548, normalized size = 3.86 \begin{align*} \frac{1}{16} \, a^{2}{\left (\frac{{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} -{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} +{\left (4 \, a^{2} x^{2} - 3 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right ) - 4\right )} \log \left (a x + 1\right )^{2} - 4 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4 \, a x +{\left (3 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 8 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )}{a^{3} x^{2} - 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{16 \,{\left (\log \left (a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x\right )\right )}}{a} + \frac{16 \,{\left (\log \left (-a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x\right )\right )}}{a} + \frac{2 \, \log \left (a x + 1\right )}{a} - \frac{2 \, \log \left (a x - 1\right )}{a}\right )} - \frac{1}{8} \, a{\left (\frac{3 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 6 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 3 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4}{a^{2} x^{2} - 1} + 8 \, \log \left (a x + 1\right ) + 8 \, \log \left (a x - 1\right ) - 16 \, \log \left (x\right )\right )} \operatorname{artanh}\left (a x\right ) + \frac{1}{4} \,{\left (3 \, a \log \left (a x + 1\right ) - 3 \, a \log \left (a x - 1\right ) - \frac{2 \,{\left (3 \, a^{2} x^{2} - 2\right )}}{a^{2} x^{3} - 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)^2,x, algorithm="maxima")
[Out]
1/16*a^2*(((a^2*x^2 - 1)*log(a*x + 1)^3 - (a^2*x^2 - 1)*log(a*x - 1)^3 + (4*a^2*x^2 - 3*(a^2*x^2 - 1)*log(a*x
- 1) - 4)*log(a*x + 1)^2 - 4*(a^2*x^2 - 1)*log(a*x - 1)^2 - 4*a*x + (3*(a^2*x^2 - 1)*log(a*x - 1)^2 - 8*(a^2*x
^2 - 1)*log(a*x - 1))*log(a*x + 1))/(a^3*x^2 - a) + 16*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2
))/a - 16*(log(a*x + 1)*log(x) + dilog(-a*x))/a + 16*(log(-a*x + 1)*log(x) + dilog(a*x))/a + 2*log(a*x + 1)/a
- 2*log(a*x - 1)/a) - 1/8*a*((3*(a^2*x^2 - 1)*log(a*x + 1)^2 - 6*(a^2*x^2 - 1)*log(a*x + 1)*log(a*x - 1) + 3*(
a^2*x^2 - 1)*log(a*x - 1)^2 - 4)/(a^2*x^2 - 1) + 8*log(a*x + 1) + 8*log(a*x - 1) - 16*log(x))*arctanh(a*x) + 1
/4*(3*a*log(a*x + 1) - 3*a*log(a*x - 1) - 2*(3*a^2*x^2 - 2)/(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{\operatorname{artanh}\left (a x\right )^{2}}{a^{4} x^{6} - 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)^2,x, algorithm="fricas")
[Out]
integral(arctanh(a*x)^2/(a^4*x^6 - 2*a^2*x^4 + x^2), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}^{2}{\left (a x \right )}}{x^{2} \left (a x - 1\right )^{2} \left (a x + 1\right )^{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)**2,x)
[Out]
Integral(atanh(a*x)**2/(x**2*(a*x - 1)**2*(a*x + 1)**2), x)
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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 )}^{2} 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)^2,x, algorithm="giac")
[Out]
integrate(arctanh(a*x)^2/((a^2*x^2 - 1)^2*x^2), x)