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

Optimal. Leaf size=142 \[ \frac {a^2 x}{4 \left (1-a^2 x^2\right )}+\frac {a^2 x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}-a \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {1}{2} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}+\frac {1}{4} a \tanh ^{-1}(a x)+2 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \]

[Out]

1/4*a^2*x/(-a^2*x^2+1)+1/4*a*arctanh(a*x)-1/2*a*arctanh(a*x)/(-a^2*x^2+1)+a*arctanh(a*x)^2-arctanh(a*x)^2/x+1/
2*a^2*x*arctanh(a*x)^2/(-a^2*x^2+1)+1/2*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 = 0.32, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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])

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 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 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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.32, size = 97, normalized size = 0.68 \[ \frac {-8 a x \text {Li}_2\left (e^{-2 \tanh ^{-1}(a x)}\right )+4 a x \tanh ^{-1}(a x)^3+2 \tanh ^{-1}(a x)^2 \left (4 a x+a x \sinh \left (2 \tanh ^{-1}(a x)\right )-4\right )+a x \sinh \left (2 \tanh ^{-1}(a x)\right )-2 a x \tanh ^{-1}(a x) \left (\cosh \left (2 \tanh ^{-1}(a x)\right )-8 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )}{8 x} \]

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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fricas [F]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (a x\right )^{2}}{a^{4} x^{6} - 2 \, a^{2} x^{4} + x^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2} x^{2}}\,{d x} \]

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)

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maple [C]  time = 0.87, size = 4589, normalized size = 32.32 \[ \text {output too large to display} \]

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]

-a*arctanh(a*x)^2+1/8*a*arctanh(a*x)/(a*x-1)-1/8*a*arctanh(a*x)/(a*x+1)-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)/(1+(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))-ar
ctanh(a*x)^2/x+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))+3/8*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))*dilog((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))*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+(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))-3/8*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1
+(a*x+1)^2/(-a^2*x^2+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)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*arctanh(a*x)^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)/(1+(a*x+1)^2/(-a^2*x^2+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)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*polylog(2,(a*x+1)/(-a^2*x
^2+1)^(1/2))-1/16*a^2*x/(a*x-1)+1/16/(a*x+1)*a^2*x-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)+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))+3/8*I*a*P
i*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
*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)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*a
rctanh(a*x)^2-3/4*I*a*Pi*csgn(I/(1+(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*a
rctanh(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+(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)/(1+(a*x+1)^2/(-a^2*x^2+1))
)^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+3/4*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+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)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*dilog(1+(a*x+1)/(-a^2
*x^2+1)^(1/2))+3/4*I*a*Pi*csgn(I/(1+(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*P
i*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3/4*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-
a^2*x^2+1)))^3*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+3/4*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*dilog(1+(a*x+
1)/(-a^2*x^2+1)^(1/2))+1/2*a*arctanh(a*x)^3+a*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-a*dilog((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/(1+
(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(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)/(-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))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)
))^2*polylog(2,-(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)^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*arcta
nh(a*x)^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))+3/8*I*a*Pi*
csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+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)/(1+(a*x+1)^2/(-a^
2*x^2+1)))^2*dilog((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*dilog((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))*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+3/8*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(
I*(a*x+1)^2/(a^2*x^2-1)/(1+(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/
(1+(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))-3/4*I*a*Pi*csgn(I/(1+(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*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-3/8*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)
))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*arctanh(a*x)^2-1/16*a/(a*x-1)-1/16*a/(a*x+1)+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))+3/4*I*a*Pi*csgn(I/(1+(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+(a*x+1)^2/(-a^2*
x^2+1)))^3*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+3/4*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*arctanh(a*x)^2-3/
4*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-3/4*I*a*Pi*csgn(I/(1+(a*x
+1)^2/(-a^2*x^2+1)))^2*arctanh(a*x)^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*csg
n(I*(a*x+1)^2/(a^2*x^2-1))^3*dilog(1+(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*dilog((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*
dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-3/8*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)
)*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+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*arctanh(a*x)^2+3/8*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^
2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*dilog(1+(a*x
+1)/(-a^2*x^2+1)^(1/2))+3/8*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(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))+3/4*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/(1+(a*x+1)^2/
(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(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/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-
1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*arctanh(a*x)^2-3/8*I*a*Pi*csgn(I/(1+(a*x+1)^2/(-a
^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*dilog((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/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1
))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))

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maxima [B]  time = 0.35, size = 406, normalized size = 2.86 \[ \frac {1}{16} \, a^{2} {\left (\frac {{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + {\left (4 \, a^{2} x^{2} - 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right ) - 4\right )} \log \left (a x + 1\right )^{2} - 4 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4 \, a x + {\left (3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 8 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )}{a^{3} x^{2} - a} + \frac {16 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {16 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {16 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )}}{a} + \frac {2 \, \log \left (a x + 1\right )}{a} - \frac {2 \, \log \left (a x - 1\right )}{a}\right )} - \frac {1}{8} \, a {\left (\frac {3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4}{a^{2} x^{2} - 1} + 8 \, \log \left (a x + 1\right ) + 8 \, \log \left (a x - 1\right ) - 16 \, \log \relax (x)\right )} \operatorname {artanh}\left (a x\right ) + \frac {1}{4} \, {\left (3 \, a \log \left (a x + 1\right ) - 3 \, a \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} - 2\right )}}{a^{2} x^{3} - x}\right )} \operatorname {artanh}\left (a x\right )^{2} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x^2\,{\left (a^2\,x^2-1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x^{2} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]

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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