3.3.71 \(\int \frac {\tanh ^{-1}(a x)^2}{x^2 (1-a^2 x^2)^2} \, dx\) [271]

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

[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 = 0.23, 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 = {6177, 6129, 6037, 6135, 6079, 2497, 6095, 6103, 6141, 205, 212} \begin {gather*} \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) \end {gather*}

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 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 6103

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 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]

Rule 6141

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 6177

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 = 0.18, size = 97, normalized size = 0.68 \begin {gather*} \frac {4 a x \tanh ^{-1}(a x)^3-2 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 a x \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )+a x \sinh \left (2 \tanh ^{-1}(a x)\right )+2 \tanh ^{-1}(a x)^2 \left (-4+4 a x+a x \sinh \left (2 \tanh ^{-1}(a x)\right )\right )}{8 x} \end {gather*}

Antiderivative was successfully verified.

[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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 79.48, size = 4474, normalized size = 31.51

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4474\)
default	\(\text {Expression too large to display}\)	\(4474\)

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,method=_RETURNVERBOSE)

[Out]

a*(1/8*arctanh(a*x)*(a*x-1)/(a*x+1)+1/8*arctanh(a*x)*(a*x+1)/(a*x-1)+3/4*I*Pi*arctanh(a*x)^2-3/8*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))+polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+polylog(2,(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))+arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-3/8*I*Pi*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))^2*arctanh(a*x)*ln(1-(a*x+
1)/(-a^2*x^2+1)^(1/2))+3/8*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))+3/4*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)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3/8*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))+3/8*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))*arc
tanh(a*x)^2-3/8*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))-3/8*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))+3/8*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(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*arctanh(a*x)^3-arctanh(a*x)^
2-3/8*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))+3/4*arctanh(a*x)^2*ln(a*x+1)-3/4*arctanh(
a*x)^2*ln(a*x-1)+3/8*I*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*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+3/8*I*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*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*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*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)
^2+3/8*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-3/4*I*Pi*arctanh(a*x)*ln(1-(a*x+
1)/(-a^2*x^2+1)^(1/2))+3/4*I*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*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)^2-3/4*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*polyl
og(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3/4*I*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*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*Pi*csgn(I/((a*
x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^2-3/4*I*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))+3/4*I*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*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*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*p
olylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3/8*I*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*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*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))+dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))
-dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-3/4*I*Pi*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3/4*I*Pi*polylog(2,-(a*x+1)/
(-a^2*x^2+1)^(1/2))-3/4*I*Pi*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+3/4*I*Pi*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-ar
ctanh(a*x)^2/a/x+3/8*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))-3/8*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))+3/4*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))-3/4*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))-3/8*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*dilo
g((a*x+1)/(-a^2*x^2+1)^(1/2))+3/8*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))+3/8*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*arctanh(a*x)^2-3/8*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))-3/4*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+3/4*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*cs
gn(I*(a*x+1)^2/(a^2*x^2-1))^2*polylog(2,(a*x+1)...

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (128) = 256\).
time = 0.28, size = 406, normalized size = 2.86 \begin {gather*} \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 {gather*}

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 = 0.00, 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)^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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, 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)^2,x, algorithm="giac")

[Out]

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

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

[Out]

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

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