3.313 \(\int \frac{\tanh ^{-1}(a x)^2}{x^2 (1-a^2 x^2)^3} \, dx\)

Optimal. Leaf size=209 \[ -a \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{31 a^2 x}{64 \left (1-a^2 x^2\right )}+\frac{a^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac{a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac{7 a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}-\frac{a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}+\frac{5}{8} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{31}{64} a \tanh ^{-1}(a x)+2 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x) \]

[Out]

(a^2*x)/(32*(1 - a^2*x^2)^2) + (31*a^2*x)/(64*(1 - a^2*x^2)) + (31*a*ArcTanh[a*x])/64 - (a*ArcTanh[a*x])/(8*(1
- a^2*x^2)^2) - (7*a*ArcTanh[a*x])/(8*(1 - a^2*x^2)) + a*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/x + (a^2*x*ArcTanh[a
*x]^2)/(4*(1 - a^2*x^2)^2) + (7*a^2*x*ArcTanh[a*x]^2)/(8*(1 - a^2*x^2)) + (5*a*ArcTanh[a*x]^3)/8 + 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 = 0.492116, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {6030, 5982, 5916, 5988, 5932, 2447, 5948, 5956, 5994, 199, 206, 5964} \[ -a \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{31 a^2 x}{64 \left (1-a^2 x^2\right )}+\frac{a^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac{a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac{7 a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}-\frac{a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}+\frac{5}{8} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{31}{64} a \tanh ^{-1}(a x)+2 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)^3),x]

[Out]

(a^2*x)/(32*(1 - a^2*x^2)^2) + (31*a^2*x)/(64*(1 - a^2*x^2)) + (31*a*ArcTanh[a*x])/64 - (a*ArcTanh[a*x])/(8*(1
- a^2*x^2)^2) - (7*a*ArcTanh[a*x])/(8*(1 - a^2*x^2)) + a*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/x + (a^2*x*ArcTanh[a
*x]^2)/(4*(1 - a^2*x^2)^2) + (7*a^2*x*ArcTanh[a*x]^2)/(8*(1 - a^2*x^2)) + (5*a*ArcTanh[a*x]^3)/8 + 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])

Rule 5964

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*p*(d + e*x^2)^(
q + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q
+ 1)*(a + b*ArcTanh[c*x])^p, x], x] + Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTanh[c*
x])^(p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c
, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )^3} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}+\frac{a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac{1}{8} a^2 \int \frac{1}{\left (1-a^2 x^2\right )^3} \, dx+\frac{1}{4} \left (3 a^2\right ) \int \frac{\tanh ^{-1}(a 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}{32 \left (1-a^2 x^2\right )^2}-\frac{a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}+\frac{a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac{7}{24} a \tanh ^{-1}(a x)^3+\frac{1}{32} \left (3 a^2\right ) \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx+a^2 \int \frac{\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx-\frac{1}{4} \left (3 a^3\right ) \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^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^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac{3 a^2 x}{64 \left (1-a^2 x^2\right )}-\frac{a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac{7 a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{a^2 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac{5}{8} 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}{64} \left (3 a^2\right ) \int \frac{1}{1-a^2 x^2} \, dx+\frac{1}{8} \left (3 a^2\right ) \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx+\frac{1}{2} a^2 \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac{a^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac{31 a^2 x}{64 \left (1-a^2 x^2\right )}+\frac{3}{64} a \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac{7 a \tanh ^{-1}(a x)}{8 \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}{4 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac{5}{8} a \tanh ^{-1}(a x)^3+(2 a) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\frac{1}{16} \left (3 a^2\right ) \int \frac{1}{1-a^2 x^2} \, dx+\frac{1}{4} a^2 \int \frac{1}{1-a^2 x^2} \, dx\\ &=\frac{a^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac{31 a^2 x}{64 \left (1-a^2 x^2\right )}+\frac{31}{64} a \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac{7 a \tanh ^{-1}(a x)}{8 \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}{4 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac{5}{8} 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}{32 \left (1-a^2 x^2\right )^2}+\frac{31 a^2 x}{64 \left (1-a^2 x^2\right )}+\frac{31}{64} a \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac{7 a \tanh ^{-1}(a x)}{8 \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}{4 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac{5}{8} 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 = 0.567019, size = 127, normalized size = 0.61 \[ -a \left (\text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x)^2 \left (\frac{a x}{a^2 x^2-1}+\frac{1}{a x}-\frac{1}{32} \sinh \left (4 \tanh ^{-1}(a x)\right )-1\right )-\frac{5}{8} \tanh ^{-1}(a x)^3-\frac{1}{4} \sinh \left (2 \tanh ^{-1}(a x)\right )-\frac{1}{256} \sinh \left (4 \tanh ^{-1}(a x)\right )+\frac{1}{64} \tanh ^{-1}(a x) \left (-128 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )+32 \cosh \left (2 \tanh ^{-1}(a x)\right )+\cosh \left (4 \tanh ^{-1}(a x)\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)^3),x]

[Out]

-(a*((-5*ArcTanh[a*x]^3)/8 + (ArcTanh[a*x]*(32*Cosh[2*ArcTanh[a*x]] + Cosh[4*ArcTanh[a*x]] - 128*Log[1 - E^(-2
*ArcTanh[a*x])]))/64 + PolyLog[2, E^(-2*ArcTanh[a*x])] - Sinh[2*ArcTanh[a*x]]/4 + ArcTanh[a*x]^2*(-1 + 1/(a*x)
+ (a*x)/(-1 + a^2*x^2) - Sinh[4*ArcTanh[a*x]]/32) - Sinh[4*ArcTanh[a*x]]/256))

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Maple [C]  time = 0.602, size = 4797, normalized size = 23. \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)^3,x)

[Out]

15/32*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))-15/32*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))+15/32*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))+15/32*I*a*Pi*c
sgn(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))+15/32*
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)
)-15/32*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))-15/32*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))-15/32*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))+15/32*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))+15/16*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))+15/32*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-
a*arctanh(a*x)^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*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))-15/16*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))+15/32*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))+15/32*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))-15/32*I*a*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*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+15/32*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))+15/16*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))+15/16*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*pol
ylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-15/32*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)^2-15/32*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)^2-15/16*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)^2-15/16*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))+15/16*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))+15/32*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))+15/32*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))-15/32*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))+15/16*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))-15/16*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))-15/16*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))-15/16*I*a*Pi*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-15/32*I*a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*
arctanh(a*x)^2+15/16*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))-1/128*a*arctanh(a*x)/(a*x-1)^2-1/128*a*arctanh(a*x)/(a*x+1)^2+15/16*a*arctanh(a
*x)^2*ln(a*x+1)-15/8*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))+1/8/(a*x+1)*a^2*x-1/8*a^2*x/(a*x-1)-7/16*a*arctanh(a*
x)^2/(a*x-1)-15/16*a*arctanh(a*x)^2*ln(a*x-1)-7/16*a*arctanh(a*x)^2/(a*x+1)+1/4*arctanh(a*x)/(a*x-1)*a^2*x+1/4
*arctanh(a*x)/(a*x+1)*a^2*x+1/512*a/(a*x-1)^2-1/512*a/(a*x+1)^2-arctanh(a*x)^2/x-15/32*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+15/32*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))-15/32*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))+15/32*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))+15/32*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))+15/32*
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))+15/32*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))-15/32*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))+15/16*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x
)^2-15/16*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^2+15/32*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))+15/16*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))+15/16*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))+15/16*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))
-15/16*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))-15/16*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))-15/32*I*a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*c
sgn(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))-1/8*a/(a*x-1)-1/8*a/(a*x+1)+5/8*a*arctanh(a*x)^3-1/128*arctanh(a*x)/(a*x-1)^2*a^3*x^2-1/64*arcta
nh(a*x)/(a*x-1)^2*a^2*x-1/128*arctanh(a*x)/(a*x+1)^2*a^3*x^2+1/64*arctanh(a*x)/(a*x+1)^2*a^2*x+15/16*I*a*Pi*ar
ctanh(a*x)^2-15/16*I*a*Pi*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-15/16*I*a*Pi*polylog(2,(a*x+1)/(-a^2*x^2+1)^(
1/2))-15/16*I*a*Pi*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+15/16*I*a*Pi*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-1/512/(a
*x+1)^2*a^3*x^2+1/256/(a*x+1)^2*a^2*x+1/512/(a*x-1)^2*a^3*x^2+1/256/(a*x-1)^2*a^2*x+1/16*a*arctanh(a*x)^2/(a*x
-1)^2-1/16*a*arctanh(a*x)^2/(a*x+1)^2-15/32*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))-15/32*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))+15/32*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+15/32*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))-15/32*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*a*arctanh(a*x)/(a*x-1)-1/4
*a*arctanh(a*x)/(a*x+1)+15/32*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))

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Maxima [B]  time = 1.03817, size = 721, normalized size = 3.45 \begin{align*} \text{result too large to display} \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)^3,x, algorithm="maxima")

[Out]

-1/128*a^2*(2*(31*a^3*x^3 - 5*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^3 + 5*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x -
1)^3 - (16*a^4*x^4 - 32*a^2*x^2 - 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1) + 16)*log(a*x + 1)^2 + 16*(a^4*x^
4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 33*a*x - (15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 32*(a^4*x^4 - 2*a^
2*x^2 + 1)*log(a*x - 1))*log(a*x + 1))/(a^5*x^4 - 2*a^3*x^2 + a) - 128*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilo
g(-1/2*a*x + 1/2))/a + 128*(log(a*x + 1)*log(x) + dilog(-a*x))/a - 128*(log(-a*x + 1)*log(x) + dilog(a*x))/a -
31*log(a*x + 1)/a + 31*log(a*x - 1)/a) + 1/32*a*((28*a^2*x^2 - 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 +
30*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x - 1) - 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 32)/(a^
4*x^4 - 2*a^2*x^2 + 1) - 32*log(a*x + 1) - 32*log(a*x - 1) + 64*log(x))*arctanh(a*x) + 1/16*(15*a*log(a*x + 1)
- 15*a*log(a*x - 1) - 2*(15*a^4*x^4 - 25*a^2*x^2 + 8)/(a^4*x^5 - 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^{6} x^{8} - 3 \, a^{4} x^{6} + 3 \, 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)^3,x, algorithm="fricas")

[Out]

integral(-arctanh(a*x)^2/(a^6*x^8 - 3*a^4*x^6 + 3*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 )}}{a^{6} x^{8} - 3 a^{4} x^{6} + 3 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)**3,x)

[Out]

-Integral(atanh(a*x)**2/(a**6*x**8 - 3*a**4*x**6 + 3*a**2*x**4 - x**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 )}^{3} 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)^3,x, algorithm="giac")

[Out]

integrate(-arctanh(a*x)^2/((a^2*x^2 - 1)^3*x^2), x)