3.3.72 \(\int \frac {\tanh ^{-1}(a x)^2}{x^3 (1-a^2 x^2)^2} \, dx\) [272]

Optimal. Leaf size=205 \[ \frac {a^2}{4 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{x}-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {2}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+2 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-2 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-a^2 \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \]

[Out]

1/4*a^2/(-a^2*x^2+1)-a*arctanh(a*x)/x-1/2*a^3*x*arctanh(a*x)/(-a^2*x^2+1)+1/4*a^2*arctanh(a*x)^2-1/2*arctanh(a
*x)^2/x^2+1/2*a^2*arctanh(a*x)^2/(-a^2*x^2+1)+2/3*a^2*arctanh(a*x)^3+a^2*ln(x)-1/2*a^2*ln(-a^2*x^2+1)+2*a^2*ar
ctanh(a*x)^2*ln(2-2/(a*x+1))-2*a^2*arctanh(a*x)*polylog(2,-1+2/(a*x+1))-a^2*polylog(3,-1+2/(a*x+1))

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Rubi [A]
time = 0.49, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {6177, 6129, 6037, 272, 36, 29, 31, 6095, 6135, 6079, 6203, 6745, 6141, 6103, 267} \begin {gather*} -a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right )-2 a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {a^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+a^2 \log (x)+\frac {2}{3} a^2 \tanh ^{-1}(a x)^3+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2+2 a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{2 x^2}-\frac {a \tanh ^{-1}(a x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^2/(4*(1 - a^2*x^2)) - (a*ArcTanh[a*x])/x - (a^3*x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + (a^2*ArcTanh[a*x]^2)/4 -
ArcTanh[a*x]^2/(2*x^2) + (a^2*ArcTanh[a*x]^2)/(2*(1 - a^2*x^2)) + (2*a^2*ArcTanh[a*x]^3)/3 + a^2*Log[x] - (a^
2*Log[1 - a^2*x^2])/2 + 2*a^2*ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - 2*a^2*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 +
a*x)] - a^2*PolyLog[3, -1 + 2/(1 + a*x)]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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]

Rule 6203

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan
h[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] - Dist[b*(p/2), Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2
/(1 + c*x))^2, 0]

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{x^3 \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=2 \left (a^2 \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx\right )+a^4 \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^3} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+a \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx\right )-a^3 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+a \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx+a^3 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-\left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\right )+\frac {1}{2} a^4 \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {a^2}{4 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{x}-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+a^2 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+a^3 \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\right )\\ &=\frac {a^2}{4 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{x}-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {Li}_3\left (-1+\frac {2}{1+a x}\right )\right )+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=\frac {a^2}{4 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{x}-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {Li}_3\left (-1+\frac {2}{1+a x}\right )\right )+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^4 \text {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=\frac {a^2}{4 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{x}-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {Li}_3\left (-1+\frac {2}{1+a x}\right )\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.62, size = 146, normalized size = 0.71 \begin {gather*} a^2 \left (2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+\frac {1}{24} \left (2 i \pi ^3-16 \tanh ^{-1}(a x)^3+3 \cosh \left (2 \tanh ^{-1}(a x)\right )+6 \tanh ^{-1}(a x)^2 \left (2-\frac {2}{a^2 x^2}+\cosh \left (2 \tanh ^{-1}(a x)\right )+8 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )\right )+24 \log \left (\frac {a x}{\sqrt {1-a^2 x^2}}\right )-24 \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )-\frac {6 \tanh ^{-1}(a x) \left (4+a x \sinh \left (2 \tanh ^{-1}(a x)\right )\right )}{a x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*(2*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + ((2*I)*Pi^3 - 16*ArcTanh[a*x]^3 + 3*Cosh[2*ArcTanh[a*x]]
+ 6*ArcTanh[a*x]^2*(2 - 2/(a^2*x^2) + Cosh[2*ArcTanh[a*x]] + 8*Log[1 - E^(2*ArcTanh[a*x])]) + 24*Log[(a*x)/Sqr
t[1 - a^2*x^2]] - 24*PolyLog[3, E^(2*ArcTanh[a*x])] - (6*ArcTanh[a*x]*(4 + a*x*Sinh[2*ArcTanh[a*x]]))/(a*x))/2
4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 402.64, size = 2444, normalized size = 11.92

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2444\)
default	\(\text {Expression too large to display}\)	\(2444\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)^2/x^3/(-a^2*x^2+1)^2,x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/2*arctanh(a*x)^2/a^2/x^2+2*arctanh(a*x)^2*ln(a*x)+1/4*arctanh(a*x)^2/(a*x+1)-arctanh(a*x)^2*ln(a*x+1)-
1/4*arctanh(a*x)^2/(a*x-1)-arctanh(a*x)^2*ln(a*x-1)+2*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-2*arctanh(
a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)-1)+2*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+4*arctanh(a*x)*polylog(2
,-(a*x+1)/(-a^2*x^2+1)^(1/2))-4*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2
+1)^(1/2))+4*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-4*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/48*(
32*arctanh(a*x)^3*a^3*x^3-32*arctanh(a*x)^3*a*x+12*arctanh(a*x)^2*a*x-48*I*arctanh(a*x)^2*Pi*a^3*x^3+24*I*arct
anh(a*x)^2*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))*csgn(I/((a*x
+1)^2/(-a^2*x^2+1)+1))*Pi*a^3*x^3-48*I*arctanh(a*x)^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1
)+1))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*Pi*a^3*x^3-24*I*arctanh(a*x)^2*csg
n(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))*csgn(I/((a*x+1)^2/(-a^2*x^
2+1)+1))*Pi*a*x+48*I*arctanh(a*x)^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I/((a*x
+1)^2/(-a^2*x^2+1)+1))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*Pi*a*x+9*a*x-48*a*x*arctanh(a*x)+48*a^3*x^3*arctanh(
a*x)+24*a^2*x^2*arctanh(a*x)-48*arctanh(a*x)+3*a^3*x^3-12*arctanh(a*x)^2*a^3*x^3-24*I*arctanh(a*x)^2*csgn(I*(a
*x+1)^2/(a^2*x^2-1))^3*Pi*a^3*x^3+96*ln(2)*arctanh(a*x)^2*a*x-96*ln(2)*arctanh(a*x)^2*a^3*x^3+48*I*arctanh(a*x
)^2*Pi*a*x-48*I*arctanh(a*x)^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*a^3*x^3-48*I
*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*a^3*x^3+48*I*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^
2+1)+1))^2*Pi*a^3*x^3+24*I*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi*a*x+24*I*arctanh(a*x)^2*csgn(I*(a
*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*a*x+48*I*arctanh(a*x)^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)
/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*a*x+48*I*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*a*x-48*I*arc
tanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*a*x-24*I*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x
+1)^2/(-a^2*x^2+1)+1))^3*Pi*a^3*x^3+48*I*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x
^2+1)^(1/2))*Pi*a*x-24*I*arctanh(a*x)^2*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*Pi*a*x+24*I*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))
^2*Pi*a*x+24*I*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*csgn(I/((a*x+1)^2/(-a
^2*x^2+1)+1))*Pi*a*x-48*I*arctanh(a*x)^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*csgn(
I/((a*x+1)^2/(-a^2*x^2+1)+1))*Pi*a*x-48*I*arctanh(a*x)^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^
2+1)+1))^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*Pi*a*x-48*I*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(
I*(a*x+1)/(-a^2*x^2+1)^(1/2))*Pi*a^3*x^3+24*I*arctanh(a*x)^2*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*Pi*a^3*x^3-24*I*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a
*x+1)/(-a^2*x^2+1)^(1/2))^2*Pi*a^3*x^3-24*I*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1
)+1))^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*Pi*a^3*x^3+48*I*arctanh(a*x)^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((
a*x+1)^2/(-a^2*x^2+1)+1))^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*Pi*a^3*x^3+48*I*arctanh(a*x)^2*csgn(I*((a*x+1)^
2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*Pi*a^3*x^3)/(a*x+1)/a/x/(a*
x-1)+ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)+ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^2/x^3/(-a^2*x^2+1)^2,x, algorithm="maxima")

[Out]

1/2*a^6*integrate(x^6*log(a*x + 1)*log(-a*x + 1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x) + 1/2*a^5*integrate(x^5*log(a
*x + 1)*log(-a*x + 1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x) - 1/16*(a*(2/(a^4*x - a^3) - log(a*x + 1)/a^3 + log(a*x
- 1)/a^3) + 4*log(-a*x + 1)/(a^4*x^2 - a^2))*a^4 - 1/2*a^4*integrate(x^4*log(a*x + 1)*log(-a*x + 1)/(a^4*x^7 -
2*a^2*x^5 + x^3), x) - 1/2*a^3*integrate(x^3*log(a*x + 1)*log(-a*x + 1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x) + 1/2
*a^3*integrate(x^3*log(-a*x + 1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x) - 1/4*a^2*integrate(x^2*log(-a*x + 1)/(a^4*x^
7 - 2*a^2*x^5 + x^3), x) - 1/4*a*integrate(x*log(-a*x + 1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x) - 1/24*(2*(a^4*x^4
- a^2*x^2)*log(-a*x + 1)^3 + 3*(2*a^2*x^2 + 2*(a^4*x^4 - a^2*x^2)*log(a*x + 1) - 1)*log(-a*x + 1)^2)/(a^2*x^4
- x^2) + 1/4*integrate(log(a*x + 1)^2/(a^4*x^7 - 2*a^2*x^5 + x^3), x) - 1/2*integrate(log(a*x + 1)*log(-a*x +
1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x)

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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^3/(-a^2*x^2+1)^2,x, algorithm="fricas")

[Out]

integral(arctanh(a*x)^2/(a^4*x^7 - 2*a^2*x^5 + x^3), 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^{3} \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**3/(-a**2*x**2+1)**2,x)

[Out]

Integral(atanh(a*x)**2/(x**3*(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^3/(-a^2*x^2+1)^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x^3\,{\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^3*(a^2*x^2 - 1)^2),x)

[Out]

int(atanh(a*x)^2/(x^3*(a^2*x^2 - 1)^2), x)

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