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

Optimal. Leaf size=221 \[ -3 a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \tanh ^{-1}(a x) \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text {Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \text {Li}_3\left (e^{\tanh ^{-1}(a x)}\right )+\frac {2 a^2}{\sqrt {1-a^2 x^2}}+\frac {a^2 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-\frac {2 a^3 x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \]

[Out]

-3*a^2*arctanh((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^2-a^2*arctanh((-a^2*x^2+1)^(1/2))-3*a^2*arctanh(a*x)*p
olylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*polylog(3
,-(a*x+1)/(-a^2*x^2+1)^(1/2))-3*a^2*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+2*a^2/(-a^2*x^2+1)^(1/2)-2*a^3*x*arc
tanh(a*x)/(-a^2*x^2+1)^(1/2)+a^2*arctanh(a*x)^2/(-a^2*x^2+1)^(1/2)-a*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x-1/2*arc
tanh(a*x)^2*(-a^2*x^2+1)^(1/2)/x^2

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Rubi [A]  time = 0.78, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6030, 6026, 6008, 266, 63, 208, 6020, 4182, 2531, 2282, 6589, 5994, 5958} \[ -3 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text {PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \text {PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )+\frac {2 a^2}{\sqrt {1-a^2 x^2}}-\frac {2 a^3 x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}+\frac {a^2 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 266

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 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5958

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[b/(c*d*Sqrt[d + e*x^2]
), x] + Simp[(x*(a + b*ArcTanh[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 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 6008

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

Rule 6020

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[1/Sqrt[d], Su
bst[Int[(a + b*x)^p*Csch[x], x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGt
Q[p, 0] && GtQ[d, 0]

Rule 6026

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

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 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

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

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Mathematica [A]  time = 2.85, size = 266, normalized size = 1.20 \[ \frac {1}{8} a^2 \left (\frac {16}{\sqrt {1-a^2 x^2}}+\frac {8 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {16 a x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {2 a x \tanh ^{-1}(a x) \text {csch}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}+24 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{-\tanh ^{-1}(a x)}\right )-24 \tanh ^{-1}(a x) \text {Li}_2\left (e^{-\tanh ^{-1}(a x)}\right )+24 \text {Li}_3\left (-e^{-\tanh ^{-1}(a x)}\right )-24 \text {Li}_3\left (e^{-\tanh ^{-1}(a x)}\right )+4 \tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right ) \tanh ^{-1}(a x)+12 \tanh ^{-1}(a x)^2 \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-12 \tanh ^{-1}(a x)^2 \log \left (e^{-\tanh ^{-1}(a x)}+1\right )+8 \log \left (\tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )+\tanh ^{-1}(a x)^2 \left (-\text {csch}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )-\tanh ^{-1}(a x)^2 \text {sech}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*(16/Sqrt[1 - a^2*x^2] - (16*a*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] + (8*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] -
(2*a*x*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2)/Sqrt[1 - a^2*x^2] - ArcTanh[a*x]^2*Csch[ArcTanh[a*x]/2]^2 + 12*Arc
Tanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] - 12*ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 8*Log[Tanh[ArcTanh[a*x
]/2]] + 24*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] - 24*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] + 24*Po
lyLog[3, -E^(-ArcTanh[a*x])] - 24*PolyLog[3, E^(-ArcTanh[a*x])] - ArcTanh[a*x]^2*Sech[ArcTanh[a*x]/2]^2 + 4*Ar
cTanh[a*x]*Tanh[ArcTanh[a*x]/2]))/8

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.48, size = 313, normalized size = 1.42 \[ -\frac {a^{2} \left (\arctanh \left (a x \right )^{2}-2 \arctanh \left (a x \right )+2\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x -1\right )}+\frac {\left (\arctanh \left (a x \right )^{2}+2 \arctanh \left (a x \right )+2\right ) a^{2} \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a x +2}-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \arctanh \left (a x \right ) \left (2 a x +\arctanh \left (a x \right )\right )}{2 x^{2}}-2 a^{2} \arctanh \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {3 a^{2} \arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-3 a^{2} \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a^{2} \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {3 a^{2} \arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+3 a^{2} \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a^{2} \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2*(arctanh(a*x)^2-2*arctanh(a*x)+2)*(-(a*x-1)*(a*x+1))^(1/2)/(a*x-1)+1/2*(arctanh(a*x)^2+2*arctanh(a*x)
+2)*a^2*(-(a*x-1)*(a*x+1))^(1/2)/(a*x+1)-1/2*(-(a*x-1)*(a*x+1))^(1/2)*arctanh(a*x)*(2*a*x+arctanh(a*x))/x^2-2*
a^2*arctanh((a*x+1)/(-a^2*x^2+1)^(1/2))-3/2*a^2*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-3*a^2*arctanh(
a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3/2*a^2*arctanh(a*x)^
2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3*a^2*polylog(3,(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 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x^3\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**2/x**3/(-a**2*x**2+1)**(3/2),x)

[Out]

Integral(atanh(a*x)**2/(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)), x)

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