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

Optimal. Leaf size=171 \[ \frac{2 i \tanh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{2 i \tanh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{2 i \text{PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac{2 i \text{PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac{2 x}{a^2 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{a^2 \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{a^3 \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a^3} \]

[Out]

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

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Rubi [A]  time = 0.251751, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6028, 5952, 4180, 2531, 2282, 6589, 5962, 191} \[ \frac{2 i \tanh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{2 i \tanh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{2 i \text{PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac{2 i \text{PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac{2 x}{a^2 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{a^2 \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{a^3 \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6028

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/e, Int
[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[d/e, Int[x^(m - 2)*(d + e*x^2)^q*(a + b*A
rcTanh[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] &&
IGtQ[m, 1] && NeQ[p, -1]

Rule 5952

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

Rule 4180

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

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 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 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]

Rule 5962

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.312353, size = 193, normalized size = 1.13 \[ \frac{i \left (2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-i e^{-\tanh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,i e^{-\tanh ^{-1}(a x)}\right )-\frac{2 i a x}{\sqrt{1-a^2 x^2}}-\frac{i a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{2 i \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+\tanh ^{-1}(a x)^2 \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )}{a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.242, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2} \operatorname{artanh}\left (a x\right )^{2}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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