Integrand size = 24, antiderivative size = 246 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {6}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 x \text {arctanh}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}+\frac {x \text {arctanh}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \arctan \left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3}{a^3}+\frac {3 i \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )}{a^3}-\frac {3 i \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )}{a^3}-\frac {6 i \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )}{a^3}+\frac {6 i \text {arctanh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )}{a^3}+\frac {6 i \operatorname {PolyLog}\left (4,-i e^{\text {arctanh}(a x)}\right )}{a^3}-\frac {6 i \operatorname {PolyLog}\left (4,i e^{\text {arctanh}(a x)}\right )}{a^3} \]
-2*arctan((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^3/a^3+3*I*arctanh(a*x)^ 2*polylog(2,-I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^3-3*I*arctanh(a*x)^2*polylog( 2,I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^3-6*I*arctanh(a*x)*polylog(3,-I*(a*x+1)/ (-a^2*x^2+1)^(1/2))/a^3+6*I*arctanh(a*x)*polylog(3,I*(a*x+1)/(-a^2*x^2+1)^ (1/2))/a^3+6*I*polylog(4,-I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^3-6*I*polylog(4, I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^3-6/a^3/(-a^2*x^2+1)^(1/2)+6*x*arctanh(a*x )/a^2/(-a^2*x^2+1)^(1/2)-3*arctanh(a*x)^2/a^3/(-a^2*x^2+1)^(1/2)+x*arctanh (a*x)^3/a^2/(-a^2*x^2+1)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(541\) vs. \(2(246)=492\).
Time = 0.68 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.20 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {7 i \pi ^4-\frac {384}{\sqrt {1-a^2 x^2}}-8 \pi ^3 \text {arctanh}(a x)+\frac {384 a x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}+24 i \pi ^2 \text {arctanh}(a x)^2-\frac {192 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}+32 \pi \text {arctanh}(a x)^3+\frac {64 a x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-16 i \text {arctanh}(a x)^4-8 \pi ^3 \log \left (1+i e^{-\text {arctanh}(a x)}\right )+48 i \pi ^2 \text {arctanh}(a x) \log \left (1+i e^{-\text {arctanh}(a x)}\right )+96 \pi \text {arctanh}(a x)^2 \log \left (1+i e^{-\text {arctanh}(a x)}\right )-64 i \text {arctanh}(a x)^3 \log \left (1+i e^{-\text {arctanh}(a x)}\right )-48 i \pi ^2 \text {arctanh}(a x) \log \left (1-i e^{\text {arctanh}(a x)}\right )-96 \pi \text {arctanh}(a x)^2 \log \left (1-i e^{\text {arctanh}(a x)}\right )+8 \pi ^3 \log \left (1+i e^{\text {arctanh}(a x)}\right )+64 i \text {arctanh}(a x)^3 \log \left (1+i e^{\text {arctanh}(a x)}\right )-8 \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 i \text {arctanh}(a x))\right )\right )-48 i (\pi -2 i \text {arctanh}(a x))^2 \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )+192 i \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )-48 i \pi ^2 \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )-192 \pi \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )-192 \pi \operatorname {PolyLog}\left (3,-i e^{-\text {arctanh}(a x)}\right )+384 i \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-i e^{-\text {arctanh}(a x)}\right )-384 i \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )+192 \pi \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )+384 i \operatorname {PolyLog}\left (4,-i e^{-\text {arctanh}(a x)}\right )+384 i \operatorname {PolyLog}\left (4,-i e^{\text {arctanh}(a x)}\right )}{64 a^3} \]
((7*I)*Pi^4 - 384/Sqrt[1 - a^2*x^2] - 8*Pi^3*ArcTanh[a*x] + (384*a*x*ArcTa nh[a*x])/Sqrt[1 - a^2*x^2] + (24*I)*Pi^2*ArcTanh[a*x]^2 - (192*ArcTanh[a*x ]^2)/Sqrt[1 - a^2*x^2] + 32*Pi*ArcTanh[a*x]^3 + (64*a*x*ArcTanh[a*x]^3)/Sq rt[1 - a^2*x^2] - (16*I)*ArcTanh[a*x]^4 - 8*Pi^3*Log[1 + I/E^ArcTanh[a*x]] + (48*I)*Pi^2*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] + 96*Pi*ArcTanh[a*x] ^2*Log[1 + I/E^ArcTanh[a*x]] - (64*I)*ArcTanh[a*x]^3*Log[1 + I/E^ArcTanh[a *x]] - (48*I)*Pi^2*ArcTanh[a*x]*Log[1 - I*E^ArcTanh[a*x]] - 96*Pi*ArcTanh[ a*x]^2*Log[1 - I*E^ArcTanh[a*x]] + 8*Pi^3*Log[1 + I*E^ArcTanh[a*x]] + (64* I)*ArcTanh[a*x]^3*Log[1 + I*E^ArcTanh[a*x]] - 8*Pi^3*Log[Tan[(Pi + (2*I)*A rcTanh[a*x])/4]] - (48*I)*(Pi - (2*I)*ArcTanh[a*x])^2*PolyLog[2, (-I)/E^Ar cTanh[a*x]] + (192*I)*ArcTanh[a*x]^2*PolyLog[2, (-I)*E^ArcTanh[a*x]] - (48 *I)*Pi^2*PolyLog[2, I*E^ArcTanh[a*x]] - 192*Pi*ArcTanh[a*x]*PolyLog[2, I*E ^ArcTanh[a*x]] - 192*Pi*PolyLog[3, (-I)/E^ArcTanh[a*x]] + (384*I)*ArcTanh[ a*x]*PolyLog[3, (-I)/E^ArcTanh[a*x]] - (384*I)*ArcTanh[a*x]*PolyLog[3, (-I )*E^ArcTanh[a*x]] + 192*Pi*PolyLog[3, I*E^ArcTanh[a*x]] + (384*I)*PolyLog[ 4, (-I)/E^ArcTanh[a*x]] + (384*I)*PolyLog[4, (-I)*E^ArcTanh[a*x]])/(64*a^3 )
Time = 1.34 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6590, 6514, 3042, 4668, 3011, 6524, 6520, 7163, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6590 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx}{a^2}-\frac {\int \frac {\text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}dx}{a^2}\) |
| \(\Big \downarrow \) 6514 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx}{a^2}-\frac {\int \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3d\text {arctanh}(a x)}{a^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx}{a^2}-\frac {\int \text {arctanh}(a x)^3 \csc \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )d\text {arctanh}(a x)}{a^3}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx}{a^2}-\frac {-3 i \int \text {arctanh}(a x)^2 \log \left (1-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+3 i \int \text {arctanh}(a x)^2 \log \left (1+i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 \text {arctanh}(a x)^3 \arctan \left (e^{\text {arctanh}(a x)}\right )}{a^3}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx}{a^2}-\frac {3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^3 \arctan \left (e^{\text {arctanh}(a x)}\right )}{a^3}\) |
| \(\Big \downarrow \) 6524 |
| \(\displaystyle \frac {6 \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}}{a^2}-\frac {3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^3 \arctan \left (e^{\text {arctanh}(a x)}\right )}{a^3}\) |
| \(\Big \downarrow \) 6520 |
| \(\displaystyle \frac {\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )}{a^2}-\frac {3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^3 \arctan \left (e^{\text {arctanh}(a x)}\right )}{a^3}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )}{a^2}-\frac {3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^3 \arctan \left (e^{\text {arctanh}(a x)}\right )}{a^3}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )}{a^2}-\frac {3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )-\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^3 \arctan \left (e^{\text {arctanh}(a x)}\right )}{a^3}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )}{a^2}-\frac {2 \text {arctanh}(a x)^3 \arctan \left (e^{\text {arctanh}(a x)}\right )+3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (4,-i e^{\text {arctanh}(a x)}\right )\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (4,i e^{\text {arctanh}(a x)}\right )\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )}{a^3}\) |
((-3*ArcTanh[a*x]^2)/(a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/Sqrt[1 - a ^2*x^2] + 6*(-(1/(a*Sqrt[1 - a^2*x^2])) + (x*ArcTanh[a*x])/Sqrt[1 - a^2*x^ 2]))/a^2 - (2*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^3 + (3*I)*(-(ArcTanh[a*x ]^2*PolyLog[2, (-I)*E^ArcTanh[a*x]]) + 2*(ArcTanh[a*x]*PolyLog[3, (-I)*E^A rcTanh[a*x]] - PolyLog[4, (-I)*E^ArcTanh[a*x]])) - (3*I)*(-(ArcTanh[a*x]^2 *PolyLog[2, I*E^ArcTanh[a*x]]) + 2*(ArcTanh[a*x]*PolyLog[3, I*E^ArcTanh[a* x]] - PolyLog[4, I*E^ArcTanh[a*x]])))/a^3
3.5.5.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
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] + (-Simp[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] + Simp[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]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[1/(c*Sqrt[d]) Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcTa nh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0 ] && GtQ[d, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symb ol] :> 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]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[(-b)*p*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) ), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x] + Simp[b^2 *p*(p - 1) Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[1/e Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*A rcTanh[c*x])^p, x], x] - Simp[d/e Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcT anh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && In tegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
\[\int \frac {x^{2} \operatorname {arctanh}\left (a x \right )^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}d x\]
\[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \operatorname {atanh}^{3}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]