Integrand size = 22, antiderivative size = 192 \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {3 x^3}{128 a \left (1-a^2 x^2\right )^2}+\frac {45 x}{256 a^3 \left (1-a^2 x^2\right )}+\frac {27 \text {arctanh}(a x)}{256 a^4}+\frac {3 x^4 \text {arctanh}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {9 \text {arctanh}(a x)}{32 a^4 \left (1-a^2 x^2\right )}-\frac {3 x^3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {9 x \text {arctanh}(a x)^2}{32 a^3 \left (1-a^2 x^2\right )}-\frac {3 \text {arctanh}(a x)^3}{32 a^4}+\frac {x^4 \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2} \]
-3/128*x^3/a/(-a^2*x^2+1)^2+45/256*x/a^3/(-a^2*x^2+1)+27/256*arctanh(a*x)/ a^4+3/32*x^4*arctanh(a*x)/(-a^2*x^2+1)^2-9/32*arctanh(a*x)/a^4/(-a^2*x^2+1 )-3/16*x^3*arctanh(a*x)^2/a/(-a^2*x^2+1)^2+9/32*x*arctanh(a*x)^2/a^3/(-a^2 *x^2+1)-3/32*arctanh(a*x)^3/a^4+1/4*x^4*arctanh(a*x)^3/(-a^2*x^2+1)^2
Time = 0.15 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.70 \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {48 \left (-4+5 a^2 x^2\right ) \text {arctanh}(a x)-48 a x \left (-3+5 a^2 x^2\right ) \text {arctanh}(a x)^2+16 \left (-3+6 a^2 x^2+5 a^4 x^4\right ) \text {arctanh}(a x)^3+3 \left (30 a x-34 a^3 x^3-17 \left (-1+a^2 x^2\right )^2 \log (1-a x)+17 \left (-1+a^2 x^2\right )^2 \log (1+a x)\right )}{512 a^4 \left (-1+a^2 x^2\right )^2} \]
(48*(-4 + 5*a^2*x^2)*ArcTanh[a*x] - 48*a*x*(-3 + 5*a^2*x^2)*ArcTanh[a*x]^2 + 16*(-3 + 6*a^2*x^2 + 5*a^4*x^4)*ArcTanh[a*x]^3 + 3*(30*a*x - 34*a^3*x^3 - 17*(-1 + a^2*x^2)^2*Log[1 - a*x] + 17*(-1 + a^2*x^2)^2*Log[1 + a*x]))/( 512*a^4*(-1 + a^2*x^2)^2)
Time = 0.85 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.38, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6570, 6566, 252, 252, 219, 6562, 6556, 215, 219}
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.
| \(\displaystyle \int \frac {x^3 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6570 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3}{4} a \int \frac {x^4 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 6566 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3}{4} a \left (-\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{4 a^2}+\frac {1}{8} \int \frac {x^4}{\left (1-a^2 x^2\right )^3}dx-\frac {x^4 \text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {x^3 \text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}\right )\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3}{4} a \left (-\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{4 a^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \int \frac {x^2}{\left (1-a^2 x^2\right )^2}dx}{4 a^2}\right )-\frac {x^4 \text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {x^3 \text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}\right )\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3}{4} a \left (-\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{4 a^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{1-a^2 x^2}dx}{2 a^2}\right )}{4 a^2}\right )-\frac {x^4 \text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {x^3 \text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3}{4} a \left (-\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{4 a^2}-\frac {x^4 \text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {x^3 \text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)}{2 a^3}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 6562 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3}{4} a \left (-\frac {3 \left (-\frac {\int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{6 a^3}+\frac {x \text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}\right )}{4 a^2}-\frac {x^4 \text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {x^3 \text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)}{2 a^3}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3}{4} a \left (-\frac {3 \left (-\frac {\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}}{a}-\frac {\text {arctanh}(a x)^3}{6 a^3}+\frac {x \text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}\right )}{4 a^2}-\frac {x^4 \text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {x^3 \text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)}{2 a^3}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3}{4} a \left (-\frac {3 \left (-\frac {\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}}{a}-\frac {\text {arctanh}(a x)^3}{6 a^3}+\frac {x \text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}\right )}{4 a^2}-\frac {x^4 \text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {x^3 \text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)}{2 a^3}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3}{4} a \left (-\frac {x^4 \text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {x^3 \text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (-\frac {\text {arctanh}(a x)^3}{6 a^3}+\frac {x \text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}}{a}\right )}{4 a^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)}{2 a^3}\right )}{4 a^2}\right )\right )\) |
(x^4*ArcTanh[a*x]^3)/(4*(1 - a^2*x^2)^2) - (3*a*(-1/8*(x^4*ArcTanh[a*x])/( a*(1 - a^2*x^2)^2) + (x^3*ArcTanh[a*x]^2)/(4*a^2*(1 - a^2*x^2)^2) + (x^3/( 4*a^2*(1 - a^2*x^2)^2) - (3*(x/(2*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]/(2*a^3 )))/(4*a^2))/8 - (3*((x*ArcTanh[a*x]^2)/(2*a^2*(1 - a^2*x^2)) - ArcTanh[a* x]^3/(6*a^3) - (ArcTanh[a*x]/(2*a^2*(1 - a^2*x^2)) - (x/(2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a))/(2*a))/a))/(4*a^2)))/4
3.4.14.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
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] + Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTanh[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]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^2)/((d_) + (e_.)*(x_)^2 )^2, x_Symbol] :> Simp[-(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(2*c^2*d*(d + e*x^2))), x] - Simp[b*( p/(2*c)) Int[x*((a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; F reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_), x_Symbol] :> Simp[(-b)*p*(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*m^2)), x] + (Simp[f*(f*x)^(m - 1)*(d + e*x^2)^ (q + 1)*((a + b*ArcTanh[c*x])^p/(c^2*d*m)), x] - Simp[f^2*((m - 1)/(c^2*d*m )) Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] + Simp[b^2*p*((p - 1)/m^2) Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^ (p - 2), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(d*(m + 1))), x] - Simp[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]
Time = 0.58 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.62
| method | result | size |
| parallelrisch | \(-\frac {-40 \operatorname {arctanh}\left (a x \right )^{3} a^{4} x^{4}-51 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )+120 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}-48 \operatorname {arctanh}\left (a x \right )^{3} a^{2} x^{2}+51 a^{3} x^{3}-18 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-72 \operatorname {arctanh}\left (a x \right )^{2} a x +24 \operatorname {arctanh}\left (a x \right )^{3}-45 a x +45 \,\operatorname {arctanh}\left (a x \right )}{256 \left (a^{2} x^{2}-1\right )^{2} a^{4}}\) | \(120\) |
| risch | \(\frac {\left (5 a^{4} x^{4}+6 a^{2} x^{2}-3\right ) \ln \left (a x +1\right )^{3}}{256 a^{4} \left (a^{2} x^{2}-1\right )^{2}}-\frac {3 \left (5 a^{4} x^{4} \ln \left (-a x +1\right )+10 a^{3} x^{3}+6 x^{2} \ln \left (-a x +1\right ) a^{2}-6 a x -3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{2}}{256 a^{4} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}+\frac {3 \left (5 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+20 a^{3} x^{3} \ln \left (-a x +1\right )+6 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+20 a^{2} x^{2}-12 a x \ln \left (-a x +1\right )-3 \ln \left (-a x +1\right )^{2}-16\right ) \ln \left (a x +1\right )}{256 a^{4} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}-\frac {10 a^{4} x^{4} \ln \left (-a x +1\right )^{3}+51 \ln \left (a x -1\right ) a^{4} x^{4}-51 \ln \left (-a x -1\right ) a^{4} x^{4}+60 a^{3} x^{3} \ln \left (-a x +1\right )^{2}+12 a^{2} x^{2} \ln \left (-a x +1\right )^{3}+102 a^{3} x^{3}-102 \ln \left (a x -1\right ) a^{2} x^{2}+102 \ln \left (-a x -1\right ) a^{2} x^{2}+120 x^{2} \ln \left (-a x +1\right ) a^{2}-36 a \ln \left (-a x +1\right )^{2} x -6 \ln \left (-a x +1\right )^{3}-90 a x +51 \ln \left (a x -1\right )-51 \ln \left (-a x -1\right )-96 \ln \left (-a x +1\right )}{512 a^{4} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\) | \(468\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(889\) |
| default | \(\text {Expression too large to display}\) | \(889\) |
| parts | \(\text {Expression too large to display}\) | \(921\) |
-1/256*(-40*arctanh(a*x)^3*a^4*x^4-51*a^4*x^4*arctanh(a*x)+120*arctanh(a*x )^2*a^3*x^3-48*arctanh(a*x)^3*a^2*x^2+51*a^3*x^3-18*a^2*x^2*arctanh(a*x)-7 2*arctanh(a*x)^2*a*x+24*arctanh(a*x)^3-45*a*x+45*arctanh(a*x))/(a^2*x^2-1) ^2/a^4
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {102 \, a^{3} x^{3} - 2 \, {\left (5 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, {\left (5 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 90 \, a x - 3 \, {\left (17 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 15\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{512 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \]
-1/512*(102*a^3*x^3 - 2*(5*a^4*x^4 + 6*a^2*x^2 - 3)*log(-(a*x + 1)/(a*x - 1))^3 + 12*(5*a^3*x^3 - 3*a*x)*log(-(a*x + 1)/(a*x - 1))^2 - 90*a*x - 3*(1 7*a^4*x^4 + 6*a^2*x^2 - 15)*log(-(a*x + 1)/(a*x - 1)))/(a^8*x^4 - 2*a^6*x^ 2 + a^4)
\[ \int \frac {x^3 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=- \int \frac {x^{3} \operatorname {atanh}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (167) = 334\).
Time = 0.20 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.28 \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {3}{64} \, a {\left (\frac {2 \, {\left (5 \, a^{2} x^{3} - 3 \, x\right )}}{a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}} - \frac {5 \, \log \left (a x + 1\right )}{a^{5}} + \frac {5 \, \log \left (a x - 1\right )}{a^{5}}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {{\left (2 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{3}}{4 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} - \frac {1}{512} \, {\left (\frac {{\left (102 \, a^{3} x^{3} - 10 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 30 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 10 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} - 90 \, a x - 3 \, {\left (17 \, a^{4} x^{4} - 34 \, a^{2} x^{2} + 10 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 17\right )} \log \left (a x + 1\right ) + 51 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{a^{11} x^{4} - 2 \, a^{9} x^{2} + a^{7}} - \frac {12 \, {\left (20 \, a^{2} x^{2} - 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 10 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a \operatorname {artanh}\left (a x\right )}{a^{10} x^{4} - 2 \, a^{8} x^{2} + a^{6}}\right )} a \]
-3/64*a*(2*(5*a^2*x^3 - 3*x)/(a^8*x^4 - 2*a^6*x^2 + a^4) - 5*log(a*x + 1)/ a^5 + 5*log(a*x - 1)/a^5)*arctanh(a*x)^2 + 1/4*(2*a^2*x^2 - 1)*arctanh(a*x )^3/(a^8*x^4 - 2*a^6*x^2 + a^4) - 1/512*((102*a^3*x^3 - 10*(a^4*x^4 - 2*a^ 2*x^2 + 1)*log(a*x + 1)^3 + 30*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2*lo g(a*x - 1) + 10*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^3 - 90*a*x - 3*(17* a^4*x^4 - 34*a^2*x^2 + 10*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 + 17)*l og(a*x + 1) + 51*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*a^2/(a^11*x^4 - 2 *a^9*x^2 + a^7) - 12*(20*a^2*x^2 - 5*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1 )^2 + 10*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x - 1) - 5*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 16)*a*arctanh(a*x)/(a^10*x^4 - 2*a^8*x^2 + a^6))*a
Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (167) = 334\).
Time = 0.29 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.78 \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {1}{2048} \, {\left (4 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {4 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} + \frac {4 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 6 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {8 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} - \frac {8 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 6 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {16 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} + \frac {16 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {3 \, {\left (a x - 1\right )}^{2} {\left (\frac {32 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} - \frac {3 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} - \frac {96 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} a \]
1/2048*(4*((a*x - 1)^2*(4*(a*x + 1)/(a*x - 1) + 1)/((a*x + 1)^2*a^5) + (a* x + 1)^2/((a*x - 1)^2*a^5) + 4*(a*x + 1)/((a*x - 1)*a^5))*log(-(a*x + 1)/( a*x - 1))^3 + 6*((a*x - 1)^2*(8*(a*x + 1)/(a*x - 1) + 1)/((a*x + 1)^2*a^5) - (a*x + 1)^2/((a*x - 1)^2*a^5) - 8*(a*x + 1)/((a*x - 1)*a^5))*log(-(a*x + 1)/(a*x - 1))^2 + 6*((a*x - 1)^2*(16*(a*x + 1)/(a*x - 1) + 1)/((a*x + 1) ^2*a^5) + (a*x + 1)^2/((a*x - 1)^2*a^5) + 16*(a*x + 1)/((a*x - 1)*a^5))*lo g(-(a*x + 1)/(a*x - 1)) + 3*(a*x - 1)^2*(32*(a*x + 1)/(a*x - 1) + 1)/((a*x + 1)^2*a^5) - 3*(a*x + 1)^2/((a*x - 1)^2*a^5) - 96*(a*x + 1)/((a*x - 1)*a ^5))*a
Time = 6.25 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.16 \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {48\,\ln \left (1-a\,x\right )-48\,\ln \left (a\,x+1\right )+51\,\mathrm {atanh}\left (a\,x\right )+45\,a\,x-3\,{\ln \left (a\,x+1\right )}^3+3\,{\ln \left (1-a\,x\right )}^3-9\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2+9\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )-51\,a^3\,x^3+6\,a^2\,x^2\,{\ln \left (a\,x+1\right )}^3-6\,a^2\,x^2\,{\ln \left (1-a\,x\right )}^3-30\,a^3\,x^3\,{\ln \left (a\,x+1\right )}^2-30\,a^3\,x^3\,{\ln \left (1-a\,x\right )}^2+5\,a^4\,x^4\,{\ln \left (a\,x+1\right )}^3-5\,a^4\,x^4\,{\ln \left (1-a\,x\right )}^3-102\,a^2\,x^2\,\mathrm {atanh}\left (a\,x\right )+51\,a^4\,x^4\,\mathrm {atanh}\left (a\,x\right )+18\,a\,x\,{\ln \left (a\,x+1\right )}^2+18\,a\,x\,{\ln \left (1-a\,x\right )}^2+60\,a^2\,x^2\,\ln \left (a\,x+1\right )-60\,a^2\,x^2\,\ln \left (1-a\,x\right )-36\,a\,x\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )+18\,a^2\,x^2\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2-18\,a^2\,x^2\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )+15\,a^4\,x^4\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2-15\,a^4\,x^4\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )+60\,a^3\,x^3\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{256\,a^4\,{\left (a^2\,x^2-1\right )}^2} \]
(48*log(1 - a*x) - 48*log(a*x + 1) + 51*atanh(a*x) + 45*a*x - 3*log(a*x + 1)^3 + 3*log(1 - a*x)^3 - 9*log(a*x + 1)*log(1 - a*x)^2 + 9*log(a*x + 1)^2 *log(1 - a*x) - 51*a^3*x^3 + 6*a^2*x^2*log(a*x + 1)^3 - 6*a^2*x^2*log(1 - a*x)^3 - 30*a^3*x^3*log(a*x + 1)^2 - 30*a^3*x^3*log(1 - a*x)^2 + 5*a^4*x^4 *log(a*x + 1)^3 - 5*a^4*x^4*log(1 - a*x)^3 - 102*a^2*x^2*atanh(a*x) + 51*a ^4*x^4*atanh(a*x) + 18*a*x*log(a*x + 1)^2 + 18*a*x*log(1 - a*x)^2 + 60*a^2 *x^2*log(a*x + 1) - 60*a^2*x^2*log(1 - a*x) - 36*a*x*log(a*x + 1)*log(1 - a*x) + 18*a^2*x^2*log(a*x + 1)*log(1 - a*x)^2 - 18*a^2*x^2*log(a*x + 1)^2* log(1 - a*x) + 15*a^4*x^4*log(a*x + 1)*log(1 - a*x)^2 - 15*a^4*x^4*log(a*x + 1)^2*log(1 - a*x) + 60*a^3*x^3*log(a*x + 1)*log(1 - a*x))/(256*a^4*(a^2 *x^2 - 1)^2)