Integrand size = 19, antiderivative size = 88 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\frac {x}{4 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{4 a}-\frac {\text {arctanh}(a x)}{2 a \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a} \]
1/4*x/(-a^2*x^2+1)+1/4*arctanh(a*x)/a-1/2*arctanh(a*x)/a/(-a^2*x^2+1)+1/2* x*arctanh(a*x)^2/(-a^2*x^2+1)+1/6*arctanh(a*x)^3/a
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.06 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\frac {12 \text {arctanh}(a x)-12 a x \text {arctanh}(a x)^2+4 \left (-1+a^2 x^2\right ) \text {arctanh}(a x)^3-3 \left (2 a x+\left (-1+a^2 x^2\right ) \log (1-a x)+\left (1-a^2 x^2\right ) \log (1+a x)\right )}{24 a \left (-1+a^2 x^2\right )} \]
(12*ArcTanh[a*x] - 12*a*x*ArcTanh[a*x]^2 + 4*(-1 + a^2*x^2)*ArcTanh[a*x]^3 - 3*(2*a*x + (-1 + a^2*x^2)*Log[1 - a*x] + (1 - a^2*x^2)*Log[1 + a*x]))/( 24*a*(-1 + a^2*x^2))
Time = 0.33 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6518, 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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle -a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle -a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle -a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\) |
(x*ArcTanh[a*x]^2)/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^3/(6*a) - a*(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))
3.3.69.3.1 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[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sy mbol] :> Simp[x*((a + b*ArcTanh[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2) Int[x*( (a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
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]
Time = 0.39 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.82
| method | result | size |
| parallelrisch | \(-\frac {-2 \operatorname {arctanh}\left (a x \right )^{3} a^{2} x^{2}-3 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+6 \operatorname {arctanh}\left (a x \right )^{2} a x +2 \operatorname {arctanh}\left (a x \right )^{3}+3 a x -3 \,\operatorname {arctanh}\left (a x \right )}{12 \left (a^{2} x^{2}-1\right ) a}\) | \(72\) |
| risch | \(\frac {\ln \left (a x +1\right )^{3}}{48 a}-\frac {\left (x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x -\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{2}}{16 \left (a^{2} x^{2}-1\right ) a}+\frac {\left (a^{2} x^{2} \ln \left (-a x +1\right )^{2}+4 a x \ln \left (-a x +1\right )-\ln \left (-a x +1\right )^{2}+4\right ) \ln \left (a x +1\right )}{16 a \left (a x -1\right ) \left (a x +1\right )}-\frac {a^{2} x^{2} \ln \left (-a x +1\right )^{3}+6 \ln \left (a x -1\right ) a^{2} x^{2}-6 \ln \left (-a x -1\right ) a^{2} x^{2}+6 a \ln \left (-a x +1\right )^{2} x -\ln \left (-a x +1\right )^{3}+12 a x -6 \ln \left (a x -1\right )+6 \ln \left (-a x -1\right )+12 \ln \left (-a x +1\right )}{48 a \left (a x -1\right ) \left (a x +1\right )}\) | \(251\) |
| derivativedivides | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 \left (a x -1\right )}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2}}{8}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}{8}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2}}{8}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}{8}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{4}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2}}{4}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3}}{4}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2}}{4}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}}{8}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{3}}{8}+\frac {\operatorname {arctanh}\left (a x \right )^{3}}{6}+\frac {\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{8 a x -8}-\frac {a x +1}{16 \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{8 a x +8}+\frac {a x -1}{16 a x +16}}{a}\) | \(727\) |
| default | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 \left (a x -1\right )}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2}}{8}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}{8}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2}}{8}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}{8}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{4}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2}}{4}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3}}{4}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2}}{4}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}}{8}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{3}}{8}+\frac {\operatorname {arctanh}\left (a x \right )^{3}}{6}+\frac {\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{8 a x -8}-\frac {a x +1}{16 \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{8 a x +8}+\frac {a x -1}{16 a x +16}}{a}\) | \(727\) |
| parts | \(-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 \left (a x +1\right ) a}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{4 a}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a \left (a x -1\right )}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{4 a}-\frac {a \left (\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{3}}{4 a^{2}}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2}}{2 a^{2}}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}{4 a^{2}}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}}{4 a^{2}}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}{4 a^{2}}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}{4 a^{2}}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}}{2 a^{2}}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3}}{2 a^{2}}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2}}{4 a^{2}}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2}}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2}}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{3 a^{2}}-\frac {\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right ) a^{2}}-\frac {\left (a x -1\right ) \operatorname {arctanh}\left (a x \right )}{4 \left (a x +1\right ) a^{2}}+\frac {a x +1}{8 \left (a x -1\right ) a^{2}}-\frac {a x -1}{8 \left (a x +1\right ) a^{2}}\right )}{2}\) | \(786\) |
-1/12*(-2*arctanh(a*x)^3*a^2*x^2-3*a^2*x^2*arctanh(a*x)+6*arctanh(a*x)^2*a *x+2*arctanh(a*x)^3+3*a*x-3*arctanh(a*x))/(a^2*x^2-1)/a
Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.08 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {6 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - {\left (a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, a x - 6 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{48 \, {\left (a^{3} x^{2} - a\right )}} \]
-1/48*(6*a*x*log(-(a*x + 1)/(a*x - 1))^2 - (a^2*x^2 - 1)*log(-(a*x + 1)/(a *x - 1))^3 + 12*a*x - 6*(a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1)))/(a^3*x^2 - a)
\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (75) = 150\).
Time = 0.19 (sec) , antiderivative size = 268, normalized size of antiderivative = 3.05 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, x}{a^{2} x^{2} - 1} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} - 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} - 12 \, a x + 3 \, {\left (2 \, a^{2} x^{2} + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2\right )} \log \left (a x + 1\right ) - 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{48 \, {\left (a^{5} x^{2} - a^{3}\right )}} - \frac {{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4\right )} a \operatorname {artanh}\left (a x\right )}{8 \, {\left (a^{4} x^{2} - a^{2}\right )}} \]
-1/4*(2*x/(a^2*x^2 - 1) - log(a*x + 1)/a + log(a*x - 1)/a)*arctanh(a*x)^2 + 1/48*((a^2*x^2 - 1)*log(a*x + 1)^3 - 3*(a^2*x^2 - 1)*log(a*x + 1)^2*log( a*x - 1) - (a^2*x^2 - 1)*log(a*x - 1)^3 - 12*a*x + 3*(2*a^2*x^2 + (a^2*x^2 - 1)*log(a*x - 1)^2 - 2)*log(a*x + 1) - 6*(a^2*x^2 - 1)*log(a*x - 1))*a^2 /(a^5*x^2 - a^3) - 1/8*((a^2*x^2 - 1)*log(a*x + 1)^2 - 2*(a^2*x^2 - 1)*log (a*x + 1)*log(a*x - 1) + (a^2*x^2 - 1)*log(a*x - 1)^2 - 4)*a*arctanh(a*x)/ (a^4*x^2 - a^2)
Time = 1.40 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\frac {1}{16} \, a^{2} {\left (\frac {{\left (a x - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{{\left (a x + 1\right )} a^{4}} + \frac {2 \, {\left (a x - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{{\left (a x + 1\right )} a^{4}} + \frac {2 \, {\left (a x - 1\right )}}{{\left (a x + 1\right )} a^{4}}\right )} \]
1/16*a^2*((a*x - 1)*log(-(a*x + 1)/(a*x - 1))^2/((a*x + 1)*a^4) + 2*(a*x - 1)*log(-(a*x + 1)/(a*x - 1))/((a*x + 1)*a^4) + 2*(a*x - 1)/((a*x + 1)*a^4 ))
Time = 4.09 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.42 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\frac {{\ln \left (a\,x+1\right )}^3}{48\,a}-\frac {\ln \left (a\,x+1\right )}{4\,\left (a-a^3\,x^2\right )}-\frac {{\ln \left (1-a\,x\right )}^3}{48\,a}-\frac {x}{4\,a^2\,x^2-4}+\frac {\ln \left (1-a\,x\right )}{4\,a-4\,a^3\,x^2}+\frac {\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{16\,a}-\frac {{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{16\,a}-\frac {x\,{\ln \left (a\,x+1\right )}^2}{8\,\left (a^2\,x^2-1\right )}-\frac {x\,{\ln \left (1-a\,x\right )}^2}{2\,\left (4\,a^2\,x^2-4\right )}+\frac {x\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4\,a^2\,x^2-4}-\frac {\mathrm {atan}\left (a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,a} \]
log(a*x + 1)^3/(48*a) - log(a*x + 1)/(4*(a - a^3*x^2)) - log(1 - a*x)^3/(4 8*a) - x/(4*a^2*x^2 - 4) - (atan(a*x*1i)*1i)/(4*a) + log(1 - a*x)/(4*a - 4 *a^3*x^2) + (log(a*x + 1)*log(1 - a*x)^2)/(16*a) - (log(a*x + 1)^2*log(1 - a*x))/(16*a) - (x*log(a*x + 1)^2)/(8*(a^2*x^2 - 1)) - (x*log(1 - a*x)^2)/ (2*(4*a^2*x^2 - 4)) + (x*log(a*x + 1)*log(1 - a*x))/(4*a^2*x^2 - 4)