Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\frac {1}{a x \text {arctanh}(a x)}-\frac {a x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {3}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \text {Chi}(4 \text {arctanh}(a x))-\frac {\text {Int}\left (\frac {1}{x^2 \text {arctanh}(a x)},x\right )}{a} \]
-1/a/x/arctanh(a*x)-a*x/(-a^2*x^2+1)^2/arctanh(a*x)-a*x/(-a^2*x^2+1)/arcta nh(a*x)+3/2*Chi(2*arctanh(a*x))+1/2*Chi(4*arctanh(a*x))-Unintegrable(1/x^2 /arctanh(a*x),x)/a
Not integrable
Time = 3.57 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx \]
Not integrable
Time = 2.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {6592, 6592, 6552, 6468, 6594, 6530, 3042, 3793, 2009, 6596, 3042, 25, 3793, 2009, 5971, 2009}
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.
| \(\displaystyle \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx+\int \frac {1}{x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx+a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx+\int \frac {1}{x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}dx\) |
| \(\Big \downarrow \) 6552 |
| \(\displaystyle a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx+a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 6468 |
| \(\displaystyle a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx+a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 6594 |
| \(\displaystyle a^2 \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx}{a}+3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx}{a}+a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 6530 |
| \(\displaystyle a^2 \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )+a^2 \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^4}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle a^2 \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {1}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )+a^2 \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}+\frac {3}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 6596 |
| \(\displaystyle a^2 \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (\frac {\int \frac {a^2 x^2}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (\frac {\int -\frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a^2 \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (-\frac {\int \frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle a^2 \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (-\frac {\int \left (\frac {1}{2 \text {arctanh}(a x)}-\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}+a^2 \left (\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle a^2 \left (\frac {3 \int \left (\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}-\frac {1}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}+a^2 \left (\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{a x \text {arctanh}(a x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\int \frac {1}{x^2 \text {arctanh}(a x)}dx}{a}+a^2 \left (\frac {3 \left (\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))-\frac {1}{8} \log (\text {arctanh}(a x))\right )}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )+a^2 \left (\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{a x \text {arctanh}(a x)}\) |
3.4.35.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_S ymbol] :> Unintegrable[(d*x)^m*(a + b*ArcTanh[c*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x _Symbol] :> Simp[d^q/c Subst[Int[(a + b*x)^p/Cosh[x]^(2*(q + 1)), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && I LtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcTanh[c*x])^(p + 1)/(b*c*d *(p + 1))), x] - Simp[f*(m/(b*c*d*(p + 1))) Int[(f*x)^(m - 1)*(a + b*ArcT anh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[1/d Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh [c*x])^p, x], x] - Simp[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] && Integers Q[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(q_), x_Symbol] :> Simp[x^m*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^( p + 1)/(b*c*d*(p + 1))), x] + (Simp[c*((m + 2*q + 2)/(b*(p + 1))) Int[x^( m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x] - Simp[m/(b*c*(p + 1)) Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x]) / ; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, - 1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1) Subst[Int[(a + b*x)^p*(Sinh[x]^ m/Cosh[x]^(m + 2*(q + 1))), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (In tegerQ[q] || GtQ[d, 0])
Not integrable
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}d x\]
Not integrable
Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )^{2}} \,d x } \]
Not integrable
Time = 2.48 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=- \int \frac {1}{a^{6} x^{7} \operatorname {atanh}^{2}{\left (a x \right )} - 3 a^{4} x^{5} \operatorname {atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atanh}^{2}{\left (a x \right )} - x \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]
-Integral(1/(a**6*x**7*atanh(a*x)**2 - 3*a**4*x**5*atanh(a*x)**2 + 3*a**2* x**3*atanh(a*x)**2 - x*atanh(a*x)**2), x)
Not integrable
Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 6.95 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )^{2}} \,d x } \]
-2/((a^5*x^5 - 2*a^3*x^3 + a*x)*log(a*x + 1) - (a^5*x^5 - 2*a^3*x^3 + a*x) *log(-a*x + 1)) + integrate(-2*(5*a^2*x^2 - 1)/((a^7*x^8 - 3*a^5*x^6 + 3*a ^3*x^4 - a*x^2)*log(a*x + 1) - (a^7*x^8 - 3*a^5*x^6 + 3*a^3*x^4 - a*x^2)*l og(-a*x + 1)), x)
Not integrable
Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )^{2}} \,d x } \]
Not integrable
Time = 3.70 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^3} \,d x \]