Integrand size = 22, antiderivative size = 94 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\frac {x}{4 a^2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{4 a^3}-\frac {\text {arctanh}(a x)}{2 a^3 \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)^3}{6 a^3} \] Output:
1/4*x/a^2/(-a^2*x^2+1)+1/4*arctanh(a*x)/a^3-1/2*arctanh(a*x)/a^3/(-a^2*x^2 +1)+1/2*x*arctanh(a*x)^2/a^2/(-a^2*x^2+1)-1/6*arctanh(a*x)^3/a^3
Time = 0.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\frac {12 \text {arctanh}(a x)-12 a x \text {arctanh}(a x)^2+\left (4-4 a^2 x^2\right ) \text {arctanh}(a x)^3-3 \left (2 a x+\left (-1+a^2 x^2\right ) \log (1-a x)+\left (1-a^2 x^2\right ) \log (1+a x)\right )}{24 a^3 \left (-1+a^2 x^2\right )} \] Input:
Integrate[(x^2*ArcTanh[a*x]^2)/(1 - a^2*x^2)^2,x]
Output:
(12*ArcTanh[a*x] - 12*a*x*ArcTanh[a*x]^2 + (4 - 4*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]))/(2 4*a^3*(-1 + a^2*x^2))
Time = 0.39 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {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.
| \(\displaystyle \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6562 |
| \(\displaystyle -\frac {\int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{6 a^3}+\frac {x \text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle -\frac {\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}}{a}-\frac {\text {arctanh}(a x)^3}{6 a^3}+\frac {x \text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle -\frac {\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}}{a}-\frac {\text {arctanh}(a x)^3}{6 a^3}+\frac {x \text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\text {arctanh}(a x)^3}{6 a^3}+\frac {x \text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}}{a}\) |
Input:
Int[(x^2*ArcTanh[a*x]^2)/(1 - a^2*x^2)^2,x]
Output:
(x*ArcTanh[a*x]^2)/(2*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^3/(6*a^3) - (ArcTa nh[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
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_.)*(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]
Time = 22.65 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.77
| method | result | size |
| parallelrisch | \(-\frac {2 \operatorname {arctanh}\left (a x \right )^{3} a^{2} x^{2}-3 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+6 \operatorname {arctanh}\left (a x \right )^{2} a x -2 \operatorname {arctanh}\left (a x \right )^{3}+3 a x -3 \,\operatorname {arctanh}\left (a x \right )}{12 \left (a^{2} x^{2}-1\right ) a^{3}}\) | \(72\) |
| risch | \(-\frac {\ln \left (a x +1\right )^{3}}{48 a^{3}}+\frac {\left (x^{2} \ln \left (-a x +1\right ) a^{2}-2 a x -\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{2}}{16 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\left (a^{2} x^{2} \ln \left (-a x +1\right )^{2}-4 a x \ln \left (-a x +1\right )-\ln \left (-a x +1\right )^{2}-4\right ) \ln \left (a x +1\right )}{16 a^{3} \left (a x -1\right ) \left (a x +1\right )}+\frac {a^{2} x^{2} \ln \left (-a x +1\right )^{3}+6 \ln \left (-a x -1\right ) a^{2} x^{2}-6 \ln \left (a x -1\right ) a^{2} x^{2}-6 a \ln \left (-a x +1\right )^{2} x -\ln \left (-a x +1\right )^{3}-12 a x -6 \ln \left (-a x -1\right )+6 \ln \left (a x -1\right )-12 \ln \left (-a x +1\right )}{48 a^{3} \left (a x -1\right ) \left (a x +1\right )}\) | \(251\) |
| 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 {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{2}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {i \pi \,\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} \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{8}-\frac {i \pi \,\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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {i \pi {\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 ) \operatorname {arctanh}\left (a x \right )^{2}}{8}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{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^{3}}\) | \(727\) |
| 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 {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{2}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {i \pi \,\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} \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{8}-\frac {i \pi \,\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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {i \pi {\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 ) \operatorname {arctanh}\left (a x \right )^{2}}{8}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{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^{3}}\) | \(727\) |
| parts | \(-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a^{3} \left (a x +1\right )}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{4 a^{3}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a^{3} \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{4 a^{3}}-\frac {a \left (-\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2 a^{4}}+\frac {i \pi \,\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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4}}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4}}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4}}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4}}+\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{2}}{2 a^{4}}-\frac {i \pi \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 ) \operatorname {arctanh}\left (a x \right )^{2}}{2 a^{4}}+\frac {i \pi \,\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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4}}-\frac {i \pi \,\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} \operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4}}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2}}{2 a^{4}}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4}}+\frac {\operatorname {arctanh}\left (a x \right )^{3}}{3 a^{4}}-\frac {\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right ) a^{4}}-\frac {\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{4 \left (a x +1\right ) a^{4}}+\frac {a x +1}{8 \left (a x -1\right ) a^{4}}-\frac {a x -1}{8 \left (a x +1\right ) a^{4}}\right )}{2}\) | \(787\) |
Input:
int(x^2*arctanh(a*x)^2/(-a^2*x^2+1)^2,x,method=_RETURNVERBOSE)
Output:
-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^3
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {6 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + {\left (a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, a x - 6 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{48 \, {\left (a^{5} x^{2} - a^{3}\right )}} \] Input:
integrate(x^2*arctanh(a*x)^2/(-a^2*x^2+1)^2,x, algorithm="fricas")
Output:
-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^5*x^2 - a^3)
\[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\int \frac {x^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \] Input:
integrate(x**2*atanh(a*x)**2/(-a**2*x**2+1)**2,x)
Output:
Integral(x**2*atanh(a*x)**2/((a*x - 1)**2*(a*x + 1)**2), x)
Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (81) = 162\).
Time = 0.04 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.90 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, x}{a^{4} x^{2} - a^{2}} + \frac {\log \left (a x + 1\right )}{a^{3}} - \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {artanh}\left (a x\right )^{2} - \frac {{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} - 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 12 \, a x - 3 \, {\left (2 \, a^{2} x^{2} - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2\right )} \log \left (a x + 1\right ) + 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{48 \, {\left (a^{7} x^{2} - a^{5}\right )}} + \frac {{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 4\right )} a \operatorname {artanh}\left (a x\right )}{8 \, {\left (a^{6} x^{2} - a^{4}\right )}} \] Input:
integrate(x^2*arctanh(a*x)^2/(-a^2*x^2+1)^2,x, algorithm="maxima")
Output:
-1/4*(2*x/(a^4*x^2 - a^2) + log(a*x + 1)/a^3 - log(a*x - 1)/a^3)*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^7*x^2 - a^5) + 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^6*x^2 - a^4)
\[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\int { \frac {x^{2} \operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate(x^2*arctanh(a*x)^2/(-a^2*x^2+1)^2,x, algorithm="giac")
Output:
integrate(x^2*arctanh(a*x)^2/(a^2*x^2 - 1)^2, x)
Time = 4.41 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.46 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\frac {\ln \left (1-a\,x\right )}{4\,a^3-4\,a^5\,x^2}-\frac {{\ln \left (a\,x+1\right )}^3}{48\,a^3}+\frac {{\ln \left (1-a\,x\right )}^3}{48\,a^3}+\frac {x}{4\,a^2-4\,a^4\,x^2}-\frac {\ln \left (a\,x+1\right )}{4\,\left (a^3-a^5\,x^2\right )}+\frac {x\,{\ln \left (1-a\,x\right )}^2}{8\,a^2-8\,a^4\,x^2}-\frac {\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{16\,a^3}+\frac {{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{16\,a^3}+\frac {x\,{\ln \left (a\,x+1\right )}^2}{8\,\left (a^2-a^4\,x^2\right )}-\frac {x\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4\,a^2-4\,a^4\,x^2}-\frac {\mathrm {atan}\left (a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,a^3} \] Input:
int((x^2*atanh(a*x)^2)/(a^2*x^2 - 1)^2,x)
Output:
log(1 - a*x)/(4*a^3 - 4*a^5*x^2) - log(a*x + 1)^3/(48*a^3) + log(1 - a*x)^ 3/(48*a^3) + x/(4*a^2 - 4*a^4*x^2) - (atan(a*x*1i)*1i)/(4*a^3) - log(a*x + 1)/(4*(a^3 - a^5*x^2)) + (x*log(1 - a*x)^2)/(8*a^2 - 8*a^4*x^2) - (log(a* x + 1)*log(1 - a*x)^2)/(16*a^3) + (log(a*x + 1)^2*log(1 - a*x))/(16*a^3) + (x*log(a*x + 1)^2)/(8*(a^2 - a^4*x^2)) - (x*log(a*x + 1)*log(1 - a*x))/(4 *a^2 - 4*a^4*x^2)
Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.29 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\frac {-4 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}+4 \mathit {atanh} \left (a x \right )^{3}-12 \mathit {atanh} \left (a x \right )^{2} a x +12 \mathit {atanh} \left (a x \right ) a^{2} x^{2}+3 \,\mathrm {log}\left (a^{2} x -a \right ) a^{2} x^{2}-3 \,\mathrm {log}\left (a^{2} x -a \right )-3 \,\mathrm {log}\left (a^{2} x +a \right ) a^{2} x^{2}+3 \,\mathrm {log}\left (a^{2} x +a \right )-6 a x}{24 a^{3} \left (a^{2} x^{2}-1\right )} \] Input:
int(x^2*atanh(a*x)^2/(-a^2*x^2+1)^2,x)
Output:
( - 4*atanh(a*x)**3*a**2*x**2 + 4*atanh(a*x)**3 - 12*atanh(a*x)**2*a*x + 1 2*atanh(a*x)*a**2*x**2 + 3*log(a**2*x - a)*a**2*x**2 - 3*log(a**2*x - a) - 3*log(a**2*x + a)*a**2*x**2 + 3*log(a**2*x + a) - 6*a*x)/(24*a**3*(a**2*x **2 - 1))