Integrand size = 22, antiderivative size = 127 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {x^4}{32 \left (1-a^2 x^2\right )^2}-\frac {3}{32 a^4 \left (1-a^2 x^2\right )}-\frac {x^3 \text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arctanh}(a x)}{16 a^3 \left (1-a^2 x^2\right )}-\frac {3 \text {arctanh}(a x)^2}{32 a^4}+\frac {x^4 \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2} \]
1/32*x^4/(-a^2*x^2+1)^2-3/32/a^4/(-a^2*x^2+1)-1/8*x^3*arctanh(a*x)/a/(-a^2 *x^2+1)^2+3/16*x*arctanh(a*x)/a^3/(-a^2*x^2+1)-3/32*arctanh(a*x)^2/a^4+1/4 *x^4*arctanh(a*x)^2/(-a^2*x^2+1)^2
Time = 0.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.56 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {-4+5 a^2 x^2+\left (6 a x-10 a^3 x^3\right ) \text {arctanh}(a x)+\left (-3+6 a^2 x^2+5 a^4 x^4\right ) \text {arctanh}(a x)^2}{32 a^4 \left (-1+a^2 x^2\right )^2} \]
(-4 + 5*a^2*x^2 + (6*a*x - 10*a^3*x^3)*ArcTanh[a*x] + (-3 + 6*a^2*x^2 + 5* a^4*x^4)*ArcTanh[a*x]^2)/(32*a^4*(-1 + a^2*x^2)^2)
Time = 0.63 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6570, 6564, 6560, 6510}
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.
| \(\displaystyle \int \frac {x^3 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6570 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{2} a \int \frac {x^4 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 6564 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{2} a \left (-\frac {3 \int \frac {x^2 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{4 a^2}+\frac {x^3 \text {arctanh}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x^4}{16 a \left (1-a^2 x^2\right )^2}\right )\) |
| \(\Big \downarrow \) 6560 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{2} a \left (-\frac {3 \left (-\frac {\int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx}{2 a^2}+\frac {x \text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {1}{4 a^3 \left (1-a^2 x^2\right )}\right )}{4 a^2}+\frac {x^3 \text {arctanh}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x^4}{16 a \left (1-a^2 x^2\right )^2}\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {x^4 \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{2} a \left (\frac {x^3 \text {arctanh}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x^4}{16 a \left (1-a^2 x^2\right )^2}-\frac {3 \left (-\frac {\text {arctanh}(a x)^2}{4 a^3}+\frac {x \text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {1}{4 a^3 \left (1-a^2 x^2\right )}\right )}{4 a^2}\right )\) |
(x^4*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)^2) - (a*(-1/16*x^4/(a*(1 - a^2*x^2)^ 2) + (x^3*ArcTanh[a*x])/(4*a^2*(1 - a^2*x^2)^2) - (3*(-1/4*1/(a^3*(1 - a^2 *x^2)) + (x*ArcTanh[a*x])/(2*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^2/(4*a^3))) /(4*a^2)))/2
3.4.8.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^2*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c^3*d*(q + 1)^2)), x] + (-Si mp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(2*c^2*d*(q + 1))), x] + Sim p[1/(2*c^2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x ]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q , -5/2]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(q_), x_Symbol] :> Simp[(-b)*(f*x)^m*((d + e*x^2)^(q + 1)/(c*d*m^2)), x] + (Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(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]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c ^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -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 = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71
| method | result | size |
| parallelrisch | \(-\frac {-5 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}-4 a^{4} x^{4}+10 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-6 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}+3 a^{2} x^{2}-6 a x \,\operatorname {arctanh}\left (a x \right )+3 \operatorname {arctanh}\left (a x \right )^{2}}{32 \left (a^{2} x^{2}-1\right )^{2} a^{4}}\) | \(90\) |
| derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x -1\right )^{2}}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x +1\right )^{2}}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x +1\right )}-\frac {\operatorname {arctanh}\left (a x \right )}{32 \left (a x -1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x -1\right )}-\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{32}+\frac {\operatorname {arctanh}\left (a x \right )}{32 \left (a x +1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x +1\right )}+\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{32}+\frac {5 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{64}-\frac {5 \ln \left (a x -1\right )^{2}}{128}-\frac {5 \ln \left (a x +1\right )^{2}}{128}+\frac {5 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{64}+\frac {1}{128 \left (a x -1\right )^{2}}+\frac {9}{128 \left (a x -1\right )}+\frac {1}{128 \left (a x +1\right )^{2}}-\frac {9}{128 \left (a x +1\right )}}{a^{4}}\) | \(238\) |
| default | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x -1\right )^{2}}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x +1\right )^{2}}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x +1\right )}-\frac {\operatorname {arctanh}\left (a x \right )}{32 \left (a x -1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x -1\right )}-\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{32}+\frac {\operatorname {arctanh}\left (a x \right )}{32 \left (a x +1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x +1\right )}+\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{32}+\frac {5 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{64}-\frac {5 \ln \left (a x -1\right )^{2}}{128}-\frac {5 \ln \left (a x +1\right )^{2}}{128}+\frac {5 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{64}+\frac {1}{128 \left (a x -1\right )^{2}}+\frac {9}{128 \left (a x -1\right )}+\frac {1}{128 \left (a x +1\right )^{2}}-\frac {9}{128 \left (a x +1\right )}}{a^{4}}\) | \(238\) |
| risch | \(\frac {\left (5 a^{4} x^{4}+6 a^{2} x^{2}-3\right ) \ln \left (a x +1\right )^{2}}{128 a^{4} \left (a^{2} x^{2}-1\right )^{2}}-\frac {\left (5 a^{4} x^{4} \ln \left (-a x +1\right )+10 a^{3} x^{3}+6 x^{2} \ln \left (-a x +1\right ) a^{2}-6 a x -3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{64 a^{4} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}+\frac {5 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+20 a^{3} x^{3} \ln \left (-a x +1\right )+6 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+20 a^{2} x^{2}-12 a x \ln \left (-a x +1\right )-3 \ln \left (-a x +1\right )^{2}-16}{128 a^{4} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\) | \(243\) |
| parts | \(\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 a^{4} \left (a x +1\right )^{2}}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{16 a^{4} \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 a^{4} \left (a x -1\right )^{2}}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{16 a^{4} \left (a x -1\right )}-\frac {\operatorname {arctanh}\left (a x \right )}{32 a^{4} \left (a x -1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 a^{4} \left (a x -1\right )}-\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{32 a^{4}}+\frac {\operatorname {arctanh}\left (a x \right )}{32 a^{4} \left (a x +1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 a^{4} \left (a x +1\right )}+\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{32 a^{4}}-\frac {-\frac {5 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {5 \ln \left (a x -1\right )^{2}}{4}+\frac {5 \ln \left (a x +1\right )^{2}}{4}-\frac {5 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {1}{4 \left (a x -1\right )^{2}}-\frac {9}{4 \left (a x -1\right )}-\frac {1}{4 \left (a x +1\right )^{2}}+\frac {9}{4 \left (a x +1\right )}}{32 a^{4}}\) | \(270\) |
-1/32*(-5*a^4*x^4*arctanh(a*x)^2-4*a^4*x^4+10*a^3*x^3*arctanh(a*x)-6*a^2*x ^2*arctanh(a*x)^2+3*a^2*x^2-6*a*x*arctanh(a*x)+3*arctanh(a*x)^2)/(a^2*x^2- 1)^2/a^4
Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.78 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {20 \, a^{2} x^{2} + {\left (5 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4 \, {\left (5 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 16}{128 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \]
1/128*(20*a^2*x^2 + (5*a^4*x^4 + 6*a^2*x^2 - 3)*log(-(a*x + 1)/(a*x - 1))^ 2 - 4*(5*a^3*x^3 - 3*a*x)*log(-(a*x + 1)/(a*x - 1)) - 16)/(a^8*x^4 - 2*a^6 *x^2 + a^4)
\[ \int \frac {x^3 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=- \int \frac {x^{3} \operatorname {atanh}^{2}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (110) = 220\).
Time = 0.19 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.78 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {1}{32} \, a {\left (\frac {2 \, {\left (5 \, a^{2} x^{3} - 3 \, x\right )}}{a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}} - \frac {5 \, \log \left (a x + 1\right )}{a^{5}} + \frac {5 \, \log \left (a x - 1\right )}{a^{5}}\right )} \operatorname {artanh}\left (a x\right ) + \frac {{\left (20 \, a^{2} x^{2} - 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 10 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a^{2}}{128 \, {\left (a^{10} x^{4} - 2 \, a^{8} x^{2} + a^{6}\right )}} + \frac {{\left (2 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{4 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \]
-1/32*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) + 1/128*(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^2/(a^10* x^4 - 2*a^8*x^2 + a^6) + 1/4*(2*a^2*x^2 - 1)*arctanh(a*x)^2/(a^8*x^4 - 2*a ^6*x^2 + a^4)
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (110) = 220\).
Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.97 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {1}{512} \, {\left (2 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {4 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} + \frac {4 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 2 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {8 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} - \frac {8 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {{\left (a x - 1\right )}^{2} {\left (\frac {16 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} + \frac {16 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} a \]
1/512*(2*((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))^2 + 2*((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)) + (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))*a
Time = 4.49 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.93 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx={\ln \left (a\,x+1\right )}^2\,\left (\frac {5}{128\,a^4}-\frac {\frac {1}{16\,a^5}-\frac {x^2}{8\,a^3}}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4}\right )-\ln \left (1-a\,x\right )\,\left (\frac {\frac {3\,x}{8}+a\,x^2-\frac {3}{4\,a}-\frac {5\,a^2\,x^3}{8}}{8\,a^7\,x^4-16\,a^5\,x^2+8\,a^3}+\frac {\frac {3\,x}{8}-a\,x^2+\frac {3}{4\,a}-\frac {5\,a^2\,x^3}{8}}{8\,a^7\,x^4-16\,a^5\,x^2+8\,a^3}-\ln \left (a\,x+1\right )\,\left (\frac {\frac {1}{4\,a^4}-\frac {x^2}{2\,a^2}}{2\,a^4\,x^4-4\,a^2\,x^2+2}-\frac {5\,\left (a^4\,x^4-2\,a^2\,x^2+1\right )}{32\,a^4\,\left (2\,a^4\,x^4-4\,a^2\,x^2+2\right )}\right )\right )-{\ln \left (1-a\,x\right )}^2\,\left (\frac {\frac {1}{4\,a^4}-\frac {x^2}{2\,a^2}}{4\,a^4\,x^4-8\,a^2\,x^2+4}-\frac {5}{128\,a^4}\right )-\frac {\frac {2}{a^2}-\frac {5\,x^2}{2}}{16\,a^6\,x^4-32\,a^4\,x^2+16\,a^2}+\frac {\ln \left (a\,x+1\right )\,\left (\frac {3\,x}{32\,a^4}-\frac {5\,x^3}{32\,a^2}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4} \]
log(a*x + 1)^2*(5/(128*a^4) - (1/(16*a^5) - x^2/(8*a^3))/(1/a - 2*a*x^2 + a^3*x^4)) - log(1 - a*x)*(((3*x)/8 + a*x^2 - 3/(4*a) - (5*a^2*x^3)/8)/(8*a ^3 - 16*a^5*x^2 + 8*a^7*x^4) + ((3*x)/8 - a*x^2 + 3/(4*a) - (5*a^2*x^3)/8) /(8*a^3 - 16*a^5*x^2 + 8*a^7*x^4) - log(a*x + 1)*((1/(4*a^4) - x^2/(2*a^2) )/(2*a^4*x^4 - 4*a^2*x^2 + 2) - (5*(a^4*x^4 - 2*a^2*x^2 + 1))/(32*a^4*(2*a ^4*x^4 - 4*a^2*x^2 + 2)))) - log(1 - a*x)^2*((1/(4*a^4) - x^2/(2*a^2))/(4* a^4*x^4 - 8*a^2*x^2 + 4) - 5/(128*a^4)) - (2/a^2 - (5*x^2)/2)/(16*a^2 - 32 *a^4*x^2 + 16*a^6*x^4) + (log(a*x + 1)*((3*x)/(32*a^4) - (5*x^3)/(32*a^2)) )/(1/a - 2*a*x^2 + a^3*x^4)