Integrand size = 20, antiderivative size = 172 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^3} \, dx=-\frac {a \text {arctanh}(a x)}{x}+\frac {1}{2} a^2 \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)^2}{2 x^2}-2 a^2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )-a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )-\frac {1}{2} a^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )+\frac {1}{2} a^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-a x}\right ) \]
-a*arctanh(a*x)/x+1/2*a^2*arctanh(a*x)^2-1/2*arctanh(a*x)^2/x^2+2*a^2*arct anh(a*x)^2*arctanh(-1+2/(-a*x+1))+a^2*ln(x)-1/2*a^2*ln(-a^2*x^2+1)+a^2*arc tanh(a*x)*polylog(2,1-2/(-a*x+1))-a^2*arctanh(a*x)*polylog(2,-1+2/(-a*x+1) )-1/2*a^2*polylog(3,1-2/(-a*x+1))+1/2*a^2*polylog(3,-1+2/(-a*x+1))
Time = 0.25 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.11 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^3} \, dx=-\frac {a \text {arctanh}(a x)}{x}+\frac {\left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2}{2 x^2}+\frac {2}{3} a^2 \text {arctanh}(a x)^3+a^2 \text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )-a^2 \text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )-a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )-a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )-\frac {1}{2} a^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )+\frac {1}{2} a^2 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right ) \]
-((a*ArcTanh[a*x])/x) + ((-1 + a^2*x^2)*ArcTanh[a*x]^2)/(2*x^2) + (2*a^2*A rcTanh[a*x]^3)/3 + a^2*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] - a^2*A rcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] + a^2*Log[x] - (a^2*Log[1 - a^2* x^2])/2 - a^2*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] - a^2*ArcTanh[ a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] - (a^2*PolyLog[3, -E^(-2*ArcTanh[a*x]) ])/2 + (a^2*PolyLog[3, E^(2*ArcTanh[a*x])])/2
Time = 1.41 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {6576, 6448, 6452, 6544, 6452, 243, 47, 14, 16, 6510, 6614, 6620, 7164}
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.
| \(\displaystyle \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 6576 |
| \(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^3}dx-a^2 \int \frac {\text {arctanh}(a x)^2}{x}dx\) |
| \(\Big \downarrow \) 6448 |
| \(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^3}dx-a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x) \text {arctanh}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle -\left (a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x) \text {arctanh}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )\right )+a \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle -\left (a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x) \text {arctanh}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )\right )+a \left (a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+\int \frac {\text {arctanh}(a x)}{x^2}dx\right )-\frac {\text {arctanh}(a x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle -\left (a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x) \text {arctanh}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )\right )+a \left (a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+a \int \frac {1}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{x}\right )-\frac {\text {arctanh}(a x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\left (a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x) \text {arctanh}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )\right )+a \left (a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )}dx^2-\frac {\text {arctanh}(a x)}{x}\right )-\frac {\text {arctanh}(a x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle -\left (a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x) \text {arctanh}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )\right )+a \left (a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx^2+\int \frac {1}{x^2}dx^2\right )-\frac {\text {arctanh}(a x)}{x}\right )-\frac {\text {arctanh}(a x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle -\left (a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x) \text {arctanh}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )\right )+a \left (a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx^2+\log \left (x^2\right )\right )-\frac {\text {arctanh}(a x)}{x}\right )-\frac {\text {arctanh}(a x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\left (a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x) \text {arctanh}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )\right )+a \left (a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )-\frac {\text {arctanh}(a x)}{x}\right )-\frac {\text {arctanh}(a x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle -\left (a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x) \text {arctanh}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )+\frac {1}{2} a \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)}{x}\right )-\frac {\text {arctanh}(a x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 6614 |
| \(\displaystyle -\left (a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \left (\frac {1}{2} \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {1}{2} \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )+\frac {1}{2} a \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)}{x}\right )-\frac {\text {arctanh}(a x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle -\left (a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \left (\frac {1}{2} \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )+\frac {1}{2} \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )\right )\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )+\frac {1}{2} a \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)}{x}\right )-\frac {\text {arctanh}(a x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle -\left (a^2 \left (2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-4 a \left (\frac {1}{2} \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}\right )+\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{4 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )\right )\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )+\frac {1}{2} a \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)}{x}\right )-\frac {\text {arctanh}(a x)^2}{2 x^2}\) |
-1/2*ArcTanh[a*x]^2/x^2 + a*(-(ArcTanh[a*x]/x) + (a*ArcTanh[a*x]^2)/2 + (a *(Log[x^2] - Log[1 - a^2*x^2]))/2) - a^2*(2*ArcTanh[a*x]^2*ArcTanh[1 - 2/( 1 - a*x)] - 4*a*(((ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/(2*a) - PolyL og[3, 1 - 2/(1 - a*x)]/(4*a))/2 + (-1/2*(ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 - a*x)])/a + PolyLog[3, -1 + 2/(1 - a*x)]/(4*a))/2))
3.2.79.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1 - c*x)], x] - Simp[2*b*c*p Int[(a + b *ArcTanh[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 - c*x)]/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
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_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x ], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x ^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x ^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ [p, 1] && IntegerQ[q]))
Int[(ArcTanh[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*( x_)^2), x_Symbol] :> Simp[1/2 Int[Log[1 + u]*((a + b*ArcTanh[c*x])^p/(d + e*x^2)), x], x] - Simp[1/2 Int[Log[1 - u]*((a + b*ArcTanh[c*x])^p/(d + e *x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/( d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 32.29 (sec) , antiderivative size = 736, normalized size of antiderivative = 4.28
| method | result | size |
| derivativedivides | \(a^{2} \left (-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2} x^{2}}+\ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}}{2}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (i \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 {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}{2}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}}{2}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}}{2}+\ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )-\frac {\operatorname {arctanh}\left (a x \right ) \left (a x +\sqrt {-a^{2} x^{2}+1}+1\right )}{2 a x}-\frac {\left (a x -\sqrt {-a^{2} x^{2}+1}+1\right ) \operatorname {arctanh}\left (a x \right )}{2 a x}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}\right )\) | \(736\) |
| default | \(a^{2} \left (-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2} x^{2}}+\ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}}{2}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (i \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 {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}{2}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}}{2}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2} \operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}}{2}+\ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )-\frac {\operatorname {arctanh}\left (a x \right ) \left (a x +\sqrt {-a^{2} x^{2}+1}+1\right )}{2 a x}-\frac {\left (a x -\sqrt {-a^{2} x^{2}+1}+1\right ) \operatorname {arctanh}\left (a x \right )}{2 a x}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}\right )\) | \(736\) |
| parts | \(\text {Expression too large to display}\) | \(1134\) |
a^2*(-arctanh(a*x)^2*ln(a*x)-1/2*arctanh(a*x)^2/a^2/x^2+ln(1+(a*x+1)/(-a^2 *x^2+1)^(1/2))+1/2*arctanh(a*x)^2-1/2*I*Pi*arctanh(a*x)^2*csgn(I*(-(a*x+1) ^2/(a^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^3-1/2*I*Pi*arctanh(a*x)^2*cs gn(I*(-(a*x+1)^2/(a^2*x^2-1)-1))*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))*csgn(I *(-(a*x+1)^2/(a^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))+1/2*I*Pi*arctanh(a *x)^2*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1) /(-(a*x+1)^2/(a^2*x^2-1)+1))^2+1/2*I*Pi*arctanh(a*x)^2*csgn(I*(-(a*x+1)^2/ (a^2*x^2-1)-1))*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+ 1))^2+ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)-1/2*arctanh(a*x)*(a*x+(-a^2*x^2+1)^ (1/2)+1)/a/x-1/2*(a*x-(-a^2*x^2+1)^(1/2)+1)/a/x*arctanh(a*x)+arctanh(a*x)^ 2*ln((a*x+1)^2/(-a^2*x^2+1)-1)-arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1 /2))-2*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+2*polylog(3,(a*x +1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-2* arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*polylog(3,-(a*x+1)/( -a^2*x^2+1)^(1/2))+arctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))-1/2*pol ylog(3,-(a*x+1)^2/(-a^2*x^2+1)))
\[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^3} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x^{3}} \,d x } \]
\[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^3} \, dx=- \int \left (- \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x^{3}}\right )\, dx - \int \frac {a^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x}\, dx \]
\[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^3} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x^{3}} \,d x } \]
-1/8*log(-a*x + 1)^2/x^2 + 1/4*integrate(-((a^3*x^3 - a^2*x^2 - a*x + 1)*l og(a*x + 1)^2 - (a*x + 2*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1))*log(- a*x + 1))/(a*x^4 - x^3), x)
\[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^3} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^3} \, dx=-\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right )}{x^3} \,d x \]