Integrand size = 22, antiderivative size = 66 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx=a \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)^2}{x}+\frac {1}{3} a \text {arctanh}(a x)^3+2 a \text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
a*arctanh(a*x)^2-arctanh(a*x)^2/x+1/3*a*arctanh(a*x)^3+2*a*arctanh(a*x)*ln (2-2/(a*x+1))-a*polylog(2,-1+2/(a*x+1))
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx=-a \left (-\frac {1}{3} \text {arctanh}(a x) \left (-\frac {3 \text {arctanh}(a x)}{a x}+\text {arctanh}(a x) (3+\text {arctanh}(a x))+6 \log \left (1-e^{-2 \text {arctanh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )\right ) \]
-(a*(-1/3*(ArcTanh[a*x]*((-3*ArcTanh[a*x])/(a*x) + ArcTanh[a*x]*(3 + ArcTa nh[a*x]) + 6*Log[1 - E^(-2*ArcTanh[a*x])])) + PolyLog[2, E^(-2*ArcTanh[a*x ])]))
Time = 0.66 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6544, 6452, 6510, 6550, 6494, 2897}
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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^2}{1-a^2 x^2}dx+\int \frac {\text {arctanh}(a x)^2}{x^2}dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^2}{1-a^2 x^2}dx+2 a \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle 2 a \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle 2 a \left (\int \frac {\text {arctanh}(a x)}{x (a x+1)}dx+\frac {1}{2} \text {arctanh}(a x)^2\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle 2 a \left (-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle 2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\) |
-(ArcTanh[a*x]^2/x) + (a*ArcTanh[a*x]^3)/3 + 2*a*(ArcTanh[a*x]^2/2 + ArcTa nh[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog[2, -1 + 2/(1 + a*x)]/2)
3.3.39.3.1 Defintions of rubi rules used
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
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_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] /(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c ^2*d^2 - e^2, 0]
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_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.54 (sec) , antiderivative size = 4380, normalized size of antiderivative = 66.36
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(4380\) |
| default | \(\text {Expression too large to display}\) | \(4380\) |
| parts | \(\text {Expression too large to display}\) | \(4383\) |
a*(-1/2*I*Pi*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^2/a/x-1/4*I*Pi *csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I/(1-(a*x+1) ^2/(a^2*x^2-1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*dilog((a*x+1)/(-a^2*x^2+1)^ (1/2))-1/4*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))*cs gn(I/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arctanh(a*x) *ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1 -(a*x+1)^2/(a^2*x^2-1)))^3*arctanh(a*x)^2-1/4*I*Pi*csgn(I*(a*x+1)^2/(a^2*x ^2-1))^3*arctanh(a*x)^2+1/2*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^3*arcta nh(a*x)^2-1/2*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^2*arctanh(a*x)^2+1/2* I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^2*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x ^2+1)^(1/2))-1/4*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1 )))^2*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))*dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2) )+polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+polylog(2,(a*x+1)/(-a^2*x^2+1)^(1 /2))+1/3*arctanh(a*x)^3-arctanh(a*x)^2+1/4*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2- 1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4* I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3*polylog(2,- (a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+ 1)^2/(a^2*x^2-1)))^3*dilog((a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*Pi*csgn(I*(a* x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3*dilog(1+(a*x+1)/(-a^2*x^2+ 1)^(1/2))-1/2*I*Pi*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I*...
\[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{2}} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx=- \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{a^{2} x^{4} - x^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (63) = 126\).
Time = 0.19 (sec) , antiderivative size = 237, normalized size of antiderivative = 3.59 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx=-\frac {1}{24} \, a^{2} {\left (\frac {3 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} - 3 \, {\left (\log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 6 \, \log \left (a x - 1\right )^{2}}{a} - \frac {24 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} + \frac {24 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} - \frac {24 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} + \frac {1}{4} \, {\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right ) + 8 \, \log \left (x\right )\right )} a \operatorname {artanh}\left (a x\right ) + \frac {1}{2} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} \operatorname {artanh}\left (a x\right )^{2} \]
-1/24*a^2*((3*(log(a*x - 1) - 2)*log(a*x + 1)^2 - log(a*x + 1)^3 + log(a*x - 1)^3 - 3*(log(a*x - 1)^2 - 4*log(a*x - 1))*log(a*x + 1) + 6*log(a*x - 1 )^2)/a - 24*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a + 24*(log(a*x + 1)*log(x) + dilog(-a*x))/a - 24*(log(-a*x + 1)*log(x) + dilo g(a*x))/a) + 1/4*(2*(log(a*x - 1) - 2)*log(a*x + 1) - log(a*x + 1)^2 - log (a*x - 1)^2 - 4*log(a*x - 1) + 8*log(x))*a*arctanh(a*x) + 1/2*(a*log(a*x + 1) - a*log(a*x - 1) - 2/x)*arctanh(a*x)^2
\[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx=-\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x^2\,\left (a^2\,x^2-1\right )} \,d x \]