Integrand size = 20, antiderivative size = 91 \[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx=-\frac {a x}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \text {arctanh}(a x)+\frac {\text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
-1/4*a*x/(-a^2*x^2+1)-1/4*arctanh(a*x)+1/2*arctanh(a*x)/(-a^2*x^2+1)+1/2*a rctanh(a*x)^2+arctanh(a*x)*ln(2-2/(a*x+1))-1/2*polylog(2,-1+2/(a*x+1))
Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.69 \[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx=\frac {1}{8} \left (4 \text {arctanh}(a x)^2+2 \text {arctanh}(a x) \left (\cosh (2 \text {arctanh}(a x))+4 \log \left (1-e^{-2 \text {arctanh}(a x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )-\sinh (2 \text {arctanh}(a x))\right ) \]
(4*ArcTanh[a*x]^2 + 2*ArcTanh[a*x]*(Cosh[2*ArcTanh[a*x]] + 4*Log[1 - E^(-2 *ArcTanh[a*x])]) - 4*PolyLog[2, E^(-2*ArcTanh[a*x])] - Sinh[2*ArcTanh[a*x] ])/8
Time = 0.67 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6592, 6550, 6494, 2897, 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.
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 6592 |
\(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx\) |
\(\Big \downarrow \) 6550 |
\(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\int \frac {\text {arctanh}(a x)}{x (a x+1)}dx+\frac {1}{2} \text {arctanh}(a x)^2\) |
\(\Big \downarrow \) 6494 |
\(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )\) |
\(\Big \downarrow \) 2897 |
\(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\) |
\(\Big \downarrow \) 6556 |
\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\) |
\(\Big \downarrow \) 215 |
\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\) |
ArcTanh[a*x]^2/2 + a^2*(ArcTanh[a*x]/(2*a^2*(1 - a^2*x^2)) - (x/(2*(1 - a^ 2*x^2)) + ArcTanh[a*x]/(2*a))/(2*a)) + ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog[2, -1 + 2/(1 + a*x)]/2
3.3.63.3.1 Defintions of rubi rules used
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[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_)]*(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_.)/((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]
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_)^(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]
Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(81)=162\).
Time = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.09
method | result | size |
derivativedivides | \(\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right )}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right )}{4 a x +4}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\operatorname {dilog}\left (a x \right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {1}{8 a x -8}+\frac {\ln \left (a x -1\right )}{8}+\frac {1}{8 a x +8}-\frac {\ln \left (a x +1\right )}{8}\) | \(190\) |
default | \(\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right )}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right )}{4 a x +4}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\operatorname {dilog}\left (a x \right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {1}{8 a x -8}+\frac {\ln \left (a x -1\right )}{8}+\frac {1}{8 a x +8}-\frac {\ln \left (a x +1\right )}{8}\) | \(190\) |
risch | \(\frac {\ln \left (a x -1\right )}{16}-\frac {\ln \left (a x +1\right ) \left (a x +1\right )}{16 \left (a x -1\right )}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )}{8 a x +8}+\frac {1}{8 a x +8}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\ln \left (-a x -1\right )}{16}+\frac {\left (-a x +1\right ) \ln \left (-a x +1\right )}{-16 a x -16}+\frac {\left (\ln \left (-a x +1\right )-\ln \left (-\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\operatorname {dilog}\left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (-a x +1\right )}{8 \left (-a x +1\right )}-\frac {1}{8 \left (-a x +1\right )}+\frac {\operatorname {dilog}\left (-a x +1\right )}{2}+\frac {\ln \left (-a x +1\right )^{2}}{8}\) | \(220\) |
parts | \(\operatorname {arctanh}\left (a x \right ) \ln \left (x \right )-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right )}{4 a x +4}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right )}-\frac {a \left (-\frac {2 \left (\ln \left (x \right )-\ln \left (a x \right )\right ) \ln \left (-a x +1\right )}{a}+\frac {2 \operatorname {dilog}\left (a x \right )}{a}+\frac {2 \operatorname {dilog}\left (a x +1\right )}{a}+\frac {2 \ln \left (x \right ) \ln \left (a x +1\right )}{a}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{a}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{a}+\frac {\ln \left (a x -1\right )^{2}}{2 a}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )-\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )-\frac {\ln \left (a x +1\right )^{2}}{2}}{a}-\frac {1}{2 a \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{2 a}-\frac {1}{2 \left (a x +1\right ) a}+\frac {\ln \left (a x +1\right )}{2 a}\right )}{4}\) | \(256\) |
arctanh(a*x)*ln(a*x)-1/4*arctanh(a*x)/(a*x-1)-1/2*arctanh(a*x)*ln(a*x-1)+1 /4*arctanh(a*x)/(a*x+1)-1/2*arctanh(a*x)*ln(a*x+1)-1/2*dilog(a*x)-1/2*dilo g(a*x+1)-1/2*ln(a*x)*ln(a*x+1)-1/8*ln(a*x-1)^2+1/2*dilog(1/2*a*x+1/2)+1/4* ln(a*x-1)*ln(1/2*a*x+1/2)-1/4*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2) +1/8*ln(a*x+1)^2+1/8/(a*x-1)+1/8*ln(a*x-1)+1/8/(a*x+1)-1/8*ln(a*x+1)
\[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )}^{2} x} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx=\int \frac {\operatorname {atanh}{\left (a x \right )}}{x \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (78) = 156\).
Time = 0.18 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.26 \[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx=\frac {1}{8} \, a {\left (\frac {{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 2 \, a x}{a^{3} x^{2} - a} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {4 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {4 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}\right )} - \frac {1}{2} \, {\left (\frac {1}{a^{2} x^{2} - 1} + \log \left (a^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} \operatorname {artanh}\left (a x\right ) \]
1/8*a*(((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 + 2*a*x)/(a^3*x^2 - a) + 4*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a - 4*(log(a*x + 1)*log(x ) + dilog(-a*x))/a + 4*(log(-a*x + 1)*log(x) + dilog(a*x))/a - log(a*x + 1 )/a + log(a*x - 1)/a) - 1/2*(1/(a^2*x^2 - 1) + log(a^2*x^2 - 1) - log(x^2) )*arctanh(a*x)
\[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )}{x\,{\left (a^2\,x^2-1\right )}^2} \,d x \]