Integrand size = 18, antiderivative size = 54 \[ \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2} \, dx=-\frac {\text {arctanh}(a x)^2}{2 a^2}+\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a^2}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^2} \]
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2} \, dx=-\frac {-\text {arctanh}(a x) \left (\text {arctanh}(a x)+2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )}{2 a^2} \]
-1/2*(-(ArcTanh[a*x]*(ArcTanh[a*x] + 2*Log[1 + E^(-2*ArcTanh[a*x])])) + Po lyLog[2, -E^(-2*ArcTanh[a*x])])/a^2
Time = 0.36 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6546, 6470, 2849, 2752}
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.
\(\displaystyle \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2} \, dx\) |
\(\Big \downarrow \) 6546 |
\(\displaystyle \frac {\int \frac {\text {arctanh}(a x)}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 6470 |
\(\displaystyle \frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 2849 |
\(\displaystyle \frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle \frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\) |
-1/2*ArcTanh[a*x]^2/a^2 + ((ArcTanh[a*x]*Log[2/(1 - a*x)])/a + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a))/a
3.3.29.3.1 Defintions of rubi rules used
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[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*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.67
method | result | size |
risch | \(\frac {\ln \left (\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (-a x +1\right )}{4 a^{2}}-\frac {\operatorname {dilog}\left (-\frac {a x}{2}+\frac {1}{2}\right )}{4 a^{2}}+\frac {\ln \left (-a x +1\right )^{2}}{8 a^{2}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{4 a^{2}}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{4 a^{2}}-\frac {\ln \left (a x +1\right )^{2}}{8 a^{2}}\) | \(90\) |
derivativedivides | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}}{a^{2}}\) | \(99\) |
default | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}}{a^{2}}\) | \(99\) |
parts | \(-\frac {\ln \left (a^{2} x^{2}-1\right ) \operatorname {arctanh}\left (a x \right )}{2 a^{2}}+\frac {\frac {\ln \left (a x +1\right ) \ln \left (a^{2} x^{2}-1\right )}{2 a}-\frac {\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x +1\right )^{2}}{4}}{a}-\frac {\ln \left (a x -1\right ) \ln \left (a^{2} x^{2}-1\right )}{2 a}+\frac {\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right )^{2}}{4}}{a}}{2 a}\) | \(158\) |
1/4/a^2*ln(1/2*a*x+1/2)*ln(-a*x+1)-1/4/a^2*dilog(-1/2*a*x+1/2)+1/8/a^2*ln( -a*x+1)^2-1/4/a^2*ln(-1/2*a*x+1/2)*ln(a*x+1)+1/4/a^2*dilog(1/2*a*x+1/2)-1/ 8/a^2*ln(a*x+1)^2
\[ \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2} \, dx=\int { -\frac {x \operatorname {artanh}\left (a x\right )}{a^{2} x^{2} - 1} \,d x } \]
\[ \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2} \, dx=- \int \frac {x \operatorname {atanh}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (47) = 94\).
Time = 0.18 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.31 \[ \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2} \, dx=-\frac {1}{8} \, a {\left (\frac {\log \left (a x + 1\right )^{2} + 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2}}{a^{3}} - \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{3}}\right )} + \frac {{\left (\frac {\log \left (a x + 1\right )}{a} - \frac {\log \left (a x - 1\right )}{a}\right )} \log \left (a^{2} x^{2} - 1\right )}{4 \, a} - \frac {\operatorname {artanh}\left (a x\right ) \log \left (a^{2} x^{2} - 1\right )}{2 \, a^{2}} \]
-1/8*a*((log(a*x + 1)^2 + 2*log(a*x + 1)*log(a*x - 1) - log(a*x - 1)^2)/a^ 3 - 4*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a^3) + 1/4 *(log(a*x + 1)/a - log(a*x - 1)/a)*log(a^2*x^2 - 1)/a - 1/2*arctanh(a*x)*l og(a^2*x^2 - 1)/a^2
\[ \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2} \, dx=\int { -\frac {x \operatorname {artanh}\left (a x\right )}{a^{2} x^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2} \, dx=-\int \frac {x\,\mathrm {atanh}\left (a\,x\right )}{a^2\,x^2-1} \,d x \]