\(\int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx\) [56]

Optimal result

Integrand size = 13, antiderivative size = 37 \[ \int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx=-\frac {1}{2} \coth ^{-1}(x)^2+\coth ^{-1}(x) \log \left (\frac {2}{1-x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {1+x}{-1+x}\right ) \]

[Out]

-1/2*arccoth(x)^2+arccoth(x)*ln(2/(1-x))+1/2*polylog(2,(1+x)/(-1+x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6132, 6056, 2449, 2352} \[ \int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx=\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {x+1}{x-1}\right )-\frac {1}{2} \coth ^{-1}(x)^2+\log \left (\frac {2}{1-x}\right ) \coth ^{-1}(x) \]

[In]

Int[(x*ArcCoth[x])/(1 - x^2),x]

[Out]

-1/2*ArcCoth[x]^2 + ArcCoth[x]*Log[2/(1 - x)] + PolyLog[2, (1 + x)/(-1 + x)]/2

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-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]

Rule 6056

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcCoth[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]

Rule 6132

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} \coth ^{-1}(x)^2+\int \frac {\coth ^{-1}(x)}{1-x} \, dx \\ & = -\frac {1}{2} \coth ^{-1}(x)^2+\coth ^{-1}(x) \log \left (\frac {2}{1-x}\right )-\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx \\ & = -\frac {1}{2} \coth ^{-1}(x)^2+\coth ^{-1}(x) \log \left (\frac {2}{1-x}\right )+\text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-x}\right ) \\ & = -\frac {1}{2} \coth ^{-1}(x)^2+\coth ^{-1}(x) \log \left (\frac {2}{1-x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {1+x}{-1+x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx=-\frac {1}{2} \coth ^{-1}(x)^2+\coth ^{-1}(x) \log \left (1-e^{2 \coth ^{-1}(x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(x)}\right ) \]

[In]

Integrate[(x*ArcCoth[x])/(1 - x^2),x]

[Out]

-1/2*ArcCoth[x]^2 + ArcCoth[x]*Log[1 - E^(2*ArcCoth[x])] + PolyLog[2, E^(2*ArcCoth[x])]/2

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.59

method	result	size
risch	\(-\frac {\ln \left (1+x \right )^{2}}{8}-\frac {\left (\ln \left (1+x \right )-\ln \left (\frac {1}{2}+\frac {x}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {x}{2}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {1}{2}+\frac {x}{2}\right )}{2}+\frac {\ln \left (x -1\right ) \ln \left (\frac {1}{2}+\frac {x}{2}\right )}{4}+\frac {\ln \left (x -1\right )^{2}}{8}\)	\(59\)
default	\(-\frac {\operatorname {arccoth}\left (x \right ) \ln \left (x -1\right )}{2}-\frac {\operatorname {arccoth}\left (x \right ) \ln \left (1+x \right )}{2}+\frac {\ln \left (1+x \right )^{2}}{8}-\frac {\left (\ln \left (1+x \right )-\ln \left (\frac {1}{2}+\frac {x}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {x}{2}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {1}{2}+\frac {x}{2}\right )}{2}+\frac {\ln \left (x -1\right ) \ln \left (\frac {1}{2}+\frac {x}{2}\right )}{4}-\frac {\ln \left (x -1\right )^{2}}{8}\)	\(75\)
parts	\(-\frac {\operatorname {arccoth}\left (x \right ) \ln \left (x -1\right )}{2}-\frac {\operatorname {arccoth}\left (x \right ) \ln \left (1+x \right )}{2}+\frac {\ln \left (1+x \right )^{2}}{8}-\frac {\left (\ln \left (1+x \right )-\ln \left (\frac {1}{2}+\frac {x}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {x}{2}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {1}{2}+\frac {x}{2}\right )}{2}+\frac {\ln \left (x -1\right ) \ln \left (\frac {1}{2}+\frac {x}{2}\right )}{4}-\frac {\ln \left (x -1\right )^{2}}{8}\)	\(75\)

[In]

int(x*arccoth(x)/(-x^2+1),x,method=_RETURNVERBOSE)

[Out]

-1/8*ln(1+x)^2-1/4*(ln(1+x)-ln(1/2+1/2*x))*ln(1/2-1/2*x)+1/2*dilog(1/2+1/2*x)+1/4*ln(x-1)*ln(1/2+1/2*x)+1/8*ln
(x-1)^2

Fricas [F]

\[ \int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx=\int { -\frac {x \operatorname {arcoth}\left (x\right )}{x^{2} - 1} \,d x } \]

[In]

integrate(x*arccoth(x)/(-x^2+1),x, algorithm="fricas")

[Out]

integral(-x*arccoth(x)/(x^2 - 1), x)

Sympy [F]

\[ \int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx=- \int \frac {x \operatorname {acoth}{\left (x \right )}}{x^{2} - 1}\, dx \]

[In]

integrate(x*acoth(x)/(-x**2+1),x)

[Out]

-Integral(x*acoth(x)/(x**2 - 1), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (30) = 60\).

Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.05 \[ \int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx=\frac {1}{4} \, {\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )} \log \left (x^{2} - 1\right ) - \frac {1}{2} \, \operatorname {arcoth}\left (x\right ) \log \left (x^{2} - 1\right ) - \frac {1}{8} \, \log \left (x + 1\right )^{2} - \frac {1}{4} \, \log \left (x + 1\right ) \log \left (x - 1\right ) + \frac {1}{8} \, \log \left (x - 1\right )^{2} + \frac {1}{2} \, \log \left (x - 1\right ) \log \left (\frac {1}{2} \, x + \frac {1}{2}\right ) + \frac {1}{2} \, {\rm Li}_2\left (-\frac {1}{2} \, x + \frac {1}{2}\right ) \]

[In]

integrate(x*arccoth(x)/(-x^2+1),x, algorithm="maxima")

[Out]

1/4*(log(x + 1) - log(x - 1))*log(x^2 - 1) - 1/2*arccoth(x)*log(x^2 - 1) - 1/8*log(x + 1)^2 - 1/4*log(x + 1)*l
og(x - 1) + 1/8*log(x - 1)^2 + 1/2*log(x - 1)*log(1/2*x + 1/2) + 1/2*dilog(-1/2*x + 1/2)

Giac [F]

\[ \int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx=\int { -\frac {x \operatorname {arcoth}\left (x\right )}{x^{2} - 1} \,d x } \]

[In]

integrate(x*arccoth(x)/(-x^2+1),x, algorithm="giac")

[Out]

integrate(-x*arccoth(x)/(x^2 - 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx=-\int \frac {x\,\mathrm {acoth}\left (x\right )}{x^2-1} \,d x \]

[In]

int(-(x*acoth(x))/(x^2 - 1),x)

[Out]

-int((x*acoth(x))/(x^2 - 1), x)