Integrand size = 6, antiderivative size = 58 \[ \int \coth ^{-1}(a x)^2 \, dx=\frac {\coth ^{-1}(a x)^2}{a}+x \coth ^{-1}(a x)^2-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a} \]
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.79 \[ \int \coth ^{-1}(a x)^2 \, dx=\frac {\coth ^{-1}(a x) \left ((-1+a x) \coth ^{-1}(a x)-2 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{a} \]
(ArcCoth[a*x]*((-1 + a*x)*ArcCoth[a*x] - 2*Log[1 - E^(-2*ArcCoth[a*x])]) + PolyLog[2, E^(-2*ArcCoth[a*x])])/a
Time = 0.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6437, 6547, 6471, 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.
\(\displaystyle \int \coth ^{-1}(a x)^2 \, dx\) |
\(\Big \downarrow \) 6437 |
\(\displaystyle x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx\) |
\(\Big \downarrow \) 6547 |
\(\displaystyle x \coth ^{-1}(a x)^2-2 a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a x}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )\) |
\(\Big \downarrow \) 6471 |
\(\displaystyle x \coth ^{-1}(a x)^2-2 a \left (\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )\) |
\(\Big \downarrow \) 2849 |
\(\displaystyle x \coth ^{-1}(a x)^2-2 a \left (\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle x \coth ^{-1}(a x)^2-2 a \left (\frac {\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )\) |
x*ArcCoth[a*x]^2 - 2*a*(-1/2*ArcCoth[a*x]^2/a^2 + ((ArcCoth[a*x]*Log[2/(1 - a*x)])/a + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a))/a)
3.1.17.3.1 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_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcCoth[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
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] + Simp[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]
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] + Simp[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]
Time = 0.50 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.00
method | result | size |
derivativedivides | \(\frac {\operatorname {arccoth}\left (a x \right )^{2} \left (a x -1\right )+2 \operatorname {arccoth}\left (a x \right )^{2}-2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}\) | \(116\) |
default | \(\frac {\operatorname {arccoth}\left (a x \right )^{2} \left (a x -1\right )+2 \operatorname {arccoth}\left (a x \right )^{2}-2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}\) | \(116\) |
risch | \(\frac {\ln \left (a x -1\right )^{2} x}{4}-\frac {\ln \left (a x -1\right ) x}{2}-\frac {\ln \left (a x -1\right )^{2}}{4 a}+\frac {\ln \left (a x -1\right )}{2 a}+\frac {1}{a}+\frac {\ln \left (a x +1\right )^{2} x}{4}-\frac {x \ln \left (a x +1\right )}{2}+\frac {\ln \left (a x +1\right )^{2}}{4 a}+\frac {\ln \left (a x +1\right )}{2 a}-\frac {\left (-1+\ln \left (a x -1\right )\right ) \left (a x -1\right ) \ln \left (a x +1\right )}{2 a}+\frac {\ln \left (a x -1\right ) \left (a x -1\right )}{2 a}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{a}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{a}\) | \(163\) |
1/a*(arccoth(a*x)^2*(a*x-1)+2*arccoth(a*x)^2-2*arccoth(a*x)*ln(1-1/((a*x-1 )/(a*x+1))^(1/2))-2*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))-2*arccoth(a*x)*ln (1+1/((a*x-1)/(a*x+1))^(1/2))-2*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2)))
\[ \int \coth ^{-1}(a x)^2 \, dx=\int { \operatorname {arcoth}\left (a x\right )^{2} \,d x } \]
\[ \int \coth ^{-1}(a x)^2 \, dx=\int \operatorname {acoth}^{2}{\left (a x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (55) = 110\).
Time = 0.21 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.33 \[ \int \coth ^{-1}(a x)^2 \, dx=x \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{4} \, {\left (a {\left (\frac {\log \left (a x + 1\right )^{2} + 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2}}{a^{3}} - \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{3}}\right )} - \frac {2 \, {\left (\frac {\log \left (a x + 1\right )}{a} - \frac {\log \left (a x - 1\right )}{a}\right )} \log \left (a^{2} x^{2} - 1\right )}{a}\right )} a + \frac {\operatorname {arcoth}\left (a x\right ) \log \left (a^{2} x^{2} - 1\right )}{a} \]
x*arccoth(a*x)^2 + 1/4*(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) - 2*(log(a*x + 1)/a - log(a*x - 1)/a)*log(a^2*x^2 - 1)/a)*a + arccoth(a*x)*log(a^2*x^2 - 1)/a
\[ \int \coth ^{-1}(a x)^2 \, dx=\int { \operatorname {arcoth}\left (a x\right )^{2} \,d x } \]
Timed out. \[ \int \coth ^{-1}(a x)^2 \, dx=\int {\mathrm {acoth}\left (a\,x\right )}^2 \,d x \]