Integrand size = 10, antiderivative size = 55 \[ \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx=a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
a*arccoth(a*x)^2-arccoth(a*x)^2/x+2*a*arccoth(a*x)*ln(2-2/(a*x+1))-a*polyl og(2,-1+2/(a*x+1))
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx=\frac {(-1+a x) \coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )-a \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right ) \]
((-1 + a*x)*ArcCoth[a*x]^2)/x + 2*a*ArcCoth[a*x]*Log[1 + E^(-2*ArcCoth[a*x ])] - a*PolyLog[2, -E^(-2*ArcCoth[a*x])]
Time = 0.42 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6453, 6551, 6495, 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.
\(\displaystyle \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx\) |
\(\Big \downarrow \) 6453 |
\(\displaystyle 2 a \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)^2}{x}\) |
\(\Big \downarrow \) 6551 |
\(\displaystyle 2 a \left (\int \frac {\coth ^{-1}(a x)}{x (a x+1)}dx+\frac {1}{2} \coth ^{-1}(a x)^2\right )-\frac {\coth ^{-1}(a x)^2}{x}\) |
\(\Big \downarrow \) 6495 |
\(\displaystyle 2 a \left (-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \coth ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)\right )-\frac {\coth ^{-1}(a x)^2}{x}\) |
\(\Big \downarrow \) 2897 |
\(\displaystyle 2 a \left (-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {1}{2} \coth ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)\right )-\frac {\coth ^{-1}(a x)^2}{x}\) |
-(ArcCoth[a*x]^2/x) + 2*a*(ArcCoth[a*x]^2/2 + ArcCoth[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog[2, -1 + 2/(1 + a*x)]/2)
3.1.19.3.1 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_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcCoth[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcCoth[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_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcCoth[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcCoth[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_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcCoth[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]
Leaf count of result is larger than twice the leaf count of optimal. \(144\) vs. \(2(55)=110\).
Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.64
method | result | size |
derivativedivides | \(a \left (-\frac {\operatorname {arccoth}\left (a x \right )^{2}}{a x}-\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )+2 \ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )-\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )+\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (a x -1\right )^{2}}{4}+\frac {\ln \left (a x +1\right )^{2}}{4}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\operatorname {dilog}\left (a x +1\right )-\ln \left (a x \right ) \ln \left (a x +1\right )-\operatorname {dilog}\left (a x \right )\right )\) | \(145\) |
default | \(a \left (-\frac {\operatorname {arccoth}\left (a x \right )^{2}}{a x}-\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )+2 \ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )-\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )+\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (a x -1\right )^{2}}{4}+\frac {\ln \left (a x +1\right )^{2}}{4}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\operatorname {dilog}\left (a x +1\right )-\ln \left (a x \right ) \ln \left (a x +1\right )-\operatorname {dilog}\left (a x \right )\right )\) | \(145\) |
parts | \(-\frac {\operatorname {arccoth}\left (a x \right )^{2}}{x}-2 a \left (\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{2}-\ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\operatorname {dilog}\left (a x +1\right )}{2}+\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\operatorname {dilog}\left (a x \right )}{2}\right )\) | \(146\) |
a*(-1/a/x*arccoth(a*x)^2-arccoth(a*x)*ln(a*x+1)+2*ln(a*x)*arccoth(a*x)-arc coth(a*x)*ln(a*x-1)+dilog(1/2*a*x+1/2)+1/2*ln(a*x-1)*ln(1/2*a*x+1/2)-1/4*l n(a*x-1)^2+1/4*ln(a*x+1)^2-1/2*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2 )-dilog(a*x+1)-ln(a*x)*ln(a*x+1)-dilog(a*x))
\[ \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{2}}{x^{2}} \,d x } \]
\[ \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx=\int \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (54) = 108\).
Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.65 \[ \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx=\frac {1}{4} \, a^{2} {\left (\frac {\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2}}{a} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {4 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {4 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} - a {\left (\log \left (a^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} \operatorname {arcoth}\left (a x\right ) - \frac {\operatorname {arcoth}\left (a x\right )^{2}}{x} \]
1/4*a^2*((log(a*x + 1)^2 - 2*log(a*x + 1)*log(a*x - 1) - log(a*x - 1)^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 ) - a*(log(a^2*x^2 - 1) - log(x^2))*arccoth(a*x) - arccoth(a*x)^2/x
\[ \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^2}{x^2} \,d x \]