\(\int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx\) [19]

Optimal result

Integrand size = 10, antiderivative size = 55 \[ \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx=a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6038, 6136, 6080, 2497} \[ \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx=-a \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+2 a \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x) \]

[In]

Int[ArcCoth[a*x]^2/x^2,x]

[Out]

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

Rule 2497

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[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 6038

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] - Dist[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
]

Rule 6080

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] - Dist[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]

Rule 6136

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] + Dist[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]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx=\frac {(-1+a x) \coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )-a \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right ) \]

[In]

Integrate[ArcCoth[a*x]^2/x^2,x]

[Out]

((-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])]

Maple [B] (verified)

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	\(-\frac {\operatorname {arccoth}\left (a x \right )^{2}}{x}-2 a \left (\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{2}-\ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\operatorname {dilog}\left (a x +1\right )}{2}+\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\operatorname {dilog}\left (a x \right )}{2}\right )\)	\(146\)

[In]

int(arccoth(a*x)^2/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

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

[In]

integrate(arccoth(a*x)^2/x^2,x, algorithm="fricas")

[Out]

integral(arccoth(a*x)^2/x^2, x)

Sympy [F]

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

[In]

integrate(acoth(a*x)**2/x**2,x)

[Out]

Integral(acoth(a*x)**2/x**2, x)

Maxima [B] (verification not implemented)

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 \[ \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx=\frac {1}{4} \, a^{2} {\left (\frac {\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2}}{a} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {4 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {4 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} - a {\left (\log \left (a^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} \operatorname {arcoth}\left (a x\right ) - \frac {\operatorname {arcoth}\left (a x\right )^{2}}{x} \]

[In]

integrate(arccoth(a*x)^2/x^2,x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(arccoth(a*x)^2/x^2,x, algorithm="giac")

[Out]

integrate(arccoth(a*x)^2/x^2, x)

Mupad [F(-1)]

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

[In]

int(acoth(a*x)^2/x^2,x)

[Out]

int(acoth(a*x)^2/x^2, x)