\(\int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx\) [21]

Optimal result

Integrand size = 10, antiderivative size = 103 \[ \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx=-\frac {a^2}{3 x}-\frac {a \coth ^{-1}(a x)}{3 x^2}+\frac {1}{3} a^3 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^3 \text {arctanh}(a x)+\frac {2}{3} a^3 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{3} a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]

[Out]

-1/3*a^2/x-1/3*a*arccoth(a*x)/x^2+1/3*a^3*arccoth(a*x)^2-1/3*arccoth(a*x)^2/x^3+1/3*a^3*arctanh(a*x)+2/3*a^3*a
rccoth(a*x)*ln(2-2/(a*x+1))-1/3*a^3*polylog(2,-1+2/(a*x+1))

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6038, 6130, 331, 212, 6136, 6080, 2497} \[ \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx=\frac {1}{3} a^3 \text {arctanh}(a x)-\frac {1}{3} a^3 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {1}{3} a^3 \coth ^{-1}(a x)^2+\frac {2}{3} a^3 \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\frac {a^2}{3 x}-\frac {\coth ^{-1}(a x)^2}{3 x^3}-\frac {a \coth ^{-1}(a x)}{3 x^2} \]

[In]

Int[ArcCoth[a*x]^2/x^4,x]

[Out]

-1/3*a^2/x - (a*ArcCoth[a*x])/(3*x^2) + (a^3*ArcCoth[a*x]^2)/3 - ArcCoth[a*x]^2/(3*x^3) + (a^3*ArcTanh[a*x])/3
+ (2*a^3*ArcCoth[a*x]*Log[2 - 2/(1 + a*x)])/3 - (a^3*PolyLog[2, -1 + 2/(1 + a*x)])/3

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, 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 6130

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

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}{3 x^3}+\frac {1}{3} (2 a) \int \frac {\coth ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx \\ & = -\frac {\coth ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} (2 a) \int \frac {\coth ^{-1}(a x)}{x^3} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx \\ & = -\frac {a \coth ^{-1}(a x)}{3 x^2}+\frac {1}{3} a^3 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^2 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\coth ^{-1}(a x)}{x (1+a x)} \, dx \\ & = -\frac {a^2}{3 x}-\frac {a \coth ^{-1}(a x)}{3 x^2}+\frac {1}{3} a^3 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{3} a^4 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{3} \left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx \\ & = -\frac {a^2}{3 x}-\frac {a \coth ^{-1}(a x)}{3 x^2}+\frac {1}{3} a^3 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^3 \text {arctanh}(a x)+\frac {2}{3} a^3 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{3} a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx=\frac {-a^2 x^2+\left (-1+a^3 x^3\right ) \coth ^{-1}(a x)^2+a x \coth ^{-1}(a x) \left (-1+a^2 x^2+2 a^2 x^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )\right )-a^3 x^3 \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )}{3 x^3} \]

[In]

Integrate[ArcCoth[a*x]^2/x^4,x]

[Out]

(-(a^2*x^2) + (-1 + a^3*x^3)*ArcCoth[a*x]^2 + a*x*ArcCoth[a*x]*(-1 + a^2*x^2 + 2*a^2*x^2*Log[1 + E^(-2*ArcCoth
[a*x])]) - a^3*x^3*PolyLog[2, -E^(-2*ArcCoth[a*x])])/(3*x^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(183\) vs. \(2(89)=178\).

Time = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.79

method	result	size
parts	\(-\frac {\operatorname {arccoth}\left (a x \right )^{2}}{3 x^{3}}-\frac {2 a^{3} \left (\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {\operatorname {arccoth}\left (a x \right )}{2 a^{2} x^{2}}-\ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\ln \left (a x -1\right )}{4}-\frac {\ln \left (a x +1\right )}{4}+\frac {1}{2 a x}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\operatorname {dilog}\left (a x +1\right )}{2}+\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\operatorname {dilog}\left (a x \right )}{2}\right )}{3}\)	\(184\)
derivativedivides	\(a^{3} \left (-\frac {\operatorname {arccoth}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{3}-\frac {\operatorname {arccoth}\left (a x \right )}{3 a^{2} x^{2}}+\frac {2 \ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )}{3}-\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}-\frac {1}{3 a x}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{6}-\frac {\ln \left (a x -1\right )^{2}}{12}+\frac {\ln \left (a x +1\right )^{2}}{12}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{6}-\frac {\operatorname {dilog}\left (a x +1\right )}{3}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {\operatorname {dilog}\left (a x \right )}{3}\right )\)	\(185\)
default	\(a^{3} \left (-\frac {\operatorname {arccoth}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{3}-\frac {\operatorname {arccoth}\left (a x \right )}{3 a^{2} x^{2}}+\frac {2 \ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )}{3}-\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}-\frac {1}{3 a x}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{6}-\frac {\ln \left (a x -1\right )^{2}}{12}+\frac {\ln \left (a x +1\right )^{2}}{12}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{6}-\frac {\operatorname {dilog}\left (a x +1\right )}{3}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {\operatorname {dilog}\left (a x \right )}{3}\right )\)	\(185\)

[In]

int(arccoth(a*x)^2/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*arccoth(a*x)^2/x^3-2/3*a^3*(1/2*arccoth(a*x)*ln(a*x-1)+1/2*arccoth(a*x)/a^2/x^2-ln(a*x)*arccoth(a*x)+1/2*
arccoth(a*x)*ln(a*x+1)+1/4*ln(a*x-1)-1/4*ln(a*x+1)+1/2/a/x-1/2*dilog(1/2*a*x+1/2)-1/4*ln(a*x-1)*ln(1/2*a*x+1/2
)+1/8*ln(a*x-1)^2-1/8*ln(a*x+1)^2+1/4*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)+1/2*dilog(a*x+1)+1/2*ln(a*x
)*ln(a*x+1)+1/2*dilog(a*x))

Fricas [F]

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

[In]

integrate(arccoth(a*x)^2/x^4,x, algorithm="fricas")

[Out]

integral(arccoth(a*x)^2/x^4, x)

Sympy [F]

\[ \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx=\int \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{x^{4}}\, dx \]

[In]

integrate(acoth(a*x)**2/x**4,x)

[Out]

Integral(acoth(a*x)**2/x**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.71 \[ \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx=\frac {1}{12} \, {\left (4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a - 4 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a + 4 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a + 2 \, a \log \left (a x + 1\right ) - 2 \, a \log \left (a x - 1\right ) + \frac {a x \log \left (a x + 1\right )^{2} - 2 \, a x \log \left (a x + 1\right ) \log \left (a x - 1\right ) - a x \log \left (a x - 1\right )^{2} - 4}{x}\right )} a^{2} - \frac {1}{3} \, {\left (a^{2} \log \left (a^{2} x^{2} - 1\right ) - a^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} a \operatorname {arcoth}\left (a x\right ) - \frac {\operatorname {arcoth}\left (a x\right )^{2}}{3 \, x^{3}} \]

[In]

integrate(arccoth(a*x)^2/x^4,x, algorithm="maxima")

[Out]

1/12*(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 + 2*a*log(a*x + 1) - 2*a*log(a*x - 1) + (a*x*log(a*x + 1)^2 - 2*a*x*
log(a*x + 1)*log(a*x - 1) - a*x*log(a*x - 1)^2 - 4)/x)*a^2 - 1/3*(a^2*log(a^2*x^2 - 1) - a^2*log(x^2) + 1/x^2)
*a*arccoth(a*x) - 1/3*arccoth(a*x)^2/x^3

Giac [F]

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

[In]

integrate(arccoth(a*x)^2/x^4,x, algorithm="giac")

[Out]

integrate(arccoth(a*x)^2/x^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^2}{x^4} \,d x \]

[In]

int(acoth(a*x)^2/x^4,x)

[Out]

int(acoth(a*x)^2/x^4, x)