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

Optimal result

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

-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*arccoth(a*x)*ln(2-2/(a*x+1))-1/3*a^3*po 
lylog(2,-1+2/(a*x+1))
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx=\frac {-a^2 x^2+\left (-1+a^3 x^3\right ) \coth ^{-1}(a x)^2+a x \coth ^{-1}(a x) \left (-1+a^2 x^2+2 a^2 x^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )\right )-a^3 x^3 \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )}{3 x^3} \] Input:

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

Output:

(-(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*A 
rcCoth[a*x])])/(3*x^3)
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6453, 6545, 6453, 264, 219, 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.

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

\(\Big \downarrow \) 6453

\(\displaystyle \frac {2}{3} a \int \frac {\coth ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)^2}{3 x^3}\)

\(\Big \downarrow \) 6545

\(\displaystyle \frac {2}{3} a \left (a^2 \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx+\int \frac {\coth ^{-1}(a x)}{x^3}dx\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\)

\(\Big \downarrow \) 6453

\(\displaystyle \frac {2}{3} a \left (\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )}dx+a^2 \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {2}{3} a \left (\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx-\frac {1}{x}\right )+a^2 \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2}{3} a \left (a^2 \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {\coth ^{-1}(a x)}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\)

\(\Big \downarrow \) 6551

\(\displaystyle \frac {2}{3} a \left (a^2 \left (\int \frac {\coth ^{-1}(a x)}{x (a x+1)}dx+\frac {1}{2} \coth ^{-1}(a x)^2\right )+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {\coth ^{-1}(a x)}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\)

\(\Big \downarrow \) 6495

\(\displaystyle \frac {2}{3} a \left (a^2 \left (-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \coth ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)\right )+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {\coth ^{-1}(a x)}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\)

\(\Big \downarrow \) 2897

\(\displaystyle \frac {2}{3} a \left (a^2 \left (-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {1}{2} \coth ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)\right )+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {\coth ^{-1}(a x)}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 2897

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]]
 

rule 6453

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]
 

rule 6495

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]
 

rule 6545

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( 
e_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x 
], x] - Simp[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 6551

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]
 

Maple [B] (verified)

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

Time = 0.16 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.79

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(arccoth(a*x)^2/x^4, x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.71 \[ \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx=\frac {1}{12} \, {\left (4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a - 4 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a + 4 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a + 2 \, a \log \left (a x + 1\right ) - 2 \, a \log \left (a x - 1\right ) + \frac {a x \log \left (a x + 1\right )^{2} - 2 \, a x \log \left (a x + 1\right ) \log \left (a x - 1\right ) - a x \log \left (a x - 1\right )^{2} - 4}{x}\right )} a^{2} - \frac {1}{3} \, {\left (a^{2} \log \left (a^{2} x^{2} - 1\right ) - a^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} a \operatorname {arcoth}\left (a x\right ) - \frac {\operatorname {arcoth}\left (a x\right )^{2}}{3 \, x^{3}} \] Input:

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

Output:

1/12*(4*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a - 4*(l 
og(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*lo 
g(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]

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

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

Output:

integrate(arccoth(a*x)^2/x^4, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx=\frac {-\mathit {acoth} \left (a x \right )^{2}-\mathit {acoth} \left (a x \right ) a^{3} x^{3}+\mathit {acoth} \left (a x \right ) a x +2 \left (\int \frac {\mathit {acoth} \left (a x \right )}{a^{2} x^{3}-x}d x \right ) a^{3} x^{3}-a^{2} x^{2}}{3 x^{3}} \] Input:

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

Output:

( - acoth(a*x)**2 - acoth(a*x)*a**3*x**3 + acoth(a*x)*a*x + 2*int(acoth(a* 
x)/(a**2*x**3 - x),x)*a**3*x**3 - a**2*x**2)/(3*x**3)