\(\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx\) [31]

Optimal result

Integrand size = 10, antiderivative size = 95 \[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\frac {3}{2} a^2 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+3 a^2 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\frac {1}{2} \left (\frac {\coth ^{-1}(a x) \left (3 a x (-1+a x) \coth ^{-1}(a x)+\left (-1+a^2 x^2\right ) \coth ^{-1}(a x)^2+6 a^2 x^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )\right )}{x^2}-3 a^2 \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )\right ) \] Input:

Integrate[ArcCoth[a*x]^3/x^3,x]
 

Output:

((ArcCoth[a*x]*(3*a*x*(-1 + a*x)*ArcCoth[a*x] + (-1 + a^2*x^2)*ArcCoth[a*x 
]^2 + 6*a^2*x^2*Log[1 + E^(-2*ArcCoth[a*x])]))/x^2 - 3*a^2*PolyLog[2, -E^( 
-2*ArcCoth[a*x])])/2
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6453, 6545, 6453, 6511, 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.

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

\(\Big \downarrow \) 6453

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

\(\Big \downarrow \) 6545

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

\(\Big \downarrow \) 6453

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

\(\Big \downarrow \) 6511

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

\(\Big \downarrow \) 6551

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

\(\Big \downarrow \) 6495

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

\(\Big \downarrow \) 2897

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

Input:

Int[ArcCoth[a*x]^3/x^3,x]
 

Output:

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

Defintions of rubi rules used

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 6511

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b 
, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
 

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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 3.79 (sec) , antiderivative size = 3498, normalized size of antiderivative = 36.82

method	result	size
parts	\(\text {Expression too large to display}\)	\(3498\)
derivativedivides	\(\text {Expression too large to display}\)	\(3502\)
default	\(\text {Expression too large to display}\)	\(3502\)

Input:

int(arccoth(x*a)^3/x^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*arccoth(x*a)^3/x^2-3/2*a^2*(1/8*I*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/ 
(a*x-1)-1))^3*polylog(2,-(a*x+1)/(a*x-1))-1/4*I*Pi*csgn(I/(a*x-1)*(a*x+1)/ 
((a*x+1)/(a*x-1)-1))^3*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(I/ 
(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^3*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))- 
1/4*I*Pi*csgn(I*(a*x+1)/(a*x-1))^3*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))-1/4* 
I*Pi*csgn(I*(a*x+1)/(a*x-1))^3*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))+1/8*I*Pi 
*csgn(I*(a*x+1)/(a*x-1))^3*polylog(2,-(a*x+1)/(a*x-1))+1/x/a*arccoth(x*a)^ 
2-1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x 
-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))*arccoth(x*a)*ln(1+I/((a*x-1)/(a*x+1))^(1/ 
2))-1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a 
*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))*arccoth(x*a)*ln(1-I/((a*x-1)/(a*x+1))^( 
1/2))+1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/ 
(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))*arccoth(x*a)*ln(1+(a*x+1)/(a*x-1))-1/ 
2*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*arccoth(x 
*a)*ln(1+(a*x+1)/(a*x-1))+1/4*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn( 
I*(a*x+1)/(a*x-1))*arccoth(x*a)*ln(1+(a*x+1)/(a*x-1))+1/8*I*Pi*csgn(I/((a* 
x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/( 
a*x-1)-1))*polylog(2,-(a*x+1)/(a*x-1))-1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1) 
)*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))*dilo 
g(1+I/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csg...
 

Fricas [F]

\[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{3}} \,d x } \] Input:

integrate(arccoth(a*x)^3/x^3,x, algorithm="fricas")
 

Output:

integral(arccoth(a*x)^3/x^3, x)
 

Sympy [F]

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

integrate(acoth(a*x)**3/x**3,x)
 

Output:

Integral(acoth(a*x)**3/x**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (84) = 168\).

Time = 0.04 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.65 \[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\frac {3}{4} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} a \operatorname {arcoth}\left (a x\right )^{2} - \frac {1}{16} \, {\left (a^{2} {\left (\frac {3 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} - 3 \, {\left (\log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 6 \, \log \left (a x - 1\right )^{2}}{a} - \frac {24 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} + \frac {24 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} - \frac {24 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} - 6 \, {\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right ) + 8 \, \log \left (x\right )\right )} a \operatorname {arcoth}\left (a x\right )\right )} a - \frac {\operatorname {arcoth}\left (a x\right )^{3}}{2 \, x^{2}} \] Input:

integrate(arccoth(a*x)^3/x^3,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

3/4*(a*log(a*x + 1) - a*log(a*x - 1) - 2/x)*a*arccoth(a*x)^2 - 1/16*(a^2*( 
(3*(log(a*x - 1) - 2)*log(a*x + 1)^2 - log(a*x + 1)^3 + log(a*x - 1)^3 - 3 
*(log(a*x - 1)^2 - 4*log(a*x - 1))*log(a*x + 1) + 6*log(a*x - 1)^2)/a - 24 
*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a + 24*(log(a*x 
 + 1)*log(x) + dilog(-a*x))/a - 24*(log(-a*x + 1)*log(x) + dilog(a*x))/a) 
- 6*(2*(log(a*x - 1) - 2)*log(a*x + 1) - log(a*x + 1)^2 - log(a*x - 1)^2 - 
 4*log(a*x - 1) + 8*log(x))*a*arccoth(a*x))*a - 1/2*arccoth(a*x)^3/x^2
 

Giac [F]

\[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{3}} \,d x } \] Input:

integrate(arccoth(a*x)^3/x^3,x, algorithm="giac")
 

Output:

integrate(arccoth(a*x)^3/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^3} \,d x \] Input:

int(acoth(a*x)^3/x^3,x)
 

Output:

int(acoth(a*x)^3/x^3, x)
 

Reduce [F]

\[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\frac {\mathit {acoth} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acoth} \left (a x \right )^{3}+3 \mathit {acoth} \left (a x \right )^{2} a x -3 \mathit {acoth} \left (a x \right ) a^{2} x^{2}+3 \mathit {acoth} \left (a x \right )-6 \left (\int \frac {\mathit {acoth} \left (a x \right )}{a^{2} x^{5}-x^{3}}d x \right ) x^{2}-3 a x}{2 x^{2}} \] Input:

int(acoth(a*x)^3/x^3,x)
 

Output:

(acoth(a*x)**3*a**2*x**2 - acoth(a*x)**3 + 3*acoth(a*x)**2*a*x - 3*acoth(a 
*x)*a**2*x**2 + 3*acoth(a*x) - 6*int(acoth(a*x)/(a**2*x**5 - x**3),x)*x**2 
 - 3*a*x)/(2*x**2)