\(\int x \coth ^{-1}(a x)^3 \, dx\) [27]

Optimal result

Integrand size = 8, antiderivative size = 95 \[ \int x \coth ^{-1}(a x)^3 \, dx=\frac {3 \coth ^{-1}(a x)^2}{2 a^2}+\frac {3 x \coth ^{-1}(a x)^2}{2 a}-\frac {\coth ^{-1}(a x)^3}{2 a^2}+\frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^2}-\frac {3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.72 \[ \int x \coth ^{-1}(a x)^3 \, dx=\frac {\coth ^{-1}(a x) \left (3 (-1+a x) \coth ^{-1}(a x)+\left (-1+a^2 x^2\right ) \coth ^{-1}(a x)^2-6 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+3 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{2 a^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6453, 6543, 6437, 6511, 6547, 6471, 2849, 2752}

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.

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

\(\Big \downarrow \) 6453

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

\(\Big \downarrow \) 6543

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

\(\Big \downarrow \) 6437

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

\(\Big \downarrow \) 6511

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

\(\Big \downarrow \) 6547

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

\(\Big \downarrow \) 6471

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 6437

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
 + b*ArcCoth[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcCoth[c*x^n]) 
^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] 
 && (EqQ[n, 1] || EqQ[p, 1])
 

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 6471

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcCoth[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[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 6543

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

rule 6547

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

Maple [C] (warning: unable to verify)

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

Time = 3.55 (sec) , antiderivative size = 2910, normalized size of antiderivative = 30.63

method	result	size
parts	\(\text {Expression too large to display}\)	\(2910\)
derivativedivides	\(\text {Expression too large to display}\)	\(2916\)
default	\(\text {Expression too large to display}\)	\(2916\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (82) = 164\).

Time = 0.04 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.26 \[ \int x \coth ^{-1}(a x)^3 \, dx=\frac {1}{2} \, x^{2} \operatorname {arcoth}\left (a x\right )^{3} + \frac {3}{4} \, a {\left (\frac {2 \, x}{a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{16} \, a {\left (\frac {\frac {3 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} - 3 \, {\left (\log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 6 \, \log \left (a x - 1\right )^{2}}{a} - \frac {24 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a}}{a^{2}} - \frac {6 \, {\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \operatorname {arcoth}\left (a x\right )}{a^{3}}\right )} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x \coth ^{-1}(a x)^3 \, dx=\frac {\mathit {acoth} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acoth} \left (a x \right )^{3}-3 \mathit {acoth} \left (a x \right )^{2} a x -3 \mathit {acoth} \left (a x \right ) a^{2} x^{2}+3 \mathit {acoth} \left (a x \right )+6 \left (\int \frac {\mathit {acoth} \left (a x \right ) x^{3}}{a^{2} x^{2}-1}d x \right ) a^{4}+3 a x}{2 a^{2}} \] Input:

int(x*acoth(a*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)*x**3)/(a**2*x**2 - 1),x)* 
a**4 + 3*a*x)/(2*a**2)