3.27 \(\int x \coth ^{-1}(a x)^3 \, dx\)
Optimal. Leaf size=95 \[ -\frac{3 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{2 a^2}-\frac{\coth ^{-1}(a x)^3}{2 a^2}+\frac{3 \coth ^{-1}(a x)^2}{2 a^2}-\frac{3 \log \left (\frac{2}{1-a x}\right ) \coth ^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \coth ^{-1}(a x)^3+\frac{3 x \coth ^{-1}(a x)^2}{2 a} \]
[Out]
(3*ArcCoth[a*x]^2)/(2*a^2) + (3*x*ArcCoth[a*x]^2)/(2*a) - ArcCoth[a*x]^3/(2*a^2) + (x^2*ArcCoth[a*x]^3)/2 - (3
*ArcCoth[a*x]*Log[2/(1 - a*x)])/a^2 - (3*PolyLog[2, 1 - 2/(1 - a*x)])/(2*a^2)
________________________________________________________________________________________
Rubi [A] time = 0.183365, antiderivative size = 95, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 8, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used =
{5917, 5981, 5911, 5985, 5919, 2402, 2315, 5949} \[ -\frac{3 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{2 a^2}-\frac{\coth ^{-1}(a x)^3}{2 a^2}+\frac{3 \coth ^{-1}(a x)^2}{2 a^2}-\frac{3 \log \left (\frac{2}{1-a x}\right ) \coth ^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \coth ^{-1}(a x)^3+\frac{3 x \coth ^{-1}(a x)^2}{2 a} \]
Antiderivative was successfully verified.
[In]
Int[x*ArcCoth[a*x]^3,x]
[Out]
(3*ArcCoth[a*x]^2)/(2*a^2) + (3*x*ArcCoth[a*x]^2)/(2*a) - ArcCoth[a*x]^3/(2*a^2) + (x^2*ArcCoth[a*x]^3)/2 - (3
*ArcCoth[a*x]*Log[2/(1 - a*x)])/a^2 - (3*PolyLog[2, 1 - 2/(1 - a*x)])/(2*a^2)
Rule 5917
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
oth[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]
Rule 5981
Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[(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 5911
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcCoth[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]
Rule 5985
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] + Dist[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]
Rule 5919
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] + Dist[(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 2402
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[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 2315
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]
Rule 5949
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> 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]
Rubi steps
\begin{align*} \int x \coth ^{-1}(a x)^3 \, dx &=\frac{1}{2} x^2 \coth ^{-1}(a x)^3-\frac{1}{2} (3 a) \int \frac{x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=\frac{1}{2} x^2 \coth ^{-1}(a x)^3+\frac{3 \int \coth ^{-1}(a x)^2 \, dx}{2 a}-\frac{3 \int \frac{\coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a}\\ &=\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)^3-3 \int \frac{x \coth ^{-1}(a x)}{1-a^2 x^2} \, 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 \int \frac{\coth ^{-1}(a x)}{1-a x} \, dx}{a}\\ &=\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 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a}\\ &=\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{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{a^2}\\ &=\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 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.140715, size = 68, normalized size = 0.72 \[ \frac{3 \text{PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )+\coth ^{-1}(a x) \left (\left (a^2 x^2-1\right ) \coth ^{-1}(a x)^2+3 (a x-1) \coth ^{-1}(a x)-6 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[x*ArcCoth[a*x]^3,x]
[Out]
(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)
________________________________________________________________________________________
Maple [C] time = 0.404, size = 3070, normalized size = 32.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*arccoth(a*x)^3,x)
[Out]
3/2*arccoth(a*x)^2/a^2+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3*dilog(1/((a*x-1)/(a*x+1))^(1
/2))-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3*dilog(1+1/((a*x-1)/(a*x+1))^(1/2))+3/8*I/a^2*P
i*csgn(I*(a*x+1)/(a*x-1))^3*dilog(1/((a*x-1)/(a*x+1))^(1/2))-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1))^3*dilog(1+1/
((a*x-1)/(a*x+1))^(1/2))+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1))^3*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))+3/8*I/a^2
*Pi*csgn(I*(a*x+1)/(a*x-1))^3*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+
1)/(a*x-1)-1))^3*arccoth(a*x)^2-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1))^3*arccoth(a*x)^2+3/4/a^2*arccoth(a*x)^2*l
n(a*x-1)-3/4/a^2*arccoth(a*x)^2*ln(a*x+1)-3/4/a^2*arccoth(a*x)^2*ln((a*x-1)/(a*x+1))+3/8*I/a^2*Pi*csgn(I/((a*x
-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))-3/4*I/a^2*Pi*csgn(I/((a*x-1
)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))+3/4*I/a^2*Pi*csgn(I/((a*x-1)/
(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*arccoth(a*x)^2+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-
1))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth(a*x)^2-3/8*I/a^2*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*
x-1))*arccoth(a*x)^2+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*csgn(I/((a*x+1)/(a*x-1)-1))*ar
ccoth(a*x)^2-3/4*I/a^2*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*polylog(2,-1/((a*x-1)/(a*x
+1))^(1/2))-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*csgn(I/((a*x+1)/(a*x-1)-1))*dilog(1/((a
*x-1)/(a*x+1))^(1/2))+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*csgn(I/((a*x+1)/(a*x-1)-1))*d
ilog(1+1/((a*x-1)/(a*x+1))^(1/2))-3/4*I/a^2*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*dilog
(1/((a*x-1)/(a*x+1))^(1/2))+3/4*I/a^2*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*dilog(1+1/(
(a*x-1)/(a*x+1))^(1/2))-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*csgn(I*(a*x+1)/(a*x-1))*pol
ylog(2,-1/((a*x-1)/(a*x+1))^(1/2))-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*csgn(I*(a*x+1)/(
a*x-1))*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*csgn(I
/((a*x+1)/(a*x-1)-1))*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1
)-1))^3*arccoth(a*x)*ln(1-1/((a*x-1)/(a*x+1))^(1/2))+3/8*I/a^2*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*
x+1)/(a*x-1))*dilog(1/((a*x-1)/(a*x+1))^(1/2))-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*csgn
(I*(a*x+1)/(a*x-1))*dilog(1/((a*x-1)/(a*x+1))^(1/2))+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^
2*csgn(I*(a*x+1)/(a*x-1))*dilog(1+1/((a*x-1)/(a*x+1))^(1/2))-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x
-1)-1))^2*csgn(I/((a*x+1)/(a*x-1)-1))*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))-3/8*I/a^2*Pi*csgn(I/((a*x-1)/(a*x+
1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*dilog(1+1/((a*x-1)/(a*x+1))^(1/2))+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1))^3
*arccoth(a*x)*ln(1-1/((a*x-1)/(a*x+1))^(1/2))+3/8*I/a^2*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a
*x-1))*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))*csgn(I*(
a*x+1)/(a*x-1))*csgn(I/((a*x+1)/(a*x-1)-1))*arccoth(a*x)*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-3/a^2*arccoth(a*x)*ln
(1+1/((a*x-1)/(a*x+1))^(1/2))-3/2/a^2*arccoth(a*x)*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-1/2*arccoth(a*x)^3/a^2+1/2*
x^2*arccoth(a*x)^3+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3*polylog(2,-1/((a*x-1)/(a*x+1))^(
1/2))+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))-3/4*I/a^
2*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*arccoth(a*x)*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-3/
8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*csgn(I/((a*x+1)/(a*x-1)-1))*arccoth(a*x)*ln(1-1/((a*x
-1)/(a*x+1))^(1/2))-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth
(a*x)*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-3/2/a^2*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))-3/2/a^2*dilog(1+1/((a*x-1)/
(a*x+1))^(1/2))-3/2/a^2*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))+3/2/a^2*dilog(1/((a*x-1)/(a*x+1))^(1/2))+3/8*I/a
^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/((a*x+1)/(a*x-1)-1))*dilog(1/
((a*x-1)/(a*x+1))^(1/2))-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn
(I/((a*x+1)/(a*x-1)-1))*arccoth(a*x)^2-3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)
/(a*x-1))*csgn(I/((a*x+1)/(a*x-1)-1))*dilog(1+1/((a*x-1)/(a*x+1))^(1/2))+3/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/(
(a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/((a*x+1)/(a*x-1)-1))*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))+3
/8*I/a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/((a*x+1)/(a*x-1)-1))*po
lylog(2,-1/((a*x-1)/(a*x+1))^(1/2))+3/2*x*arccoth(a*x)^2/a+3/8*I/a^2*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn
(I*(a*x+1)/(a*x-1))*arccoth(a*x)*ln(1-1/((a*x-1)/(a*x+1))^(1/2))
________________________________________________________________________________________
Maxima [B] time = 0.999338, size = 290, normalized size = 3.05 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*arccoth(a*x)^3,x, algorithm="maxima")
[Out]
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*(l
og(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)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \operatorname{arcoth}\left (a x\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*arccoth(a*x)^3,x, algorithm="fricas")
[Out]
integral(x*arccoth(a*x)^3, x)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{acoth}^{3}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*acoth(a*x)**3,x)
[Out]
Integral(x*acoth(a*x)**3, x)
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{arcoth}\left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*arccoth(a*x)^3,x, algorithm="giac")
[Out]
integrate(x*arccoth(a*x)^3, x)