3.32 \(\int \frac{\coth ^{-1}(a x)^3}{x^4} \, dx\)
Optimal. Leaf size=154 \[ -\frac{1}{2} a^3 \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-a^3 \coth ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{1}{2} a^3 \log \left (1-a^2 x^2\right )+a^3 \log (x)+\frac{1}{3} a^3 \coth ^{-1}(a x)^3+\frac{1}{2} a^3 \coth ^{-1}(a x)^2-\frac{a^2 \coth ^{-1}(a x)}{x}+a^3 \log \left (2-\frac{2}{a x+1}\right ) \coth ^{-1}(a x)^2-\frac{a \coth ^{-1}(a x)^2}{2 x^2}-\frac{\coth ^{-1}(a x)^3}{3 x^3} \]
[Out]
-((a^2*ArcCoth[a*x])/x) + (a^3*ArcCoth[a*x]^2)/2 - (a*ArcCoth[a*x]^2)/(2*x^2) + (a^3*ArcCoth[a*x]^3)/3 - ArcCo
th[a*x]^3/(3*x^3) + a^3*Log[x] - (a^3*Log[1 - a^2*x^2])/2 + a^3*ArcCoth[a*x]^2*Log[2 - 2/(1 + a*x)] - a^3*ArcC
oth[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (a^3*PolyLog[3, -1 + 2/(1 + a*x)])/2
________________________________________________________________________________________
Rubi [A] time = 0.368827, antiderivative size = 154, normalized size of antiderivative = 1.,
number of steps used = 14, number of rules used = 11, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.1, Rules used =
{5917, 5983, 266, 36, 29, 31, 5949, 5989, 5933, 6057, 6610} \[ -\frac{1}{2} a^3 \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-a^3 \coth ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{1}{2} a^3 \log \left (1-a^2 x^2\right )+a^3 \log (x)+\frac{1}{3} a^3 \coth ^{-1}(a x)^3+\frac{1}{2} a^3 \coth ^{-1}(a x)^2-\frac{a^2 \coth ^{-1}(a x)}{x}+a^3 \log \left (2-\frac{2}{a x+1}\right ) \coth ^{-1}(a x)^2-\frac{a \coth ^{-1}(a x)^2}{2 x^2}-\frac{\coth ^{-1}(a x)^3}{3 x^3} \]
Antiderivative was successfully verified.
[In]
Int[ArcCoth[a*x]^3/x^4,x]
[Out]
-((a^2*ArcCoth[a*x])/x) + (a^3*ArcCoth[a*x]^2)/2 - (a*ArcCoth[a*x]^2)/(2*x^2) + (a^3*ArcCoth[a*x]^3)/3 - ArcCo
th[a*x]^3/(3*x^3) + a^3*Log[x] - (a^3*Log[1 - a^2*x^2])/2 + a^3*ArcCoth[a*x]^2*Log[2 - 2/(1 + a*x)] - a^3*ArcC
oth[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (a^3*PolyLog[3, -1 + 2/(1 + a*x)])/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 5983
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 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 36
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Rule 29
Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
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]
Rule 5989
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]
Rule 5933
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 6057
Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcCo
th[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p)/2, Int[((a + b*ArcCoth[c*x])^(p - 1)*PolyLog[2, 1 - u])
/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2
/(1 + c*x))^2, 0]
Rule 6610
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
; !FalseQ[w]] /; FreeQ[n, x]
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(a x)^3}{x^4} \, dx &=-\frac{\coth ^{-1}(a x)^3}{3 x^3}+a \int \frac{\coth ^{-1}(a x)^2}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{\coth ^{-1}(a x)^3}{3 x^3}+a \int \frac{\coth ^{-1}(a x)^2}{x^3} \, dx+a^3 \int \frac{\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{a \coth ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} a^3 \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{3 x^3}+a^2 \int \frac{\coth ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx+a^3 \int \frac{\coth ^{-1}(a x)^2}{x (1+a x)} \, dx\\ &=-\frac{a \coth ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} a^3 \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{3 x^3}+a^3 \coth ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )+a^2 \int \frac{\coth ^{-1}(a x)}{x^2} \, dx+a^4 \int \frac{\coth ^{-1}(a x)}{1-a^2 x^2} \, dx-\left (2 a^4\right ) \int \frac{\coth ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a^2 \coth ^{-1}(a x)}{x}+\frac{1}{2} a^3 \coth ^{-1}(a x)^2-\frac{a \coth ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} a^3 \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{3 x^3}+a^3 \coth ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-a^3 \coth ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+a^3 \int \frac{1}{x \left (1-a^2 x^2\right )} \, dx+a^4 \int \frac{\text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a^2 \coth ^{-1}(a x)}{x}+\frac{1}{2} a^3 \coth ^{-1}(a x)^2-\frac{a \coth ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} a^3 \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{3 x^3}+a^3 \coth ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-a^3 \coth ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{1}{2} a^3 \text{Li}_3\left (-1+\frac{2}{1+a x}\right )+\frac{1}{2} a^3 \operatorname{Subst}\left (\int \frac{1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{a^2 \coth ^{-1}(a x)}{x}+\frac{1}{2} a^3 \coth ^{-1}(a x)^2-\frac{a \coth ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} a^3 \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{3 x^3}+a^3 \coth ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-a^3 \coth ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{1}{2} a^3 \text{Li}_3\left (-1+\frac{2}{1+a x}\right )+\frac{1}{2} a^3 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} a^5 \operatorname{Subst}\left (\int \frac{1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a^2 \coth ^{-1}(a x)}{x}+\frac{1}{2} a^3 \coth ^{-1}(a x)^2-\frac{a \coth ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} a^3 \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{3 x^3}+a^3 \log (x)-\frac{1}{2} a^3 \log \left (1-a^2 x^2\right )+a^3 \coth ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-a^3 \coth ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{1}{2} a^3 \text{Li}_3\left (-1+\frac{2}{1+a x}\right )\\ \end{align*}
Mathematica [A] time = 0.21288, size = 142, normalized size = 0.92 \[ \frac{1}{6} \left (-6 a^3 \coth ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-3 a^3 \text{PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right )+6 a^3 \log \left (\frac{1}{\sqrt{1-\frac{1}{a^2 x^2}}}\right )+2 a^3 \coth ^{-1}(a x)^3+3 a^3 \coth ^{-1}(a x)^2-\frac{6 a^2 \coth ^{-1}(a x)}{x}+6 a^3 \coth ^{-1}(a x)^2 \log \left (e^{-2 \coth ^{-1}(a x)}+1\right )-\frac{3 a \coth ^{-1}(a x)^2}{x^2}-\frac{2 \coth ^{-1}(a x)^3}{x^3}\right ) \]
Warning: Unable to verify antiderivative.
[In]
Integrate[ArcCoth[a*x]^3/x^4,x]
[Out]
((-6*a^2*ArcCoth[a*x])/x + 3*a^3*ArcCoth[a*x]^2 - (3*a*ArcCoth[a*x]^2)/x^2 + 2*a^3*ArcCoth[a*x]^3 - (2*ArcCoth
[a*x]^3)/x^3 + 6*a^3*ArcCoth[a*x]^2*Log[1 + E^(-2*ArcCoth[a*x])] + 6*a^3*Log[1/Sqrt[1 - 1/(a^2*x^2)]] - 6*a^3*
ArcCoth[a*x]*PolyLog[2, -E^(-2*ArcCoth[a*x])] - 3*a^3*PolyLog[3, -E^(-2*ArcCoth[a*x])])/6
________________________________________________________________________________________
Maple [C] time = 0.657, size = 895, normalized size = 5.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(arccoth(a*x)^3/x^4,x)
[Out]
-a^3*arccoth(a*x)-1/2*a^3*polylog(3,-(a*x+1)/(a*x-1))+a^3*ln((a*x+1)/(a*x-1)+1)-1/3*arccoth(a*x)^3/x^3-a^2*arc
coth(a*x)/x-1/2*a^3*arccoth(a*x)^2*ln(a*x-1)+a^3*arccoth(a*x)^2*ln(a*x)+a^3*arccoth(a*x)*polylog(2,-(a*x+1)/(a
*x-1))-1/2*a^3*arccoth(a*x)^2*ln(a*x+1)-1/2*a^3*arccoth(a*x)^2*ln((a*x-1)/(a*x+1))+a^3*arccoth(a*x)^2*ln(2)+1/
2*a^3*arccoth(a*x)^2-1/4*I*a^3*arccoth(a*x)^2*Pi*csgn(I*(a*x+1)/(a*x-1))^3+1/2*I*a^3*arccoth(a*x)^2*Pi*csgn(I/
((a*x+1)/(a*x-1)-1)*((a*x+1)/(a*x-1)+1))^3-1/2*a*arccoth(a*x)^2/x^2-1/3*a^3*arccoth(a*x)^3-1/4*I*a^3*arccoth(a
*x)^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3+1/2*I*a^3*arccoth(a*x)^2*Pi*csgn(I/((a*x+1)/(a*x-1)-1))
*csgn(I/((a*x+1)/(a*x-1)-1)*((a*x+1)/(a*x-1)+1))*csgn(I*((a*x+1)/(a*x-1)+1))-1/4*I*a^3*arccoth(a*x)^2*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))*csgn(I*(a*x+1)/(a*x-1))+1/4*I*a^3*arccoth(a
*x)^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))+1/4*I*a^3*arccoth(a*x)^2*Pi*csg
n(I/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2+1/2*I*a^3*arccoth(a*x)^2*Pi*csgn(I*(a*x
+1)/(a*x-1))^2*csgn(I/((a*x-1)/(a*x+1))^(1/2))-1/4*I*a^3*arccoth(a*x)^2*Pi*csgn(I*(a*x+1)/(a*x-1))*csgn(I/((a*
x-1)/(a*x+1))^(1/2))^2-1/2*I*a^3*arccoth(a*x)^2*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I/((a*x+1)/(a*x-1)-1)*((a*
x+1)/(a*x-1)+1))^2-1/2*I*a^3*arccoth(a*x)^2*Pi*csgn(I/((a*x+1)/(a*x-1)-1)*((a*x+1)/(a*x-1)+1))^2*csgn(I*((a*x+
1)/(a*x-1)+1))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccoth(a*x)^3/x^4,x, algorithm="maxima")
[Out]
1/4*a^4*integrate(x^4*log(a*x + 1)*log(a*x - 1)/(a*x^5 + x^4), x) - 1/2*a^4*integrate(x^4*log(a*x + 1)*log(x)/
(a*x^5 + x^4), x) + 1/16*(2*a^2*log(a*x + 1) - 2*a^2*log(x) - (2*a*x - 1)/x^2)*a*log(a)^3 + 3/8*a*integrate(x*
log(a*x - 1)/(a*x^5 + x^4), x)*log(a)^2 - 3/8*a*integrate(x*log(x)/(a*x^5 + x^4), x)*log(a)^2 - 1/48*(6*a^3*lo
g(a*x + 1) - 6*a^3*log(x) - (6*a^2*x^2 - 3*a*x + 2)/x^3)*log(a)^3 + 1/4*a^2*integrate(x^2*log(a*x + 1)/(a*x^5
+ x^4), x) + 3/4*a*integrate(x*log(a*x - 1)*log(x)/(a*x^5 + x^4), x)*log(a) - 3/8*a*integrate(x*log(x)^2/(a*x^
5 + x^4), x)*log(a) + 3/8*integrate(log(a*x - 1)/(a*x^5 + x^4), x)*log(a)^2 - 3/8*integrate(log(x)/(a*x^5 + x^
4), x)*log(a)^2 + 3/8*a*integrate(x*log(a*x + 1)*log(a*x - 1)^2/(a*x^5 + x^4), x) - 3/8*a*integrate(x*log(a*x
- 1)^2*log(x)/(a*x^5 + x^4), x) + 3/8*a*integrate(x*log(a*x - 1)*log(x)^2/(a*x^5 + x^4), x) - 1/8*a*integrate(
x*log(x)^3/(a*x^5 + x^4), x) - 1/4*a*integrate(x*log(a*x + 1)*log(a*x - 1)/(a*x^5 + x^4), x) - 3/8*integrate(a
*x*log(a*x - 1)^2/(a*x^5 + x^4), x)*log(a) - 3/8*integrate(log(a*x - 1)^2/(a*x^5 + x^4), x)*log(a) + 3/4*integ
rate(log(a*x - 1)*log(x)/(a*x^5 + x^4), x)*log(a) - 3/8*integrate(log(x)^2/(a*x^5 + x^4), x)*log(a) - 1/48*(6*
a^4*log(a*x - 1) - 6*a^4*log(x) + (6*a^3*x^2 + 3*a^2*x + 2*a)/x^3)*log(-1/(a*x) + 1)^2/a + 1/864*(6*(18*a^5*x^
3*log(a*x - 1)^2 + 18*a^5*x^3*log(x)^2 - 66*a^5*x^3*log(x) + 66*a^4*x^2 + 15*a^3*x + 4*a^2 - 6*(6*a^5*x^3*log(
x) - 11*a^5*x^3)*log(a*x - 1))*log(-1/(a*x) + 1)/(a*x^3) - (36*a^6*x^3*log(a*x - 1)^3 - 36*a^6*x^3*log(x)^3 +
198*a^6*x^3*log(x)^2 - 510*a^6*x^3*log(x) + 510*a^5*x^2 + 57*a^4*x + 8*a^3 - 18*(6*a^6*x^3*log(x) - 11*a^6*x^3
)*log(a*x - 1)^2 + 6*(18*a^6*x^3*log(x)^2 - 66*a^6*x^3*log(x) + 85*a^6*x^3)*log(a*x - 1))/(a^2*x^3))/a + 1/24*
log(-1/(a*x) + 1)^3/x^3 - 1/24*((a^3*x^3 + 1)*log(a*x + 1)^3 - 3*(2*a^3*x^3*log(x) - a*x - (a^3*x^3 - 1)*log(a
*x - 1))*log(a*x + 1)^2)/x^3 + 3/8*integrate(log(a*x + 1)*log(a*x - 1)^2/(a*x^5 + x^4), x) - 3/8*integrate(log
(a*x - 1)^2*log(x)/(a*x^5 + x^4), x) + 3/8*integrate(log(a*x - 1)*log(x)^2/(a*x^5 + x^4), x) - 1/8*integrate(l
og(x)^3/(a*x^5 + x^4), x)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcoth}\left (a x\right )^{3}}{x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccoth(a*x)^3/x^4,x, algorithm="fricas")
[Out]
integral(arccoth(a*x)^3/x^4, x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acoth}^{3}{\left (a x \right )}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(acoth(a*x)**3/x**4,x)
[Out]
Integral(acoth(a*x)**3/x**4, x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (a x\right )^{3}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccoth(a*x)^3/x^4,x, algorithm="giac")
[Out]
integrate(arccoth(a*x)^3/x^4, x)