3.1.32 \(\int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx\) [32]

Optimal. Leaf size=154 \[ -\frac {a^2 \coth ^{-1}(a x)}{x}+\frac {1}{2} a^3 \coth ^{-1}(a x)^2-\frac {a \coth ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{3 x^3}+a^3 \log (x)-\frac {1}{2} a^3 \log \left (1-a^2 x^2\right )+a^3 \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^3 \coth ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^3 \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6038, 6130, 272, 36, 29, 31, 6096, 6136, 6080, 6204, 6745} \begin {gather*} -\frac {1}{2} a^3 \text {Li}_3\left (\frac {2}{a x+1}-1\right )-a^3 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \coth ^{-1}(a x)+a^3 \log (x)+\frac {1}{3} a^3 \coth ^{-1}(a x)^3+\frac {1}{2} a^3 \coth ^{-1}(a x)^2+a^3 \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2-\frac {a^2 \coth ^{-1}(a x)}{x}-\frac {1}{2} a^3 \log \left (1-a^2 x^2\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}-\frac {a \coth ^{-1}(a x)^2}{2 x^2} \end {gather*}

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 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 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 272

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 6038

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] - Dist[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 6080

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 6096

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 6130

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 6136

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 6204

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCot
h[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 6745

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 \text {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 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^5 \text {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 = 0.16, size = 142, normalized size = 0.92 \begin {gather*} \frac {1}{6} \left (-\frac {6 a^2 \coth ^{-1}(a x)}{x}+3 a^3 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{x^2}+2 a^3 \coth ^{-1}(a x)^3-\frac {2 \coth ^{-1}(a x)^3}{x^3}+6 a^3 \coth ^{-1}(a x)^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )+6 a^3 \log \left (\frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )-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 )\right ) \end {gather*}

Antiderivative was successfully verified.

[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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 5.10, size = 840, normalized size = 5.45 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x)^3/x^4,x,method=_RETURNVERBOSE)

[Out]

a^3*(-1/3/a^3/x^3*arccoth(a*x)^3-1/2*arccoth(a*x)^2*ln(a*x+1)-1/2/a^2/x^2*arccoth(a*x)^2+arccoth(a*x)^2*ln(a*x
)-1/2*arccoth(a*x)^2*ln(a*x-1)-1/2*arccoth(a*x)^2*ln((a*x-1)/(a*x+1))+arccoth(a*x)*polylog(2,-1/(a*x-1)*(a*x+1
))-1/2*polylog(3,-1/(a*x-1)*(a*x+1))-1/12*arccoth(a*x)*(6*I*arccoth(a*x)*Pi*csgn(I/(1/(a*x-1)*(a*x+1)-1))*csgn
(I/(1/(a*x-1)*(a*x+1)-1)*(1+1/(a*x-1)*(a*x+1)))^2*a*x+3*I*arccoth(a*x)*Pi*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a
*x+1)-1))^3*a*x-3*I*arccoth(a*x)*Pi*csgn(I/(1/(a*x-1)*(a*x+1)-1))*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1)
)^2*a*x-6*I*arccoth(a*x)*Pi*csgn(I*(1+1/(a*x-1)*(a*x+1)))*csgn(I/(1/(a*x-1)*(a*x+1)-1))*csgn(I/(1/(a*x-1)*(a*x
+1)-1)*(1+1/(a*x-1)*(a*x+1)))*a*x+6*I*arccoth(a*x)*Pi*csgn(I*(1+1/(a*x-1)*(a*x+1)))*csgn(I/(1/(a*x-1)*(a*x+1)-
1)*(1+1/(a*x-1)*(a*x+1)))^2*a*x+3*I*arccoth(a*x)*Pi*csgn(I/(a*x-1)*(a*x+1))^3*a*x-3*I*arccoth(a*x)*Pi*csgn(I/(
a*x-1)*(a*x+1))*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^2*a*x+3*I*arccoth(a*x)*Pi*csgn(I/(a*x-1)*(a*x+1)
)*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*a*x+3*I*arccoth(a*x)*Pi*csgn(I/(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/(a*x-1)*(a*x+1)-1))*a*x-6*I*arccoth(a*x)*Pi*csgn(I/(a*x-1)*(a*x+1))^2*csgn(I/((a
*x-1)/(a*x+1))^(1/2))*a*x-6*I*arccoth(a*x)*Pi*csgn(I/(1/(a*x-1)*(a*x+1)-1)*(1+1/(a*x-1)*(a*x+1)))^3*a*x+4*arcc
oth(a*x)^2*a*x-12*arccoth(a*x)*ln(2)*a*x-6*a*x*arccoth(a*x)+12*a*x+12)/a/x+ln(1+1/(a*x-1)*(a*x+1)))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x^{4}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(a*x)^3/x^4,x)

[Out]

int(acoth(a*x)^3/x^4, x)

________________________________________________________________________________________