3.30 \(\int \frac{\coth ^{-1}(a x)^3}{x^2} \, dx\)
Optimal. Leaf size=79 \[ -\frac{3}{2} a \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-3 a \coth ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+a \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{x}+3 a \log \left (2-\frac{2}{a x+1}\right ) \coth ^{-1}(a x)^2 \]
[Out]
a*ArcCoth[a*x]^3 - ArcCoth[a*x]^3/x + 3*a*ArcCoth[a*x]^2*Log[2 - 2/(1 + a*x)] - 3*a*ArcCoth[a*x]*PolyLog[2, -1
+ 2/(1 + a*x)] - (3*a*PolyLog[3, -1 + 2/(1 + a*x)])/2
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Rubi [A] time = 0.199122, antiderivative size = 79, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used =
{5917, 5989, 5933, 5949, 6057, 6610} \[ -\frac{3}{2} a \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-3 a \coth ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+a \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{x}+3 a \log \left (2-\frac{2}{a x+1}\right ) \coth ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
[In]
Int[ArcCoth[a*x]^3/x^2,x]
[Out]
a*ArcCoth[a*x]^3 - ArcCoth[a*x]^3/x + 3*a*ArcCoth[a*x]^2*Log[2 - 2/(1 + a*x)] - 3*a*ArcCoth[a*x]*PolyLog[2, -1
+ 2/(1 + a*x)] - (3*a*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 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 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 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^2} \, dx &=-\frac{\coth ^{-1}(a x)^3}{x}+(3 a) \int \frac{\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx\\ &=a \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{x}+(3 a) \int \frac{\coth ^{-1}(a x)^2}{x (1+a x)} \, dx\\ &=a \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-\left (6 a^2\right ) \int \frac{\coth ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=a \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-3 a \coth ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+\left (3 a^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=a \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-3 a \coth ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} a \text{Li}_3\left (-1+\frac{2}{1+a x}\right )\\ \end{align*}
Mathematica [A] time = 0.131787, size = 72, normalized size = 0.91 \[ -3 a \coth ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-\frac{3}{2} a \text{PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right )+\frac{(a x-1) \coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (e^{-2 \coth ^{-1}(a x)}+1\right ) \]
Warning: Unable to verify antiderivative.
[In]
Integrate[ArcCoth[a*x]^3/x^2,x]
[Out]
((-1 + a*x)*ArcCoth[a*x]^3)/x + 3*a*ArcCoth[a*x]^2*Log[1 + E^(-2*ArcCoth[a*x])] - 3*a*ArcCoth[a*x]*PolyLog[2,
-E^(-2*ArcCoth[a*x])] - (3*a*PolyLog[3, -E^(-2*ArcCoth[a*x])])/2
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Maple [C] time = 0.342, size = 796, normalized size = 10.1 \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^2,x)
[Out]
-arccoth(a*x)^3/x-3/2*a*arccoth(a*x)^2*ln(a*x-1)+3*a*arccoth(a*x)^2*ln(a*x)-3/2*a*arccoth(a*x)^2*ln(a*x+1)-3/2
*a*arccoth(a*x)^2*ln((a*x-1)/(a*x+1))-a*arccoth(a*x)^3+3*a*arccoth(a*x)*polylog(2,-(a*x+1)/(a*x-1))-3/2*a*poly
log(3,-(a*x+1)/(a*x-1))+3/2*I*a*arccoth(a*x)^2*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))*csgn(I/((a*x+1)/(a*x-1)-1))*Pi-3/2*I*a*arccoth(a*x)^2*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))*Pi-3/2*I*a*arccoth(a*x)^2*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))*Pi+3/2*I*a*arccoth(a*x)^2*csgn(I/((a*x+1)/(a*x-1)-1)*((a*x+1)/(a*x-1)+1))^3*Pi+3/
2*I*a*arccoth(a*x)^2*csgn(I*(a*x+1)/(a*x-1))^2*csgn(I/((a*x-1)/(a*x+1))^(1/2))*Pi-3/4*I*a*arccoth(a*x)^2*csgn(
I*(a*x+1)/(a*x-1))^3*Pi-3/4*I*a*arccoth(a*x)^2*csgn(I*(a*x+1)/(a*x-1))*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*Pi-3/
4*I*a*arccoth(a*x)^2*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3*Pi+3/4*I*a*arccoth(a*x)^2*csgn(I*(a*x+1)/(a
*x-1))*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*Pi+3/4*I*a*arccoth(a*x)^2*csgn(I/((a*x+1)/(a*x-1)-1))*csg
n(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*Pi-3/4*I*a*arccoth(a*x)^2*csgn(I*(a*x+1)/(a*x-1))*csgn(I/((a*x+1)/(
a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))*Pi+3*a*arccoth(a*x)^2*ln(2)
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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^2,x, algorithm="maxima")
[Out]
1/8*a*(log(a*x + 1) - log(x))*log(a)^3 + 3/8*a*integrate(x*log(a*x - 1)/(a*x^3 + x^2), x)*log(a)^2 - 3/8*a*int
egrate(x*log(x)/(a*x^3 + x^2), x)*log(a)^2 - 1/8*(a*log(a*x + 1) - a*log(x) - 1/x)*log(a)^3 + 3/4*a^2*integrat
e(x^2*log(a*x + 1)*log(a*x - 1)/(a*x^3 + x^2), x) - 3/2*a^2*integrate(x^2*log(a*x + 1)*log(x)/(a*x^3 + x^2), x
) + 3/4*a*integrate(x*log(a*x - 1)*log(x)/(a*x^3 + x^2), x)*log(a) - 3/8*a*integrate(x*log(x)^2/(a*x^3 + x^2),
x)*log(a) + 3/8*integrate(log(a*x - 1)/(a*x^3 + x^2), x)*log(a)^2 - 3/8*integrate(log(x)/(a*x^3 + x^2), x)*lo
g(a)^2 + 3/8*a*integrate(x*log(a*x + 1)*log(a*x - 1)^2/(a*x^3 + x^2), x) - 3/8*a*integrate(x*log(a*x - 1)^2*lo
g(x)/(a*x^3 + x^2), x) + 3/8*a*integrate(x*log(a*x - 1)*log(x)^2/(a*x^3 + x^2), x) - 1/8*a*integrate(x*log(x)^
3/(a*x^3 + x^2), x) - 3/4*a*integrate(x*log(a*x + 1)*log(a*x - 1)/(a*x^3 + x^2), x) - 3/8*integrate(a*x*log(a*
x - 1)^2/(a*x^3 + x^2), x)*log(a) - 3/8*integrate(log(a*x - 1)^2/(a*x^3 + x^2), x)*log(a) + 3/4*integrate(log(
a*x - 1)*log(x)/(a*x^3 + x^2), x)*log(a) - 3/8*integrate(log(x)^2/(a*x^3 + x^2), x)*log(a) - 3/8*(a^2*log(a*x
- 1) - a^2*log(x) + a/x)*log(-1/(a*x) + 1)^2/a + 1/8*log(-1/(a*x) + 1)^3/x - 1/8*((a*x + 1)*log(a*x + 1)^3 - 3
*(2*a*x*log(x) - (a*x - 1)*log(a*x - 1))*log(a*x + 1)^2)/x + 1/8*(3*(a^3*x*log(a*x - 1)^2 + a^3*x*log(x)^2 - 2
*a^3*x*log(x) + 2*a^2 - 2*(a^3*x*log(x) - a^3*x)*log(a*x - 1))*log(-1/(a*x) + 1)/(a*x) - (a^4*x*log(a*x - 1)^3
- a^4*x*log(x)^3 + 3*a^4*x*log(x)^2 - 6*a^4*x*log(x) + 6*a^3 - 3*(a^4*x*log(x) - a^4*x)*log(a*x - 1)^2 + 3*(a
^4*x*log(x)^2 - 2*a^4*x*log(x) + 2*a^4*x)*log(a*x - 1))/(a^2*x))/a + 3/8*integrate(log(a*x + 1)*log(a*x - 1)^2
/(a*x^3 + x^2), x) - 3/8*integrate(log(a*x - 1)^2*log(x)/(a*x^3 + x^2), x) + 3/8*integrate(log(a*x - 1)*log(x)
^2/(a*x^3 + x^2), x) - 1/8*integrate(log(x)^3/(a*x^3 + x^2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcoth}\left (a x\right )^{3}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccoth(a*x)^3/x^2,x, algorithm="fricas")
[Out]
integral(arccoth(a*x)^3/x^2, 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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(acoth(a*x)**3/x**2,x)
[Out]
Integral(acoth(a*x)**3/x**2, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccoth(a*x)^3/x^2,x, algorithm="giac")
[Out]
integrate(arccoth(a*x)^3/x^2, x)