3.31 \(\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx\)
Optimal. Leaf size=95 \[ -\frac {3}{2} a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {1}{2} a^2 \coth ^{-1}(a x)^3+\frac {3}{2} a^2 \coth ^{-1}(a x)^2+3 a^2 \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\frac {\coth ^{-1}(a x)^3}{2 x^2}-\frac {3 a \coth ^{-1}(a x)^2}{2 x} \]
[Out]
3/2*a^2*arccoth(a*x)^2-3/2*a*arccoth(a*x)^2/x+1/2*a^2*arccoth(a*x)^3-1/2*arccoth(a*x)^3/x^2+3*a^2*arccoth(a*x)
*ln(2-2/(a*x+1))-3/2*a^2*polylog(2,-1+2/(a*x+1))
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 95, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used =
{5917, 5983, 5989, 5933, 2447, 5949} \[ -\frac {3}{2} a^2 \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {1}{2} a^2 \coth ^{-1}(a x)^3+\frac {3}{2} a^2 \coth ^{-1}(a x)^2+3 a^2 \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\frac {\coth ^{-1}(a x)^3}{2 x^2}-\frac {3 a \coth ^{-1}(a x)^2}{2 x} \]
Antiderivative was successfully verified.
[In]
Int[ArcCoth[a*x]^3/x^3,x]
[Out]
(3*a^2*ArcCoth[a*x]^2)/2 - (3*a*ArcCoth[a*x]^2)/(2*x) + (a^2*ArcCoth[a*x]^3)/2 - ArcCoth[a*x]^3/(2*x^2) + 3*a^
2*ArcCoth[a*x]*Log[2 - 2/(1 + a*x)] - (3*a^2*PolyLog[2, -1 + 2/(1 + a*x)])/2
Rule 2447
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]
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 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 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 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]
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx &=-\frac {\coth ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} (3 a) \int \frac {\coth ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\coth ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} (3 a) \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx+\frac {1}{2} \left (3 a^3\right ) \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=-\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=\frac {3}{2} a^2 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac {\coth ^{-1}(a x)}{x (1+a x)} \, dx\\ &=\frac {3}{2} a^2 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+3 a^2 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\left (3 a^3\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {3}{2} a^2 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+3 a^2 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 79, normalized size = 0.83 \[ \frac {1}{2} \left (\frac {\coth ^{-1}(a x) \left (\left (a^2 x^2-1\right ) \coth ^{-1}(a x)^2+6 a^2 x^2 \log \left (e^{-2 \coth ^{-1}(a x)}+1\right )+3 a x (a x-1) \coth ^{-1}(a x)\right )}{x^2}-3 a^2 \text {Li}_2\left (-e^{-2 \coth ^{-1}(a x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
Integrate[ArcCoth[a*x]^3/x^3,x]
[Out]
((ArcCoth[a*x]*(3*a*x*(-1 + a*x)*ArcCoth[a*x] + (-1 + a^2*x^2)*ArcCoth[a*x]^2 + 6*a^2*x^2*Log[1 + E^(-2*ArcCot
h[a*x])]))/x^2 - 3*a^2*PolyLog[2, -E^(-2*ArcCoth[a*x])])/2
________________________________________________________________________________________
fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccoth(a*x)^3/x^3,x, algorithm="fricas")
[Out]
integral(arccoth(a*x)^3/x^3, x)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccoth(a*x)^3/x^3,x, algorithm="giac")
[Out]
integrate(arccoth(a*x)^3/x^3, x)
________________________________________________________________________________________
maple [C] time = 0.86, size = 3673, normalized size = 38.66 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(arccoth(a*x)^3/x^3,x)
[Out]
1/2*a^2*arccoth(a*x)^3-3/2*a^2*arccoth(a*x)^2-3/2*a*arccoth(a*x)^2/x+3/2*a^2*arccoth(a*x)*ln(1+I/((a*x-1)/(a*x
+1))^(1/2))+3/2*a^2*arccoth(a*x)*ln(1-I/((a*x-1)/(a*x+1))^(1/2))+3/2*a^2*arccoth(a*x)*ln(1+(a*x+1)/(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)+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))*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*arccoth(a*x)*ln(1+(a*x+1)/
(a*x-1))+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+(a*x+1)/(a*x
-1))-3/8*I*a^2*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(a*x)*ln(1-I/((
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-I/((a*x-1)/(a*x+1))^(1/2))-3/8*I*a^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))^2*arccoth(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1/2))-3/8*I*a^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))^2*arccoth(a*x)*ln(1-I/((a*x-1)/(a*x+1))^(1/2))-3/8*I*a^2*Pi*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))*arccoth(a*x)*ln(1+(a*x+1)/(a*x-1))+3/8
*I*a^2*Pi*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))*arcc
oth(a*x)*ln(1-I/((a*x-1)/(a*x+1))^(1/2))+3/8*I*a^2*Pi*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))*arccoth(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1/2))-3/16*I*a^2*Pi*csgn(I*(a*x
+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3*polylog(2,-(a*x+1)/(a*x-1))+3/8*I*a^2*Pi*csgn(I*(a*x+1)/(a*x-1))^3*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))^3*arccoth(a*x)^2-3/16*I*a^2*Pi*csgn(I*(a*x+1)/
(a*x-1))^3*polylog(2,-(a*x+1)/(a*x-1))+3/8*I*a^2*Pi*csgn(I*(a*x+1)/(a*x-1))^3*dilog(1-I/((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+I/((a*x-1)/(a*x+1))^(1/2))+3/8*I*a^2*Pi*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))*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))
+3/2*a^2*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))+3/2*a^2*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))+3/4*a^2*polylog(2,-(a*x
+1)/(a*x-1))+3/8*I*a^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3*dilog(1+I/((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-I/((a*x-1)/(a*x+1))^(1/2))-1/2*arccoth(a*x)^3/
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)*ln(1+(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))*arccoth(a*x)*ln(1-I/((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+I/((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))*arccoth(a*x)*ln(1+I/((a*x-1
)/(a*x+1))^(1/2))-3/16*I*a^2*Pi*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))*polylog(2,-(a*x+1)/(a*x-1))+3/8*I*a^2*Pi*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))*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))+3/8*I*a^2*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(a*x)*ln(1+(a*x+1)/(a*x-1))+3/8*I*a^2*Pi*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))*arccoth(a*x)^2-3/8*I*a^
2*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(a*x)*ln(1+I/((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-I/((a*x-1)/(a*x+1))^(1/2))-3/8*I*a^2*Pi*csg
n(I*(a*x+1)/(a*x-1))*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*dilog(1+I/((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))*arccoth(a*x)^2-3/8*I*a^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))^2*arccoth(a*x)^2-3/8*I*a^2*Pi*csgn(I*(a*x+1)/(a*x-1))*csg
n(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*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))*dilog(1-I/((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
+(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))*dilog(1+I/((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+I/((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-I/((a*x-1)/(a*x+1))^(
1/2))-3/16*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,-(a*x+1)/(a*x-1))-3/8*
I*a^2*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*dilog(1-I/((a*x-1)/(a*x+1))^(1/
2))+3/8*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,-(a*x+1)/(a*x-1))+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+I/((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-I/((a*x-1)/(a*x+1))^(1/2))-3/8*I*a^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))^2*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))+3/16*I*
a^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))^2*polylog(2,-(a*x+1)/(a*x-1))+3
/8*I*a^2*Pi*csgn(I*(a*x+1)/(a*x-1))^3*arccoth(a*x)*ln(1+I/((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+(a*x+1)/(a*x-1))-3/8*I*a^2*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*cs
gn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))+3/16*I*a^2*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*polylog(2,-(a*x+1)/(a*x-1))-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
________________________________________________________________________________________
maxima [B] time = 0.33, size = 252, normalized size = 2.65 \[ \frac {3}{4} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} a \operatorname {arcoth}\left (a x\right )^{2} - \frac {1}{16} \, {\left (a^{2} {\left (\frac {3 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} - 3 \, {\left (\log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 6 \, \log \left (a x - 1\right )^{2}}{a} - \frac {24 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} + \frac {24 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )}}{a} - \frac {24 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} - 6 \, {\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right ) + 8 \, \log \relax (x)\right )} a \operatorname {arcoth}\left (a x\right )\right )} a - \frac {\operatorname {arcoth}\left (a x\right )^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccoth(a*x)^3/x^3,x, algorithm="maxima")
[Out]
3/4*(a*log(a*x + 1) - a*log(a*x - 1) - 2/x)*a*arccoth(a*x)^2 - 1/16*(a^2*((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 + 24*(log(a*x + 1)*log(x) + dilog(-a*x))/a - 24
*(log(-a*x + 1)*log(x) + dilog(a*x))/a) - 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) + 8*log(x))*a*arccoth(a*x))*a - 1/2*arccoth(a*x)^3/x^2
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(acoth(a*x)^3/x^3,x)
[Out]
int(acoth(a*x)^3/x^3, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(acoth(a*x)**3/x**3,x)
[Out]
Integral(acoth(a*x)**3/x**3, x)
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