∫ coth−1 (ax)3 _________________

3.1.31 \(\int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx\) [31]

Optimal. Leaf size=95 \[ \frac {3}{2} a^2 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+3 a^2 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]

[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))

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Rubi [A]
time = 0.15, 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 = {6038, 6130, 6136, 6080, 2497, 6096} \begin {gather*} -\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} \end {gather*}

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 2497

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 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]

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*}

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Mathematica [A]
time = 0.11, size = 79, normalized size = 0.83 \begin {gather*} \frac {1}{2} \left (\frac {\coth ^{-1}(a x) \left (3 a x (-1+a x) \coth ^{-1}(a x)+\left (-1+a^2 x^2\right ) \coth ^{-1}(a x)^2+6 a^2 x^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )\right )}{x^2}-3 a^2 \text {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.63, size = 3502, normalized size = 36.86


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(3502\)
default	\(\text {Expression too large to display}\)	\(3502\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(3/8*I*Pi*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^3*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))+3/8*I*Pi*csgn

(I/(a*x-1)*(a*x+1))^3*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))+3/8*I*Pi*csgn(I/(a*x-1)*(a*x+1))^3*dilog(1-I/((a*x-1)

/(a*x+1))^(1/2))+3/8*I*Pi*csgn(I/(a*x-1)*(a*x+1))^3*arccoth(a*x)^2-3/16*I*Pi*csgn(I/(a*x-1)*(a*x+1))^3*polylog

(2,-1/(a*x-1)*(a*x+1))+3/8*I*Pi*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^3*arccoth(a*x)^2-3/16*I*Pi*csgn(

I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^3*polylog(2,-1/(a*x-1)*(a*x+1))+3/8*I*Pi*csgn(I/(a*x-1)*(a*x+1)/(1/(a

*x-1)*(a*x+1)-1))^3*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))-3/2/a/x*arccoth(a*x)^2+3/8*I*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*arccoth(a*x)*ln(1+1/(a*x-1)*(a*x+1))-3/8*I*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))*arccoth(a*x)*ln(1+1/(a*x-

1)*(a*x+1))+1/2*arccoth(a*x)^3-3/2*arccoth(a*x)^2+3/8*I*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))*arccoth(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1/2))+3/4*I*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))-3/8*I*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*arccoth(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1/2))+

3/8*I*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)

)+3/8*I*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*arccoth(a*x)*ln(1+1/(

a*x-1)*(a*x+1))-3/16*I*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))*polylog(2,-1/(a*x-1)*(a*x+1))-3/8*I*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*arccoth(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1/2))-3/8*I*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))+3/8*I*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))*arccoth(a*x)^2+3/16*I*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*polylog(2,-1/(a*x-1)*(a*x+1))-3/8*I*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*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))-3/8*I*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*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))+3/8*I*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*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))*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))-

3/8*I*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*arccoth(a*x)*ln(1-I/((a*x-1)/

(a*x+1))^(1/2))+3/8*I*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))*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))-3/8*I*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*arccoth(a*x)*ln(1-I/((a*x-1)/(a*x+1))^(1/2))+3/8*I*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*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))-3/4*I*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*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))*arccoth(a*x)*ln(1-I/((a*x-1)/(a*x

+1))^(1/2))-1/2/a^2/x^2*arccoth(a*x)^3-3/4*arccoth(a*x)^2*ln(a*x-1)+3/4*arccoth(a*x)^2*ln(a*x+1)-3/4*I*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*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))-3/8*I*Pi*csgn(I/(a*x-1)*(a*x+1))^3*arccoth(a*x)*ln(1+1

/(a*x-1)*(a*x+1))-3/8*I*Pi*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^3*arccoth(a*x)*ln(1+1/(a*x-1)*(a*x+1)

)+3/8*I*Pi*csgn(I/(a*x-1)*(a*x+1)/(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*

Pi*csgn(I/(a*x-1)*(a*x+1)/(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*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*arccoth(a*x)^2-3/8*I*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*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))-3/8*I*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*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))+3/8*

I*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*Pi*csg

n(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*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))+3/2*dilog(1-I/((a*x-1)/(a*x+

1))^(1/2))+3/4*polylog(2,-1/(a*x-1)*(a*x+1))+3/2*arccoth(a*x)*ln(1+1/(a*x-1)*(a*x+1))+3/2*arccoth(a*x)*ln(1+I/

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (84) = 168\).
time = 0.27, size = 252, normalized size = 2.65 \begin {gather*} \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 \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} - \frac {24 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\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 \left (x\right )\right )} a \operatorname {arcoth}\left (a x\right )\right )} a - \frac {\operatorname {arcoth}\left (a x\right )^{3}}{2 \, x^{2}} \end {gather*}

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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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^3,x, algorithm="fricas")

[Out]

integral(arccoth(a*x)^3/x^3, x)

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

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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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^3,x, algorithm="giac")

[Out]

integrate(arccoth(a*x)^3/x^3, x)

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

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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