∫ x coth−1(ax)3 dx [27]

3.1.27 \(\int x \coth ^{-1}(a x)^3 \, dx\) [27]

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

[Out]

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

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

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6038, 6128, 6022, 6132, 6056, 2449, 2352, 6096} \begin {gather*} -\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^2}-\frac {\coth ^{-1}(a x)^3}{2 a^2}+\frac {3 \coth ^{-1}(a x)^2}{2 a^2}-\frac {3 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a^2}+\frac {1}{2} x^2 \coth ^{-1}(a x)^3+\frac {3 x \coth ^{-1}(a x)^2}{2 a} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[x*ArcCoth[a*x]^3,x]

[Out]

(3*ArcCoth[a*x]^2)/(2*a^2) + (3*x*ArcCoth[a*x]^2)/(2*a) - ArcCoth[a*x]^3/(2*a^2) + (x^2*ArcCoth[a*x]^3)/2 - (3

*ArcCoth[a*x]*Log[2/(1 - a*x)])/a^2 - (3*PolyLog[2, 1 - 2/(1 - a*x)])/(2*a^2)

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e

}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*

d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 6022

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x^n])^p, x] - Dist[b

*c*n*p, Int[x^n*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p

, 0] && (EqQ[n, 1] || EqQ[p, 1])

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 6056

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)

*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[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 6128

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2

/e, Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[d*(f^2/e), 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] && GtQ[m, 1]

Rule 6132

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

*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,

d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x \coth ^{-1}(a x)^3 \, dx &=\frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {1}{2} (3 a) \int \frac {x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=\frac {1}{2} x^2 \coth ^{-1}(a x)^3+\frac {3 \int \coth ^{-1}(a x)^2 \, dx}{2 a}-\frac {3 \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a}\\ &=\frac {3 x \coth ^{-1}(a x)^2}{2 a}-\frac {\coth ^{-1}(a x)^3}{2 a^2}+\frac {1}{2} x^2 \coth ^{-1}(a x)^3-3 \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {3 \coth ^{-1}(a x)^2}{2 a^2}+\frac {3 x \coth ^{-1}(a x)^2}{2 a}-\frac {\coth ^{-1}(a x)^3}{2 a^2}+\frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3 \int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{a}\\ &=\frac {3 \coth ^{-1}(a x)^2}{2 a^2}+\frac {3 x \coth ^{-1}(a x)^2}{2 a}-\frac {\coth ^{-1}(a x)^3}{2 a^2}+\frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^2}+\frac {3 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a}\\ &=\frac {3 \coth ^{-1}(a x)^2}{2 a^2}+\frac {3 x \coth ^{-1}(a x)^2}{2 a}-\frac {\coth ^{-1}(a x)^3}{2 a^2}+\frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^2}-\frac {3 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{a^2}\\ &=\frac {3 \coth ^{-1}(a x)^2}{2 a^2}+\frac {3 x \coth ^{-1}(a x)^2}{2 a}-\frac {\coth ^{-1}(a x)^3}{2 a^2}+\frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^2}-\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^2}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 68, normalized size = 0.72 \begin {gather*} \frac {\coth ^{-1}(a x) \left (3 (-1+a x) \coth ^{-1}(a x)+\left (-1+a^2 x^2\right ) \coth ^{-1}(a x)^2-6 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+3 \text {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*ArcCoth[a*x]^3,x]

[Out]

(ArcCoth[a*x]*(3*(-1 + a*x)*ArcCoth[a*x] + (-1 + a^2*x^2)*ArcCoth[a*x]^2 - 6*Log[1 - E^(-2*ArcCoth[a*x])]) + 3

*PolyLog[2, E^(-2*ArcCoth[a*x])])/(2*a^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.87, size = 2916, normalized size = 30.69


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(2916\)
default	\(\text {Expression too large to display}\)	\(2916\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

)+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))^(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+1/((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/((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)^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*polylog(2,1/((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(1-1/((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/(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))^(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*polylog(2,-1/((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)^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))*polylog(2,1/((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(1-1/

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

n(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/((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*arc

coth(a*x)*ln(1-1/((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))*a

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

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

ccoth(a*x)^2+1/2*a^2*x^2*arccoth(a*x)^3+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))-3/2*dilog(1+1/((a*x-1)/(a*

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

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

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

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

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

sgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I/(a*x-1)*(a*x+1))*dilog(1+1/((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/((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)^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+1/((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/((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+1/((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/((a*x-1)/(a*x+1))^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (82) = 164\).
time = 0.26, size = 215, normalized size = 2.26 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (a x\right )^{3} + \frac {3}{4} \, a {\left (\frac {2 \, x}{a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{16} \, a {\left (\frac {\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}}{a^{2}} - \frac {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 )\right )} \operatorname {arcoth}\left (a x\right )}{a^{3}}\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccoth(a*x)^3,x, algorithm="maxima")

[Out]

1/2*x^2*arccoth(a*x)^3 + 3/4*a*(2*x/a^2 - log(a*x + 1)/a^3 + log(a*x - 1)/a^3)*arccoth(a*x)^2 + 1/16*a*(((3*(l

og(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)/a^2 - 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))*arccoth(a*x)/a^3)

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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(x*arccoth(a*x)^3,x, algorithm="fricas")

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*acoth(a*x)**3,x)

[Out]

Integral(x*acoth(a*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(x*arccoth(a*x)^3,x, algorithm="giac")

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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