\(\int x^3 \coth ^{-1}(a x)^3 \, dx\) [25]

Optimal result

Integrand size = 10, antiderivative size = 139 \[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\frac {x}{4 a^3}+\frac {x^2 \coth ^{-1}(a x)}{4 a^2}+\frac {\coth ^{-1}(a x)^2}{a^4}+\frac {3 x \coth ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \coth ^{-1}(a x)^2}{4 a}-\frac {\coth ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {\text {arctanh}(a x)}{4 a^4}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6038, 6128, 327, 212, 6132, 6056, 2449, 2352, 6022, 6096} \[ \int x^3 \coth ^{-1}(a x)^3 \, dx=-\frac {\text {arctanh}(a x)}{4 a^4}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^4}-\frac {\coth ^{-1}(a x)^3}{4 a^4}+\frac {\coth ^{-1}(a x)^2}{a^4}-\frac {2 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a^4}+\frac {x}{4 a^3}+\frac {3 x \coth ^{-1}(a x)^2}{4 a^3}+\frac {x^2 \coth ^{-1}(a x)}{4 a^2}+\frac {1}{4} x^4 \coth ^{-1}(a x)^3+\frac {x^3 \coth ^{-1}(a x)^2}{4 a} \]

[In]

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

[Out]

x/(4*a^3) + (x^2*ArcCoth[a*x])/(4*a^2) + ArcCoth[a*x]^2/a^4 + (3*x*ArcCoth[a*x]^2)/(4*a^3) + (x^3*ArcCoth[a*x]
^2)/(4*a) - ArcCoth[a*x]^3/(4*a^4) + (x^4*ArcCoth[a*x]^3)/4 - ArcTanh[a*x]/(4*a^4) - (2*ArcCoth[a*x]*Log[2/(1
- a*x)])/a^4 - PolyLog[2, 1 - 2/(1 - a*x)]/a^4

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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*} \text {integral}& = \frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^4 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx \\ & = \frac {1}{4} x^4 \coth ^{-1}(a x)^3+\frac {3 \int x^2 \coth ^{-1}(a x)^2 \, dx}{4 a}-\frac {3 \int \frac {x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{4 a} \\ & = \frac {x^3 \coth ^{-1}(a x)^2}{4 a}+\frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {1}{2} \int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {3 \int \coth ^{-1}(a x)^2 \, dx}{4 a^3}-\frac {3 \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{4 a^3} \\ & = \frac {3 x \coth ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \coth ^{-1}(a x)^2}{4 a}-\frac {\coth ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^3+\frac {\int x \coth ^{-1}(a x) \, dx}{2 a^2}-\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{2 a^2}-\frac {3 \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{2 a^2} \\ & = \frac {x^2 \coth ^{-1}(a x)}{4 a^2}+\frac {\coth ^{-1}(a x)^2}{a^4}+\frac {3 x \coth ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \coth ^{-1}(a x)^2}{4 a}-\frac {\coth ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {\int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{2 a^3}-\frac {3 \int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{2 a^3}-\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{4 a} \\ & = \frac {x}{4 a^3}+\frac {x^2 \coth ^{-1}(a x)}{4 a^2}+\frac {\coth ^{-1}(a x)^2}{a^4}+\frac {3 x \coth ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \coth ^{-1}(a x)^2}{4 a}-\frac {\coth ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{4 a^3}+\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{2 a^3}+\frac {3 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{2 a^3} \\ & = \frac {x}{4 a^3}+\frac {x^2 \coth ^{-1}(a x)}{4 a^2}+\frac {\coth ^{-1}(a x)^2}{a^4}+\frac {3 x \coth ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \coth ^{-1}(a x)^2}{4 a}-\frac {\coth ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {\text {arctanh}(a x)}{4 a^4}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {\text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{2 a^4}-\frac {3 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{2 a^4} \\ & = \frac {x}{4 a^3}+\frac {x^2 \coth ^{-1}(a x)}{4 a^2}+\frac {\coth ^{-1}(a x)^2}{a^4}+\frac {3 x \coth ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \coth ^{-1}(a x)^2}{4 a}-\frac {\coth ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^3-\frac {\text {arctanh}(a x)}{4 a^4}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.63 \[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\frac {a x+\left (-4+3 a x+a^3 x^3\right ) \coth ^{-1}(a x)^2+\left (-1+a^4 x^4\right ) \coth ^{-1}(a x)^3+\coth ^{-1}(a x) \left (-1+a^2 x^2-8 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+4 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{4 a^4} \]

[In]

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

[Out]

(a*x + (-4 + 3*a*x + a^3*x^3)*ArcCoth[a*x]^2 + (-1 + a^4*x^4)*ArcCoth[a*x]^3 + ArcCoth[a*x]*(-1 + a^2*x^2 - 8*
Log[1 - E^(-2*ArcCoth[a*x])]) + 4*PolyLog[2, E^(-2*ArcCoth[a*x])])/(4*a^4)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 4.40 (sec) , antiderivative size = 871, normalized size of antiderivative = 6.27

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(871\)
default	\(\text {Expression too large to display}\)	\(871\)
parts	\(\text {Expression too large to display}\)	\(871\)

[In]

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

[Out]

1/a^4*(3/16*I*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+3/16
*I*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)^2-3/16*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/((a*x-1)/(a*x+1))^(1/2))*csgn(
I*(a*x+1)/(a*x-1))^2*arccoth(a*x)^2+3/4*arccoth(a*x)^2*a*x+1/4*a^3*x^3*arccoth(a*x)^2-3/16*I*Pi*csgn(I/((a*x+1
)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))*arccoth(a*x)^2-2*arccoth(a*x
)*ln(1+1/((a*x-1)/(a*x+1))^(1/2))-1/4*arccoth(a*x)^3-1/8*(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)-
a*x-1)*arccoth(a*x)-1/4/(((a*x-1)/(a*x+1))^(1/2)-1)*((a*x-1)/(a*x+1))^(1/2)+1/8*(((a*x-1)/(a*x+1))^(1/2)*a*x+(
(a*x-1)/(a*x+1))^(1/2)+a*x+1)*arccoth(a*x)-1/4/(((a*x-1)/(a*x+1))^(1/2)+1)*((a*x-1)/(a*x+1))^(1/2)+arccoth(a*x
)^2+1/4*a^4*x^4*arccoth(a*x)^3-1/4*(((a*x-1)/(a*x+1))^(1/2)*a*x-a*x+1)*arccoth(a*x)*(a*x+1)+1/8*(((a*x-1)/(a*x
+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)-a*x)*arccoth(a*x)*(a*x+1)+1/4*(((a*x-1)/(a*x+1))^(1/2)*a*x+a*x-1)*arcco
th(a*x)*(a*x+1)-1/8*(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)+a*x)*arccoth(a*x)*(a*x+1)-3/16*I*Pi*c
sgn(I*(a*x+1)/(a*x-1))^3*arccoth(a*x)^2-3/16*I*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^3*arccoth(a*x)^2
+2*dilog(1/((a*x-1)/(a*x+1))^(1/2))-2*dilog(1+1/((a*x-1)/(a*x+1))^(1/2))-3/8*arccoth(a*x)^2*ln((a*x-1)/(a*x+1)
)+3/8*arccoth(a*x)^2*ln(a*x-1)-3/8*arccoth(a*x)^2*ln(a*x+1))

Fricas [F]

\[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\int { x^{3} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

integral(x^3*arccoth(a*x)^3, x)

Sympy [F]

\[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\int x^{3} \operatorname {acoth}^{3}{\left (a x \right )}\, dx \]

[In]

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

[Out]

Integral(x**3*acoth(a*x)**3, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (122) = 244\).

Time = 0.20 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.88 \[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\frac {1}{4} \, x^{4} \operatorname {arcoth}\left (a x\right )^{3} + \frac {1}{8} \, a {\left (\frac {2 \, {\left (a^{2} x^{3} + 3 \, x\right )}}{a^{4}} - \frac {3 \, \log \left (a x + 1\right )}{a^{5}} + \frac {3 \, \log \left (a x - 1\right )}{a^{5}}\right )} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{32} \, a {\left (\frac {\frac {{\left (3 \, \log \left (a x - 1\right ) - 8\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} + 8 \, a x - {\left (3 \, \log \left (a x - 1\right )^{2} - 16 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 8 \, \log \left (a x - 1\right )^{2} + 4 \, \log \left (a x - 1\right )}{a} - \frac {32 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {4 \, \log \left (a x + 1\right )}{a}}{a^{4}} + \frac {2 \, {\left (4 \, a^{2} x^{2} - 2 \, {\left (3 \, \log \left (a x - 1\right ) - 8\right )} \log \left (a x + 1\right ) + 3 \, \log \left (a x + 1\right )^{2} + 3 \, \log \left (a x - 1\right )^{2} + 16 \, \log \left (a x - 1\right )\right )} \operatorname {arcoth}\left (a x\right )}{a^{5}}\right )} \]

[In]

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

[Out]

1/4*x^4*arccoth(a*x)^3 + 1/8*a*(2*(a^2*x^3 + 3*x)/a^4 - 3*log(a*x + 1)/a^5 + 3*log(a*x - 1)/a^5)*arccoth(a*x)^
2 + 1/32*a*((((3*log(a*x - 1) - 8)*log(a*x + 1)^2 - log(a*x + 1)^3 + log(a*x - 1)^3 + 8*a*x - (3*log(a*x - 1)^
2 - 16*log(a*x - 1))*log(a*x + 1) + 8*log(a*x - 1)^2 + 4*log(a*x - 1))/a - 32*(log(a*x - 1)*log(1/2*a*x + 1/2)
+ dilog(-1/2*a*x + 1/2))/a - 4*log(a*x + 1)/a)/a^4 + 2*(4*a^2*x^2 - 2*(3*log(a*x - 1) - 8)*log(a*x + 1) + 3*l
og(a*x + 1)^2 + 3*log(a*x - 1)^2 + 16*log(a*x - 1))*arccoth(a*x)/a^5)

Giac [F]

\[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\int { x^{3} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

integrate(x^3*arccoth(a*x)^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \coth ^{-1}(a x)^3 \, dx=\int x^3\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \]

[In]

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

[Out]

int(x^3*acoth(a*x)^3, x)