3.24 \(\int x^4 \coth ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=196 \[ \frac{3 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{10 a^5}-\frac{3 \coth ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{5 a^5}+\frac{x^2}{20 a^3}+\frac{\log \left (1-a^2 x^2\right )}{2 a^5}+\frac{x^3 \coth ^{-1}(a x)}{10 a^2}+\frac{3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac{9 x \coth ^{-1}(a x)}{10 a^4}+\frac{\coth ^{-1}(a x)^3}{5 a^5}-\frac{9 \coth ^{-1}(a x)^2}{20 a^5}-\frac{3 \log \left (\frac{2}{1-a x}\right ) \coth ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \coth ^{-1}(a x)^3+\frac{3 x^4 \coth ^{-1}(a x)^2}{20 a} \]

[Out]

x^2/(20*a^3) + (9*x*ArcCoth[a*x])/(10*a^4) + (x^3*ArcCoth[a*x])/(10*a^2) - (9*ArcCoth[a*x]^2)/(20*a^5) + (3*x^
2*ArcCoth[a*x]^2)/(10*a^3) + (3*x^4*ArcCoth[a*x]^2)/(20*a) + ArcCoth[a*x]^3/(5*a^5) + (x^5*ArcCoth[a*x]^3)/5 -
(3*ArcCoth[a*x]^2*Log[2/(1 - a*x)])/(5*a^5) + Log[1 - a^2*x^2]/(2*a^5) - (3*ArcCoth[a*x]*PolyLog[2, 1 - 2/(1
- a*x)])/(5*a^5) + (3*PolyLog[3, 1 - 2/(1 - a*x)])/(10*a^5)

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Rubi [A]  time = 0.580282, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 11, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.1, Rules used = {5917, 5981, 266, 43, 5911, 260, 5949, 5985, 5919, 6059, 6610} \[ \frac{3 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{10 a^5}-\frac{3 \coth ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{5 a^5}+\frac{x^2}{20 a^3}+\frac{\log \left (1-a^2 x^2\right )}{2 a^5}+\frac{x^3 \coth ^{-1}(a x)}{10 a^2}+\frac{3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac{9 x \coth ^{-1}(a x)}{10 a^4}+\frac{\coth ^{-1}(a x)^3}{5 a^5}-\frac{9 \coth ^{-1}(a x)^2}{20 a^5}-\frac{3 \log \left (\frac{2}{1-a x}\right ) \coth ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \coth ^{-1}(a x)^3+\frac{3 x^4 \coth ^{-1}(a x)^2}{20 a} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCoth[a*x]^3,x]

[Out]

x^2/(20*a^3) + (9*x*ArcCoth[a*x])/(10*a^4) + (x^3*ArcCoth[a*x])/(10*a^2) - (9*ArcCoth[a*x]^2)/(20*a^5) + (3*x^
2*ArcCoth[a*x]^2)/(10*a^3) + (3*x^4*ArcCoth[a*x]^2)/(20*a) + ArcCoth[a*x]^3/(5*a^5) + (x^5*ArcCoth[a*x]^3)/5 -
(3*ArcCoth[a*x]^2*Log[2/(1 - a*x)])/(5*a^5) + Log[1 - a^2*x^2]/(2*a^5) - (3*ArcCoth[a*x]*PolyLog[2, 1 - 2/(1
- a*x)])/(5*a^5) + (3*PolyLog[3, 1 - 2/(1 - a*x)])/(10*a^5)

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 5981

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 5911

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcCoth[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 5985

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]

Rule 5919

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 6059

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[((a + b*ArcC
oth[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 x^4 \coth ^{-1}(a x)^3 \, dx &=\frac{1}{5} x^5 \coth ^{-1}(a x)^3-\frac{1}{5} (3 a) \int \frac{x^5 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=\frac{1}{5} x^5 \coth ^{-1}(a x)^3+\frac{3 \int x^3 \coth ^{-1}(a x)^2 \, dx}{5 a}-\frac{3 \int \frac{x^3 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{5 a}\\ &=\frac{3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac{1}{5} x^5 \coth ^{-1}(a x)^3-\frac{3}{10} \int \frac{x^4 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac{3 \int x \coth ^{-1}(a x)^2 \, dx}{5 a^3}-\frac{3 \int \frac{x \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{5 a^3}\\ &=\frac{3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac{3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac{\coth ^{-1}(a x)^3}{5 a^5}+\frac{1}{5} x^5 \coth ^{-1}(a x)^3-\frac{3 \int \frac{\coth ^{-1}(a x)^2}{1-a x} \, dx}{5 a^4}+\frac{3 \int x^2 \coth ^{-1}(a x) \, dx}{10 a^2}-\frac{3 \int \frac{x^2 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{10 a^2}-\frac{3 \int \frac{x^2 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^2}\\ &=\frac{x^3 \coth ^{-1}(a x)}{10 a^2}+\frac{3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac{3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac{\coth ^{-1}(a x)^3}{5 a^5}+\frac{1}{5} x^5 \coth ^{-1}(a x)^3-\frac{3 \coth ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{5 a^5}+\frac{3 \int \coth ^{-1}(a x) \, dx}{10 a^4}-\frac{3 \int \frac{\coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{10 a^4}+\frac{3 \int \coth ^{-1}(a x) \, dx}{5 a^4}-\frac{3 \int \frac{\coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^4}+\frac{6 \int \frac{\coth ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4}-\frac{\int \frac{x^3}{1-a^2 x^2} \, dx}{10 a}\\ &=\frac{9 x \coth ^{-1}(a x)}{10 a^4}+\frac{x^3 \coth ^{-1}(a x)}{10 a^2}-\frac{9 \coth ^{-1}(a x)^2}{20 a^5}+\frac{3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac{3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac{\coth ^{-1}(a x)^3}{5 a^5}+\frac{1}{5} x^5 \coth ^{-1}(a x)^3-\frac{3 \coth ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{5 a^5}-\frac{3 \coth ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{5 a^5}+\frac{3 \int \frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4}-\frac{3 \int \frac{x}{1-a^2 x^2} \, dx}{10 a^3}-\frac{3 \int \frac{x}{1-a^2 x^2} \, dx}{5 a^3}-\frac{\operatorname{Subst}\left (\int \frac{x}{1-a^2 x} \, dx,x,x^2\right )}{20 a}\\ &=\frac{9 x \coth ^{-1}(a x)}{10 a^4}+\frac{x^3 \coth ^{-1}(a x)}{10 a^2}-\frac{9 \coth ^{-1}(a x)^2}{20 a^5}+\frac{3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac{3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac{\coth ^{-1}(a x)^3}{5 a^5}+\frac{1}{5} x^5 \coth ^{-1}(a x)^3-\frac{3 \coth ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{5 a^5}+\frac{9 \log \left (1-a^2 x^2\right )}{20 a^5}-\frac{3 \coth ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{5 a^5}+\frac{3 \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{10 a^5}-\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{a^2}-\frac{1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )}{20 a}\\ &=\frac{x^2}{20 a^3}+\frac{9 x \coth ^{-1}(a x)}{10 a^4}+\frac{x^3 \coth ^{-1}(a x)}{10 a^2}-\frac{9 \coth ^{-1}(a x)^2}{20 a^5}+\frac{3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac{3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac{\coth ^{-1}(a x)^3}{5 a^5}+\frac{1}{5} x^5 \coth ^{-1}(a x)^3-\frac{3 \coth ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{5 a^5}+\frac{\log \left (1-a^2 x^2\right )}{2 a^5}-\frac{3 \coth ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{5 a^5}+\frac{3 \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{10 a^5}\\ \end{align*}

Mathematica [C]  time = 0.566502, size = 175, normalized size = 0.89 \[ \frac{-24 \coth ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )+12 \text{PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right )+2 a^2 x^2-40 \log \left (\frac{1}{a x \sqrt{1-\frac{1}{a^2 x^2}}}\right )+8 a^5 x^5 \coth ^{-1}(a x)^3+6 a^4 x^4 \coth ^{-1}(a x)^2+4 a^3 x^3 \coth ^{-1}(a x)+12 a^2 x^2 \coth ^{-1}(a x)^2+36 a x \coth ^{-1}(a x)+8 \coth ^{-1}(a x)^3-18 \coth ^{-1}(a x)^2-24 \coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-i \pi ^3-2}{40 a^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4*ArcCoth[a*x]^3,x]

[Out]

(-2 - I*Pi^3 + 2*a^2*x^2 + 36*a*x*ArcCoth[a*x] + 4*a^3*x^3*ArcCoth[a*x] - 18*ArcCoth[a*x]^2 + 12*a^2*x^2*ArcCo
th[a*x]^2 + 6*a^4*x^4*ArcCoth[a*x]^2 + 8*ArcCoth[a*x]^3 + 8*a^5*x^5*ArcCoth[a*x]^3 - 24*ArcCoth[a*x]^2*Log[1 -
E^(2*ArcCoth[a*x])] - 40*Log[1/(a*Sqrt[1 - 1/(a^2*x^2)]*x)] - 24*ArcCoth[a*x]*PolyLog[2, E^(2*ArcCoth[a*x])]
+ 12*PolyLog[3, E^(2*ArcCoth[a*x])])/(40*a^5)

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Maple [C]  time = 1.106, size = 806, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccoth(a*x)^3,x)

[Out]

1/5*arccoth(a*x)^3/a^5-1/20/a^5-9/20*arccoth(a*x)^2/a^5+1/a^5*arccoth(a*x)+6/5/a^5*polylog(3,1/((a*x-1)/(a*x+1
))^(1/2))-1/a^5*ln(1+1/((a*x-1)/(a*x+1))^(1/2))-1/a^5*ln(1/((a*x-1)/(a*x+1))^(1/2)-1)+6/5/a^5*polylog(3,-1/((a
*x-1)/(a*x+1))^(1/2))+1/20*x^2/a^3+1/5*x^5*arccoth(a*x)^3+3/10*x^2*arccoth(a*x)^2/a^3+9/10*x*arccoth(a*x)/a^4+
1/10*x^3*arccoth(a*x)/a^2+3/20*x^4*arccoth(a*x)^2/a+3/20*I/a^5*Pi*csgn(I*(a*x+1)/(a*x-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))*arccoth(a*x)^2-3/5/a^5*arccoth(a*x)^2*ln(2)+3/10/a^5*arcco
th(a*x)^2*ln(a*x-1)+3/10/a^5*arccoth(a*x)^2*ln(a*x+1)+3/10/a^5*arccoth(a*x)^2*ln((a*x-1)/(a*x+1))+3/5/a^5*arcc
oth(a*x)^2*ln((a*x+1)/(a*x-1)-1)-3/5/a^5*arccoth(a*x)^2*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-6/5/a^5*arccoth(a*x)*p
olylog(2,1/((a*x-1)/(a*x+1))^(1/2))-3/5/a^5*arccoth(a*x)^2*ln(1+1/((a*x-1)/(a*x+1))^(1/2))-6/5/a^5*arccoth(a*x
)*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))-3/20*I/a^5*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*csgn(I/((a
*x+1)/(a*x-1)-1))*arccoth(a*x)^2-3/20*I/a^5*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^2*csgn(I*(a*x+1)/(a
*x-1))*arccoth(a*x)^2-3/10*I/a^5*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
/20*I/a^5*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth(a*x)^2+3/20*I/a^5*Pi*csgn(I*(a*
x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3*arccoth(a*x)^2+3/20*I/a^5*Pi*csgn(I*(a*x+1)/(a*x-1))^3*arccoth(a*x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (a^{5} x^{5} + 1\right )} \log \left (a x + 1\right )^{3} + 3 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} - 2 \,{\left (a^{5} x^{5} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )^{2}}{80 \, a^{5}} + \frac{1}{8} \, \int -\frac{5 \,{\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (a x - 1\right )^{3} + 3 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} - 5 \,{\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (a x - 1\right )^{2} - 2 \,{\left (a^{5} x^{5} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )}{5 \,{\left (a^{5} x + a^{4}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccoth(a*x)^3,x, algorithm="maxima")

[Out]

1/80*(2*(a^5*x^5 + 1)*log(a*x + 1)^3 + 3*(a^4*x^4 + 2*a^2*x^2 - 2*(a^5*x^5 - 1)*log(a*x - 1))*log(a*x + 1)^2)/
a^5 + 1/8*integrate(-1/5*(5*(a^5*x^5 + a^4*x^4)*log(a*x - 1)^3 + 3*(a^4*x^4 + 2*a^2*x^2 - 5*(a^5*x^5 + a^4*x^4
)*log(a*x - 1)^2 - 2*(a^5*x^5 - 1)*log(a*x - 1))*log(a*x + 1))/(a^5*x + a^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{4} \operatorname{arcoth}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccoth(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^4*arccoth(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{acoth}^{3}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acoth(a*x)**3,x)

[Out]

Integral(x**4*acoth(a*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{arcoth}\left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccoth(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^4*arccoth(a*x)^3, x)