3.1.24 \(\int x^4 \coth ^{-1}(a x)^3 \, dx\) [24]

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

[Out]

1/20*x^2/a^3+9/10*x*arccoth(a*x)/a^4+1/10*x^3*arccoth(a*x)/a^2-9/20*arccoth(a*x)^2/a^5+3/10*x^2*arccoth(a*x)^2
/a^3+3/20*x^4*arccoth(a*x)^2/a+1/5*arccoth(a*x)^3/a^5+1/5*x^5*arccoth(a*x)^3-3/5*arccoth(a*x)^2*ln(2/(-a*x+1))
/a^5+1/2*ln(-a^2*x^2+1)/a^5-3/5*arccoth(a*x)*polylog(2,1-2/(-a*x+1))/a^5+3/10*polylog(3,1-2/(-a*x+1))/a^5

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Rubi [A]
time = 0.39, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6038, 6128, 272, 45, 6022, 266, 6096, 6132, 6056, 6206, 6745} \begin {gather*} \frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{10 a^5}-\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{5 a^5}+\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 {9 x \coth ^{-1}(a x)}{10 a^4}+\frac {x^2}{20 a^3}+\frac {3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac {x^3 \coth ^{-1}(a x)}{10 a^2}+\frac {\log \left (1-a^2 x^2\right )}{2 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^3+\frac {3 x^4 \coth ^{-1}(a x)^2}{20 a} \end {gather*}

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 45

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 266

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 272

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 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]

Rule 6206

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 6745

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 {\text {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 {\text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.41, size = 175, normalized size = 0.89 \begin {gather*} \frac {-2-i \pi ^3+2 a^2 x^2+36 a x \coth ^{-1}(a x)+4 a^3 x^3 \coth ^{-1}(a x)-18 \coth ^{-1}(a x)^2+12 a^2 x^2 \coth ^{-1}(a x)^2+6 a^4 x^4 \coth ^{-1}(a x)^2+8 \coth ^{-1}(a x)^3+8 a^5 x^5 \coth ^{-1}(a x)^3-24 \coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-40 \log \left (\frac {1}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\right )-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 )}{40 a^5} \end {gather*}

Antiderivative was successfully verified.

[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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 4.97, size = 737, normalized size = 3.76 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccoth(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^5*(9/10*a*x*arccoth(a*x)+3/20*arccoth(a*x)^2*a^4*x^4+3/10*a^2*x^2*arccoth(a*x)^2+1/10*a^3*x^3*arccoth(a*x)
+3/20*I*arccoth(a*x)^2*csgn(I/(1/(a*x-1)*(a*x+1)-1))*Pi*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))*csgn(I/(
a*x-1)*(a*x+1))-3/5*arccoth(a*x)^2*ln(2)+1/20*a^2*x^2-1/20-ln(1+1/((a*x-1)/(a*x+1))^(1/2))+1/5*arccoth(a*x)^3-
9/20*arccoth(a*x)^2+arccoth(a*x)-3/20*I*arccoth(a*x)^2*csgn(I/(1/(a*x-1)*(a*x+1)-1))*Pi*csgn(I/(a*x-1)*(a*x+1)
/(1/(a*x-1)*(a*x+1)-1))^2-3/20*I*arccoth(a*x)^2*Pi*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^2*csgn(I/(a*x
-1)*(a*x+1))-3/10*I*arccoth(a*x)^2*Pi*csgn(I/(a*x-1)*(a*x+1))^2*csgn(I/((a*x-1)/(a*x+1))^(1/2))+3/20*I*arccoth
(a*x)^2*Pi*csgn(I/(a*x-1)*(a*x+1))*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2+1/5*a^5*x^5*arccoth(a*x)^3+3/5*arccoth(a*
x)^2*ln(1/(a*x-1)*(a*x+1)-1)-3/5*arccoth(a*x)^2*ln(1-1/((a*x-1)/(a*x+1))^(1/2))+3/20*I*arccoth(a*x)^2*Pi*csgn(
I/(a*x-1)*(a*x+1))^3+3/20*I*arccoth(a*x)^2*Pi*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^3+3/10*arccoth(a*x
)^2*ln(a*x-1)+3/10*arccoth(a*x)^2*ln(a*x+1)+6/5*polylog(3,1/((a*x-1)/(a*x+1))^(1/2))-ln(1/((a*x-1)/(a*x+1))^(1
/2)-1)-6/5*arccoth(a*x)*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))-3/5*arccoth(a*x)^2*ln(1+1/((a*x-1)/(a*x+1))^(1/2)
)-6/5*arccoth(a*x)*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))+6/5*polylog(3,-1/((a*x-1)/(a*x+1))^(1/2))+3/10*arccot
h(a*x)^2*ln((a*x-1)/(a*x+1)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, 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^4*arccoth(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^4*arccoth(a*x)^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \operatorname {acoth}^{3}{\left (a x \right )}\, dx \end {gather*}

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 = 0.00, 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^4*arccoth(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^4*arccoth(a*x)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*acoth(a*x)^3, x)

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