3.23 \(\int x^5 \coth ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=186 \[ -\frac{23 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{30 a^6}+\frac{x^3}{60 a^3}+\frac{x^4 \coth ^{-1}(a x)}{20 a^2}+\frac{x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac{4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac{19 x}{60 a^5}-\frac{19 \tanh ^{-1}(a x)}{60 a^6}+\frac{x \coth ^{-1}(a x)^2}{2 a^5}-\frac{\coth ^{-1}(a x)^3}{6 a^6}+\frac{23 \coth ^{-1}(a x)^2}{30 a^6}-\frac{23 \log \left (\frac{2}{1-a x}\right ) \coth ^{-1}(a x)}{15 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^3+\frac{x^5 \coth ^{-1}(a x)^2}{10 a} \]

[Out]

(19*x)/(60*a^5) + x^3/(60*a^3) + (4*x^2*ArcCoth[a*x])/(15*a^4) + (x^4*ArcCoth[a*x])/(20*a^2) + (23*ArcCoth[a*x
]^2)/(30*a^6) + (x*ArcCoth[a*x]^2)/(2*a^5) + (x^3*ArcCoth[a*x]^2)/(6*a^3) + (x^5*ArcCoth[a*x]^2)/(10*a) - ArcC
oth[a*x]^3/(6*a^6) + (x^6*ArcCoth[a*x]^3)/6 - (19*ArcTanh[a*x])/(60*a^6) - (23*ArcCoth[a*x]*Log[2/(1 - a*x)])/
(15*a^6) - (23*PolyLog[2, 1 - 2/(1 - a*x)])/(30*a^6)

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Rubi [A]  time = 0.717193, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 11, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.1, Rules used = {5917, 5981, 302, 206, 321, 5985, 5919, 2402, 2315, 5911, 5949} \[ -\frac{23 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{30 a^6}+\frac{x^3}{60 a^3}+\frac{x^4 \coth ^{-1}(a x)}{20 a^2}+\frac{x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac{4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac{19 x}{60 a^5}-\frac{19 \tanh ^{-1}(a x)}{60 a^6}+\frac{x \coth ^{-1}(a x)^2}{2 a^5}-\frac{\coth ^{-1}(a x)^3}{6 a^6}+\frac{23 \coth ^{-1}(a x)^2}{30 a^6}-\frac{23 \log \left (\frac{2}{1-a x}\right ) \coth ^{-1}(a x)}{15 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^3+\frac{x^5 \coth ^{-1}(a x)^2}{10 a} \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcCoth[a*x]^3,x]

[Out]

(19*x)/(60*a^5) + x^3/(60*a^3) + (4*x^2*ArcCoth[a*x])/(15*a^4) + (x^4*ArcCoth[a*x])/(20*a^2) + (23*ArcCoth[a*x
]^2)/(30*a^6) + (x*ArcCoth[a*x]^2)/(2*a^5) + (x^3*ArcCoth[a*x]^2)/(6*a^3) + (x^5*ArcCoth[a*x]^2)/(10*a) - ArcC
oth[a*x]^3/(6*a^6) + (x^6*ArcCoth[a*x]^3)/6 - (19*ArcTanh[a*x])/(60*a^6) - (23*ArcCoth[a*x]*Log[2/(1 - a*x)])/
(15*a^6) - (23*PolyLog[2, 1 - 2/(1 - a*x)])/(30*a^6)

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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 206

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

Rule 321

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 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 2402

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 2315

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

Rubi steps

\begin{align*} \int x^5 \coth ^{-1}(a x)^3 \, dx &=\frac{1}{6} x^6 \coth ^{-1}(a x)^3-\frac{1}{2} a \int \frac{x^6 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=\frac{1}{6} x^6 \coth ^{-1}(a x)^3+\frac{\int x^4 \coth ^{-1}(a x)^2 \, dx}{2 a}-\frac{\int \frac{x^4 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a}\\ &=\frac{x^5 \coth ^{-1}(a x)^2}{10 a}+\frac{1}{6} x^6 \coth ^{-1}(a x)^3-\frac{1}{5} \int \frac{x^5 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac{\int x^2 \coth ^{-1}(a x)^2 \, dx}{2 a^3}-\frac{\int \frac{x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a^3}\\ &=\frac{x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \coth ^{-1}(a x)^2}{10 a}+\frac{1}{6} x^6 \coth ^{-1}(a x)^3+\frac{\int \coth ^{-1}(a x)^2 \, dx}{2 a^5}-\frac{\int \frac{\coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a^5}+\frac{\int x^3 \coth ^{-1}(a x) \, dx}{5 a^2}-\frac{\int \frac{x^3 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^2}-\frac{\int \frac{x^3 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^2}\\ &=\frac{x^4 \coth ^{-1}(a x)}{20 a^2}+\frac{x \coth ^{-1}(a x)^2}{2 a^5}+\frac{x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \coth ^{-1}(a x)^2}{10 a}-\frac{\coth ^{-1}(a x)^3}{6 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^3+\frac{\int x \coth ^{-1}(a x) \, dx}{5 a^4}-\frac{\int \frac{x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^4}+\frac{\int x \coth ^{-1}(a x) \, dx}{3 a^4}-\frac{\int \frac{x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^4}-\frac{\int \frac{x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{a^4}-\frac{\int \frac{x^4}{1-a^2 x^2} \, dx}{20 a}\\ &=\frac{4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac{x^4 \coth ^{-1}(a x)}{20 a^2}+\frac{23 \coth ^{-1}(a x)^2}{30 a^6}+\frac{x \coth ^{-1}(a x)^2}{2 a^5}+\frac{x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \coth ^{-1}(a x)^2}{10 a}-\frac{\coth ^{-1}(a x)^3}{6 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^3-\frac{\int \frac{\coth ^{-1}(a x)}{1-a x} \, dx}{5 a^5}-\frac{\int \frac{\coth ^{-1}(a x)}{1-a x} \, dx}{3 a^5}-\frac{\int \frac{\coth ^{-1}(a x)}{1-a x} \, dx}{a^5}-\frac{\int \frac{x^2}{1-a^2 x^2} \, dx}{10 a^3}-\frac{\int \frac{x^2}{1-a^2 x^2} \, dx}{6 a^3}-\frac{\int \left (-\frac{1}{a^4}-\frac{x^2}{a^2}+\frac{1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx}{20 a}\\ &=\frac{19 x}{60 a^5}+\frac{x^3}{60 a^3}+\frac{4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac{x^4 \coth ^{-1}(a x)}{20 a^2}+\frac{23 \coth ^{-1}(a x)^2}{30 a^6}+\frac{x \coth ^{-1}(a x)^2}{2 a^5}+\frac{x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \coth ^{-1}(a x)^2}{10 a}-\frac{\coth ^{-1}(a x)^3}{6 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^3-\frac{23 \coth ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{15 a^6}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{20 a^5}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{10 a^5}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{6 a^5}+\frac{\int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^5}+\frac{\int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{3 a^5}+\frac{\int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^5}\\ &=\frac{19 x}{60 a^5}+\frac{x^3}{60 a^3}+\frac{4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac{x^4 \coth ^{-1}(a x)}{20 a^2}+\frac{23 \coth ^{-1}(a x)^2}{30 a^6}+\frac{x \coth ^{-1}(a x)^2}{2 a^5}+\frac{x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \coth ^{-1}(a x)^2}{10 a}-\frac{\coth ^{-1}(a x)^3}{6 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^3-\frac{19 \tanh ^{-1}(a x)}{60 a^6}-\frac{23 \coth ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{15 a^6}-\frac{\operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{5 a^6}-\frac{\operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{3 a^6}-\frac{\operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{a^6}\\ &=\frac{19 x}{60 a^5}+\frac{x^3}{60 a^3}+\frac{4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac{x^4 \coth ^{-1}(a x)}{20 a^2}+\frac{23 \coth ^{-1}(a x)^2}{30 a^6}+\frac{x \coth ^{-1}(a x)^2}{2 a^5}+\frac{x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \coth ^{-1}(a x)^2}{10 a}-\frac{\coth ^{-1}(a x)^3}{6 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^3-\frac{19 \tanh ^{-1}(a x)}{60 a^6}-\frac{23 \coth ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{15 a^6}-\frac{23 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{30 a^6}\\ \end{align*}

Mathematica [A]  time = 0.525733, size = 117, normalized size = 0.63 \[ \frac{46 \text{PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )+a x \left (a^2 x^2+19\right )+10 \left (a^6 x^6-1\right ) \coth ^{-1}(a x)^3+2 \left (3 a^5 x^5+5 a^3 x^3+15 a x-23\right ) \coth ^{-1}(a x)^2+\coth ^{-1}(a x) \left (3 a^4 x^4+16 a^2 x^2-92 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )-19\right )}{60 a^6} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^5*ArcCoth[a*x]^3,x]

[Out]

(a*x*(19 + a^2*x^2) + 2*(-23 + 15*a*x + 5*a^3*x^3 + 3*a^5*x^5)*ArcCoth[a*x]^2 + 10*(-1 + a^6*x^6)*ArcCoth[a*x]
^3 + ArcCoth[a*x]*(-19 + 16*a^2*x^2 + 3*a^4*x^4 - 92*Log[1 - E^(-2*ArcCoth[a*x])]) + 46*PolyLog[2, E^(-2*ArcCo
th[a*x])])/(60*a^6)

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Maple [C]  time = 1.901, size = 1141, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arccoth(a*x)^3,x)

[Out]

23/30*arccoth(a*x)^2/a^6+1/4/a^6*arccoth(a*x)^2*ln(a*x-1)-1/4/a^6*arccoth(a*x)^2*ln(a*x+1)-23/15/a^6*arccoth(a
*x)*ln(1+1/((a*x-1)/(a*x+1))^(1/2))-1/4/a^6*arccoth(a*x)^2*ln((a*x-1)/(a*x+1))-41/120/a^6/(-1+((a*x-1)/(a*x+1)
)^(1/2))*((a*x-1)/(a*x+1))^(1/2)-41/120/a^6/(((a*x-1)/(a*x+1))^(1/2)+1)*((a*x-1)/(a*x+1))^(1/2)+1/80/a^5/(((a*
x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)+a*x)*x-1/80/a^5/(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))
^(1/2)-a*x)*x+1/120/a^6*((a*x-1)/(a*x+1))^(1/2)/(2*((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)-2*a*x+1
)+1/120/a^6*((a*x-1)/(a*x+1))^(1/2)/(2*((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)+2*a*x-1)-1/6*arccot
h(a*x)^3/a^6+1/6*x^6*arccoth(a*x)^3+4/15*x^2*arccoth(a*x)/a^4+1/10*x^5*arccoth(a*x)^2/a+1/20*x^4*arccoth(a*x)/
a^2+1/2*x*arccoth(a*x)^2/a^5+1/6*x^3*arccoth(a*x)^2/a^3-1/80/a^6/(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1)
)^(1/2)+a*x)+1/80/a^6/(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)-a*x)-19/60/a^6*arccoth(a*x)+23/15/a
^6*dilog(1/((a*x-1)/(a*x+1))^(1/2))-23/15/a^6*dilog(1+1/((a*x-1)/(a*x+1))^(1/2))-1/8*I/a^6*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+1/4*I/a^6*Pi*cs
gn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*arccoth(a*x)^2+1/8*I/a^6*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)^2-1/8*I/a^6*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn
(I*(a*x+1)/(a*x-1))*arccoth(a*x)^2+1/8*I/a^6*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)^2-1/8*I/a^6*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3*arccoth(a*x)^2-1/8*I/a^6
*Pi*csgn(I*(a*x+1)/(a*x-1))^3*arccoth(a*x)^2-1/120/a^5*((a*x-1)/(a*x+1))^(1/2)/(2*((a*x-1)/(a*x+1))^(1/2)*a*x+
((a*x-1)/(a*x+1))^(1/2)-2*a*x+1)*x-1/120/a^5*((a*x-1)/(a*x+1))^(1/2)/(2*((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(
a*x+1))^(1/2)+2*a*x-1)*x

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Maxima [A]  time = 1.01173, size = 390, normalized size = 2.1 \begin{align*} \frac{1}{6} \, x^{6} \operatorname{arcoth}\left (a x\right )^{3} + \frac{1}{60} \, a{\left (\frac{2 \,{\left (3 \, a^{4} x^{5} + 5 \, a^{2} x^{3} + 15 \, x\right )}}{a^{6}} - \frac{15 \, \log \left (a x + 1\right )}{a^{7}} + \frac{15 \, \log \left (a x - 1\right )}{a^{7}}\right )} \operatorname{arcoth}\left (a x\right )^{2} + \frac{1}{240} \, a{\left (\frac{\frac{4 \, a^{3} x^{3} +{\left (15 \, \log \left (a x - 1\right ) - 46\right )} \log \left (a x + 1\right )^{2} - 5 \, \log \left (a x + 1\right )^{3} + 5 \, \log \left (a x - 1\right )^{3} + 76 \, a x -{\left (15 \, \log \left (a x - 1\right )^{2} - 92 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 46 \, \log \left (a x - 1\right )^{2} + 38 \, \log \left (a x - 1\right )}{a} - \frac{184 \,{\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{38 \, \log \left (a x + 1\right )}{a}}{a^{6}} + \frac{2 \,{\left (6 \, a^{4} x^{4} + 32 \, a^{2} x^{2} - 2 \,{\left (15 \, \log \left (a x - 1\right ) - 46\right )} \log \left (a x + 1\right ) + 15 \, \log \left (a x + 1\right )^{2} + 15 \, \log \left (a x - 1\right )^{2} + 92 \, \log \left (a x - 1\right )\right )} \operatorname{arcoth}\left (a x\right )}{a^{7}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x)^3,x, algorithm="maxima")

[Out]

1/6*x^6*arccoth(a*x)^3 + 1/60*a*(2*(3*a^4*x^5 + 5*a^2*x^3 + 15*x)/a^6 - 15*log(a*x + 1)/a^7 + 15*log(a*x - 1)/
a^7)*arccoth(a*x)^2 + 1/240*a*(((4*a^3*x^3 + (15*log(a*x - 1) - 46)*log(a*x + 1)^2 - 5*log(a*x + 1)^3 + 5*log(
a*x - 1)^3 + 76*a*x - (15*log(a*x - 1)^2 - 92*log(a*x - 1))*log(a*x + 1) + 46*log(a*x - 1)^2 + 38*log(a*x - 1)
)/a - 184*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a - 38*log(a*x + 1)/a)/a^6 + 2*(6*a^4*x^4
+ 32*a^2*x^2 - 2*(15*log(a*x - 1) - 46)*log(a*x + 1) + 15*log(a*x + 1)^2 + 15*log(a*x - 1)^2 + 92*log(a*x - 1)
)*arccoth(a*x)/a^7)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{5} \operatorname{arcoth}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^5*arccoth(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{acoth}^{3}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*acoth(a*x)**3,x)

[Out]

Integral(x**5*acoth(a*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{arcoth}\left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^5*arccoth(a*x)^3, x)