3.26 \(\int x^2 \coth ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=149 \[ \frac{\text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^3}-\frac{\coth ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{a^3}+\frac{\log \left (1-a^2 x^2\right )}{2 a^3}+\frac{\coth ^{-1}(a x)^3}{3 a^3}-\frac{\coth ^{-1}(a x)^2}{2 a^3}+\frac{x \coth ^{-1}(a x)}{a^2}-\frac{\log \left (\frac{2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a^3}+\frac{1}{3} x^3 \coth ^{-1}(a x)^3+\frac{x^2 \coth ^{-1}(a x)^2}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.332522, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {5917, 5981, 5911, 260, 5949, 5985, 5919, 6059, 6610} \[ \frac{\text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^3}-\frac{\coth ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{a^3}+\frac{\log \left (1-a^2 x^2\right )}{2 a^3}+\frac{\coth ^{-1}(a x)^3}{3 a^3}-\frac{\coth ^{-1}(a x)^2}{2 a^3}+\frac{x \coth ^{-1}(a x)}{a^2}-\frac{\log \left (\frac{2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a^3}+\frac{1}{3} x^3 \coth ^{-1}(a x)^3+\frac{x^2 \coth ^{-1}(a x)^2}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCoth[a*x]^3,x]

[Out]

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

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 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^2 \coth ^{-1}(a x)^3 \, dx &=\frac{1}{3} x^3 \coth ^{-1}(a x)^3-a \int \frac{x^3 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=\frac{1}{3} x^3 \coth ^{-1}(a x)^3+\frac{\int x \coth ^{-1}(a x)^2 \, dx}{a}-\frac{\int \frac{x \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{a}\\ &=\frac{x^2 \coth ^{-1}(a x)^2}{2 a}+\frac{\coth ^{-1}(a x)^3}{3 a^3}+\frac{1}{3} x^3 \coth ^{-1}(a x)^3-\frac{\int \frac{\coth ^{-1}(a x)^2}{1-a x} \, dx}{a^2}-\int \frac{x^2 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac{x^2 \coth ^{-1}(a x)^2}{2 a}+\frac{\coth ^{-1}(a x)^3}{3 a^3}+\frac{1}{3} x^3 \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{a^3}+\frac{\int \coth ^{-1}(a x) \, dx}{a^2}-\frac{\int \frac{\coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{a^2}+\frac{2 \int \frac{\coth ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2}\\ &=\frac{x \coth ^{-1}(a x)}{a^2}-\frac{\coth ^{-1}(a x)^2}{2 a^3}+\frac{x^2 \coth ^{-1}(a x)^2}{2 a}+\frac{\coth ^{-1}(a x)^3}{3 a^3}+\frac{1}{3} x^3 \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{a^3}-\frac{\coth ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{a^3}+\frac{\int \frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2}-\frac{\int \frac{x}{1-a^2 x^2} \, dx}{a}\\ &=\frac{x \coth ^{-1}(a x)}{a^2}-\frac{\coth ^{-1}(a x)^2}{2 a^3}+\frac{x^2 \coth ^{-1}(a x)^2}{2 a}+\frac{\coth ^{-1}(a x)^3}{3 a^3}+\frac{1}{3} x^3 \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{a^3}+\frac{\log \left (1-a^2 x^2\right )}{2 a^3}-\frac{\coth ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{a^3}+\frac{\text{Li}_3\left (1-\frac{2}{1-a x}\right )}{2 a^3}\\ \end{align*}

Mathematica [C]  time = 0.352743, size = 140, normalized size = 0.94 \[ \frac{-24 \coth ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )+12 \text{PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right )-24 \log \left (\frac{1}{a x \sqrt{1-\frac{1}{a^2 x^2}}}\right )+8 a^3 x^3 \coth ^{-1}(a x)^3+12 a^2 x^2 \coth ^{-1}(a x)^2+8 \coth ^{-1}(a x)^3-12 \coth ^{-1}(a x)^2+24 a x \coth ^{-1}(a x)-24 \coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-i \pi ^3}{24 a^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*ArcCoth[a*x]^3,x]

[Out]

((-I)*Pi^3 + 24*a*x*ArcCoth[a*x] - 12*ArcCoth[a*x]^2 + 12*a^2*x^2*ArcCoth[a*x]^2 + 8*ArcCoth[a*x]^3 + 8*a^3*x^
3*ArcCoth[a*x]^3 - 24*ArcCoth[a*x]^2*Log[1 - E^(2*ArcCoth[a*x])] - 24*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])])/(24*a^3)

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Maple [C]  time = 0.57, size = 765, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccoth(a*x)^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (a^{3} x^{3} + 1\right )} \log \left (a x + 1\right )^{3} + 3 \,{\left (a^{2} x^{2} -{\left (a^{3} x^{3} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )^{2}}{24 \, a^{3}} + \frac{1}{8} \, \int -\frac{{\left (a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x - 1\right )^{3} +{\left (2 \, a^{2} x^{2} - 3 \,{\left (a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x - 1\right )^{2} - 2 \,{\left (a^{3} x^{3} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )}{a^{3} x + a^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccoth(a*x)^3,x, algorithm="maxima")

[Out]

1/24*((a^3*x^3 + 1)*log(a*x + 1)^3 + 3*(a^2*x^2 - (a^3*x^3 - 1)*log(a*x - 1))*log(a*x + 1)^2)/a^3 + 1/8*integr
ate(-((a^3*x^3 + a^2*x^2)*log(a*x - 1)^3 + (2*a^2*x^2 - 3*(a^3*x^3 + a^2*x^2)*log(a*x - 1)^2 - 2*(a^3*x^3 - 1)
*log(a*x - 1))*log(a*x + 1))/(a^3*x + a^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \operatorname{arcoth}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccoth(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^2*arccoth(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{acoth}^{3}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*acoth(a*x)**3,x)

[Out]

Integral(x**2*acoth(a*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{arcoth}\left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccoth(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^2*arccoth(a*x)^3, x)