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

Optimal. Leaf size=149 \[ \frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^3}-\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a^3}+\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a^3}+\frac {x \coth ^{-1}(a x)}{a^2}+\frac {\log \left (1-a^2 x^2\right )}{2 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3+\frac {x^2 \coth ^{-1}(a x)^2}{2 a} \]

[Out]

x*arccoth(a*x)/a^2-1/2*arccoth(a*x)^2/a^3+1/2*x^2*arccoth(a*x)^2/a+1/3*arccoth(a*x)^3/a^3+1/3*x^3*arccoth(a*x)
^3-arccoth(a*x)^2*ln(2/(-a*x+1))/a^3+1/2*ln(-a^2*x^2+1)/a^3-arccoth(a*x)*polylog(2,1-2/(-a*x+1))/a^3+1/2*polyl
og(3,1-2/(-a*x+1))/a^3

________________________________________________________________________________________

Rubi [A]  time = 0.33, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, 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 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 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 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 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 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 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 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 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 = 0.38, size = 140, normalized size = 0.94 \[ \frac {8 a^3 x^3 \coth ^{-1}(a x)^3-24 \log \left (\frac {1}{a x \sqrt {1-\frac {1}{a^2 x^2}}}\right )+12 a^2 x^2 \coth ^{-1}(a x)^2-24 \coth ^{-1}(a x) \text {Li}_2\left (e^{2 \coth ^{-1}(a x)}\right )+12 \text {Li}_3\left (e^{2 \coth ^{-1}(a x)}\right )+8 \coth ^{-1}(a x)^3-12 \coth ^{-1}(a x)^2+24 a x \coth ^{-1}(a x)-24 \coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-i \pi ^3}{24 a^3} \]

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)

________________________________________________________________________________________

fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {arcoth}\left (a x\right )^{3}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {arcoth}\left (a x\right )^{3}\,{d x} \]

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)

________________________________________________________________________________________

maple [C]  time = 1.49, size = 765, normalized size = 5.13 \[ \frac {\mathrm {arccoth}\left (a x \right )}{a^{3}}-\frac {\ln \left (\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}-1\right )}{a^{3}}-\frac {\ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a^{3}}-\frac {\mathrm {arccoth}\left (a x \right )^{2}}{2 a^{3}}+\frac {\mathrm {arccoth}\left (a x \right )^{3}}{3 a^{3}}+\frac {x^{2} \mathrm {arccoth}\left (a x \right )^{2}}{2 a}+\frac {\mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2 a^{3}}+\frac {\mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2 a^{3}}+\frac {\mathrm {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2 a^{3}}+\frac {\mathrm {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x +1}{a x -1}-1\right )}{a^{3}}-\frac {\mathrm {arccoth}\left (a x \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a^{3}}-\frac {\mathrm {arccoth}\left (a x \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a^{3}}-\frac {\mathrm {arccoth}\left (a x \right )^{2} \ln \relax (2)}{a^{3}}+\frac {x \,\mathrm {arccoth}\left (a x \right )}{a^{2}}+\frac {x^{3} \mathrm {arccoth}\left (a x \right )^{3}}{3}+\frac {2 \polylog \left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a^{3}}+\frac {2 \polylog \left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a^{3}}-\frac {2 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a^{3}}-\frac {2 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a^{3}}+\frac {i \mathrm {arccoth}\left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{3}}{4 a^{3}}+\frac {i \mathrm {arccoth}\left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}}{4 a^{3}}-\frac {i \mathrm {arccoth}\left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )}{4 a^{3}}-\frac {i \mathrm {arccoth}\left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right )}{4 a^{3}}-\frac {i \mathrm {arccoth}\left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2} \mathrm {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{2 a^{3}}+\frac {i \mathrm {arccoth}\left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \mathrm {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2}}{4 a^{3}}+\frac {i \mathrm {arccoth}\left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right )}{4 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*arccoth(a*x)-1/a^3*ln(1/((a*x-1)/(a*x+1))^(1/2)-1)-1/a^3*ln(1+1/((a*x-1)/(a*x+1))^(1/2))+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))-1/2*arccoth(a*x)^2/a^3+1/3*arccoth(a*x
)^3/a^3+1/2*x^2*arccoth(a*x)^2/a+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*arc
coth(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)*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))-1/a^3*arccoth(a*x)^2*ln(2)+x*arccoth
(a*x)/a^2+1/3*x^3*arccoth(a*x)^3+1/4*I/a^3*arccoth(a*x)^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3+1/4
*I/a^3*arccoth(a*x)^2*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))+1/4*I/a^3*arccoth(a*x)^2*Pi*csgn(I*(a*x+1)/(a*x-1))^3-1/4*I/a^3*arccoth(a*x)^2*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))-1/4*I/a^3*arccoth(a*x)^2*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))-1/2*I/a^3*arccoth(a*x)^2*Pi*csgn(I*(a*x+1)/(a*x-1))^2*csgn(I/((a*
x-1)/(a*x+1))^(1/2))+1/4*I/a^3*arccoth(a*x)^2*Pi*csgn(I*(a*x+1)/(a*x-1))*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a^{3} x^{3} + 1\right )} \log \left (a x + 1\right )^{3} + 3 \, {\left (a^{2} x^{2} - {\left (a^{3} x^{3} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )^{2}}{24 \, a^{3}} + \frac {1}{8} \, \int -\frac {{\left (a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x - 1\right )^{3} + {\left (2 \, a^{2} x^{2} - 3 \, {\left (a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x - 1\right )^{2} - 2 \, {\left (a^{3} x^{3} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )}{a^{3} x + a^{2}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*acoth(a*x)^3, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {acoth}^{3}{\left (a x \right )}\, dx \]

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)

________________________________________________________________________________________