Integrand size = 10, antiderivative size = 149 \[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\frac {x \coth ^{-1}(a x)}{a^2}-\frac {\coth ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \coth ^{-1}(a x)^2}{2 a}+\frac {\coth ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {\log \left (1-a^2 x^2\right )}{2 a^3}-\frac {\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^3} \] Output:
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*arc coth(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*polylog (3,1-2/(-a*x+1))/a^3
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.94 \[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\frac {-i \pi ^3+24 a x \coth ^{-1}(a x)-12 \coth ^{-1}(a x)^2+12 a^2 x^2 \coth ^{-1}(a x)^2+8 \coth ^{-1}(a x)^3+8 a^3 x^3 \coth ^{-1}(a x)^3-24 \coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-24 \log \left (\frac {1}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\right )-24 \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right )}{24 a^3} \] Input:
Integrate[x^2*ArcCoth[a*x]^3,x]
Output:
((-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*L og[1 - E^(2*ArcCoth[a*x])] - 24*Log[1/(a*Sqrt[1 - 1/(a^2*x^2)]*x)] - 24*Ar cCoth[a*x]*PolyLog[2, E^(2*ArcCoth[a*x])] + 12*PolyLog[3, E^(2*ArcCoth[a*x ])])/(24*a^3)
Time = 1.53 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6453, 6543, 6453, 6543, 6437, 240, 6511, 6547, 6471, 6621, 7164}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \coth ^{-1}(a x)^3 \, dx\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \int \frac {x^3 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \left (\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\int x \coth ^{-1}(a x)^2dx}{a^2}\right )\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \left (\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}\right )\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \left (\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int \coth ^{-1}(a x)dx}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6437 |
| \(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \left (\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)-a \int \frac {x}{1-a^2 x^2}dx}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \left (\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6511 |
| \(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \left (\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6547 |
| \(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \left (\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a x}dx}{a}-\frac {\coth ^{-1}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6471 |
| \(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \left (\frac {\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a}-2 \int \frac {\coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\coth ^{-1}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6621 |
| \(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \left (\frac {\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a}-2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{2 a}\right )}{a}-\frac {\coth ^{-1}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \left (\frac {\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a}-2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{2 a}\right )}{a}-\frac {\coth ^{-1}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}\right )\) |
Input:
Int[x^2*ArcCoth[a*x]^3,x]
Output:
(x^3*ArcCoth[a*x]^3)/3 - a*(-(((x^2*ArcCoth[a*x]^2)/2 - a*(ArcCoth[a*x]^2/ (2*a^3) - (x*ArcCoth[a*x] + Log[1 - a^2*x^2]/(2*a))/a^2))/a^2) + (-1/3*Arc Coth[a*x]^3/a^2 + ((ArcCoth[a*x]^2*Log[2/(1 - a*x)])/a - 2*(-1/2*(ArcCoth[ a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a + PolyLog[3, 1 - 2/(1 - a*x)]/(4*a)))/ a)/a^2)
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x^n])^p, x] - Simp[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])
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] - Simp[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]
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] + Simp[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]
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> 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]
Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcCoth[c* x])^p, x], x] - Simp[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]
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] + Simp[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]
Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.34 (sec) , antiderivative size = 776, normalized size of antiderivative = 5.21
| method | result | size |
| parts | \(\frac {x^{3} \operatorname {arccoth}\left (x a \right )^{3}}{3}+\frac {\frac {x^{2} a^{2} \operatorname {arccoth}\left (x a \right )^{2}}{2}-\ln \left (1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+\frac {\operatorname {arccoth}\left (x a \right )^{3}}{3}+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a -1}{x a +1}\right )}{2}+\frac {\left (\sqrt {\frac {x a -1}{x a +1}}\, x a +\sqrt {\frac {x a -1}{x a +1}}+x a +1\right ) \operatorname {arccoth}\left (x a \right )}{2}-\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-2 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right ) \operatorname {arccoth}\left (x a \right )^{2}}{4}-\ln \left (-1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a -1\right )}{2}+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a +1\right )}{2}-\frac {\left (\sqrt {\frac {x a -1}{x a +1}}\, x a +\sqrt {\frac {x a -1}{x a +1}}-x a -1\right ) \operatorname {arccoth}\left (x a \right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{2} \operatorname {arccoth}\left (x a \right )^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {arccoth}\left (x a \right )^{2}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2} \operatorname {arccoth}\left (x a \right )^{2}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2} \operatorname {arccoth}\left (x a \right )^{2}}{4}-2 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-\ln \left (2\right ) \operatorname {arccoth}\left (x a \right )^{2}+\operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a +1}{x a -1}-1\right )-\frac {\operatorname {arccoth}\left (x a \right )^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{3} \operatorname {arccoth}\left (x a \right )^{2}}{4}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{3} \operatorname {arccoth}\left (x a \right )^{2}}{4}}{a^{3}}\) | \(776\) |
| derivativedivides | \(\frac {\frac {x^{2} a^{2} \operatorname {arccoth}\left (x a \right )^{2}}{2}-\ln \left (1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+\frac {\operatorname {arccoth}\left (x a \right )^{3}}{3}+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a -1}{x a +1}\right )}{2}+\frac {\left (\sqrt {\frac {x a -1}{x a +1}}\, x a +\sqrt {\frac {x a -1}{x a +1}}+x a +1\right ) \operatorname {arccoth}\left (x a \right )}{2}-\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-2 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right ) \operatorname {arccoth}\left (x a \right )^{2}}{4}-\ln \left (-1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a -1\right )}{2}+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a +1\right )}{2}-\frac {\left (\sqrt {\frac {x a -1}{x a +1}}\, x a +\sqrt {\frac {x a -1}{x a +1}}-x a -1\right ) \operatorname {arccoth}\left (x a \right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{2} \operatorname {arccoth}\left (x a \right )^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {arccoth}\left (x a \right )^{2}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2} \operatorname {arccoth}\left (x a \right )^{2}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2} \operatorname {arccoth}\left (x a \right )^{2}}{4}-2 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-\ln \left (2\right ) \operatorname {arccoth}\left (x a \right )^{2}+\operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a +1}{x a -1}-1\right )+\frac {\operatorname {arccoth}\left (x a \right )^{3} a^{3} x^{3}}{3}-\frac {\operatorname {arccoth}\left (x a \right )^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{3} \operatorname {arccoth}\left (x a \right )^{2}}{4}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{3} \operatorname {arccoth}\left (x a \right )^{2}}{4}}{a^{3}}\) | \(778\) |
| default | \(\frac {\frac {x^{2} a^{2} \operatorname {arccoth}\left (x a \right )^{2}}{2}-\ln \left (1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+\frac {\operatorname {arccoth}\left (x a \right )^{3}}{3}+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a -1}{x a +1}\right )}{2}+\frac {\left (\sqrt {\frac {x a -1}{x a +1}}\, x a +\sqrt {\frac {x a -1}{x a +1}}+x a +1\right ) \operatorname {arccoth}\left (x a \right )}{2}-\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-\operatorname {arccoth}\left (x a \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-2 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right ) \operatorname {arccoth}\left (x a \right )^{2}}{4}-\ln \left (-1+\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a -1\right )}{2}+\frac {\operatorname {arccoth}\left (x a \right )^{2} \ln \left (x a +1\right )}{2}-\frac {\left (\sqrt {\frac {x a -1}{x a +1}}\, x a +\sqrt {\frac {x a -1}{x a +1}}-x a -1\right ) \operatorname {arccoth}\left (x a \right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{2} \operatorname {arccoth}\left (x a \right )^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {x a -1}{x a +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {arccoth}\left (x a \right )^{2}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {x a +1}{x a -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2} \operatorname {arccoth}\left (x a \right )^{2}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right ) \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{2} \operatorname {arccoth}\left (x a \right )^{2}}{4}-2 \,\operatorname {arccoth}\left (x a \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {x a -1}{x a +1}}}\right )-\ln \left (2\right ) \operatorname {arccoth}\left (x a \right )^{2}+\operatorname {arccoth}\left (x a \right )^{2} \ln \left (\frac {x a +1}{x a -1}-1\right )+\frac {\operatorname {arccoth}\left (x a \right )^{3} a^{3} x^{3}}{3}-\frac {\operatorname {arccoth}\left (x a \right )^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{\left (x a -1\right ) \left (\frac {x a +1}{x a -1}-1\right )}\right )^{3} \operatorname {arccoth}\left (x a \right )^{2}}{4}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (x a +1\right )}{x a -1}\right )^{3} \operatorname {arccoth}\left (x a \right )^{2}}{4}}{a^{3}}\) | \(778\) |
Input:
int(x^2*arccoth(x*a)^3,x,method=_RETURNVERBOSE)
Output:
1/3*x^3*arccoth(x*a)^3+1/a^3*(1/2*x^2*a^2*arccoth(x*a)^2-ln(1+1/((a*x-1)/( a*x+1))^(1/2))+1/3*arccoth(x*a)^3+1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csg n(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))*arccoth(x *a)^2+1/2*arccoth(x*a)^2*ln((a*x-1)/(a*x+1))+1/2*(((a*x-1)/(a*x+1))^(1/2)* x*a+((a*x-1)/(a*x+1))^(1/2)+x*a+1)*arccoth(x*a)-arccoth(x*a)^2*ln(1-1/((a* x-1)/(a*x+1))^(1/2))-arccoth(x*a)^2*ln(1+1/((a*x-1)/(a*x+1))^(1/2))-2*arcc oth(x*a)*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))-ln(-1+1/((a*x-1)/(a*x+1))^( 1/2))+1/4*I*Pi*csgn(I*(a*x+1)/(a*x-1))^3*arccoth(x*a)^2+1/4*I*Pi*csgn(I/(a *x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^3*arccoth(x*a)^2+2*polylog(3,1/((a*x-1) /(a*x+1))^(1/2))+2*polylog(3,-1/((a*x-1)/(a*x+1))^(1/2))+1/2*arccoth(x*a)^ 2*ln(a*x-1)+1/2*arccoth(x*a)^2*ln(a*x+1)-1/2*(((a*x-1)/(a*x+1))^(1/2)*x*a+ ((a*x-1)/(a*x+1))^(1/2)-x*a-1)*arccoth(x*a)-2*arccoth(x*a)*polylog(2,1/((a *x-1)/(a*x+1))^(1/2))-ln(2)*arccoth(x*a)^2+arccoth(x*a)^2*ln((a*x+1)/(a*x- 1)-1)-1/2*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*a rccoth(x*a)^2-1/4*I*Pi*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(x*a)^2-1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csg n(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^2*arccoth(x*a)^2+1/4*I*Pi*csgn(I/ ((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth(x*a)^2-1/2*arcc oth(x*a)^2)
\[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\int { x^{2} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \] Input:
integrate(x^2*arccoth(a*x)^3,x, algorithm="fricas")
Output:
integral(x^2*arccoth(a*x)^3, x)
\[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\int x^{2} \operatorname {acoth}^{3}{\left (a x \right )}\, dx \] Input:
integrate(x**2*acoth(a*x)**3,x)
Output:
Integral(x**2*acoth(a*x)**3, x)
\[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\int { x^{2} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \] Input:
integrate(x^2*arccoth(a*x)^3,x, algorithm="maxima")
Output:
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*integrate(-((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)*l og(a*x - 1))*log(a*x + 1))/(a^3*x + a^2), x)
\[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\int { x^{2} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \] Input:
integrate(x^2*arccoth(a*x)^3,x, algorithm="giac")
Output:
integrate(x^2*arccoth(a*x)^3, x)
Timed out. \[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\int x^2\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \] Input:
int(x^2*acoth(a*x)^3,x)
Output:
int(x^2*acoth(a*x)^3, x)
\[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\frac {2 \mathit {acoth} \left (a x \right )^{3} a^{3} x^{3}-2 \mathit {acoth} \left (a x \right )^{3} a x -3 \mathit {acoth} \left (a x \right )^{2} a^{2} x^{2}+3 \mathit {acoth} \left (a x \right )^{2}+6 \mathit {acoth} \left (a x \right ) a x +6 \mathit {acoth} \left (a x \right )+2 \left (\int \mathit {acoth} \left (a x \right )^{3}d x \right ) a -6 \,\mathrm {log}\left (a^{2} x -a \right )}{6 a^{3}} \] Input:
int(x^2*acoth(a*x)^3,x)
Output:
(2*acoth(a*x)**3*a**3*x**3 - 2*acoth(a*x)**3*a*x - 3*acoth(a*x)**2*a**2*x* *2 + 3*acoth(a*x)**2 + 6*acoth(a*x)*a*x + 6*acoth(a*x) + 2*int(acoth(a*x)* *3,x)*a - 6*log(a**2*x - a))/(6*a**3)