Integrand size = 10, antiderivative size = 105 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {4 x^2}{45 a^4}+\frac {x^4}{60 a^2}+\frac {x \coth ^{-1}(a x)}{3 a^5}+\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^5 \coth ^{-1}(a x)}{15 a}-\frac {\coth ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2+\frac {23 \log \left (1-a^2 x^2\right )}{90 a^6} \]
4/45*x^2/a^4+1/60*x^4/a^2+1/3*x*arccoth(a*x)/a^5+1/9*x^3*arccoth(a*x)/a^3+ 1/15*x^5*arccoth(a*x)/a-1/6*arccoth(a*x)^2/a^6+1/6*x^6*arccoth(a*x)^2+23/9 0*ln(-a^2*x^2+1)/a^6
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.76 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {16 a^2 x^2+3 a^4 x^4+4 a x \left (15+5 a^2 x^2+3 a^4 x^4\right ) \coth ^{-1}(a x)+30 \left (-1+a^6 x^6\right ) \coth ^{-1}(a x)^2+46 \log \left (1-a^2 x^2\right )}{180 a^6} \]
(16*a^2*x^2 + 3*a^4*x^4 + 4*a*x*(15 + 5*a^2*x^2 + 3*a^4*x^4)*ArcCoth[a*x] + 30*(-1 + a^6*x^6)*ArcCoth[a*x]^2 + 46*Log[1 - a^2*x^2])/(180*a^6)
Time = 1.25 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.69, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {6453, 6543, 6453, 243, 49, 2009, 6543, 6453, 243, 49, 2009, 6543, 6437, 240, 6511}
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.
| \(\displaystyle \int x^5 \coth ^{-1}(a x)^2 \, dx\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \int \frac {x^6 \coth ^{-1}(a x)}{1-a^2 x^2}dx\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\int \frac {x^4 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x^4 \coth ^{-1}(a x)dx}{a^2}\right )\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\int \frac {x^4 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{5} a \int \frac {x^5}{1-a^2 x^2}dx}{a^2}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\int \frac {x^4 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \int \frac {x^4}{1-a^2 x^2}dx^2}{a^2}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\int \frac {x^4 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \int \left (-\frac {x^2}{a^2}-\frac {1}{a^4 \left (a^2 x^2-1\right )}-\frac {1}{a^4}\right )dx^2}{a^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\int \frac {x^4 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \left (-\frac {x^2}{a^4}-\frac {x^4}{2 a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^6}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x^2 \coth ^{-1}(a x)dx}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \left (-\frac {x^2}{a^4}-\frac {x^4}{2 a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^6}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{3} a \int \frac {x^3}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \left (-\frac {x^2}{a^4}-\frac {x^4}{2 a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^6}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \int \frac {x^2}{1-a^2 x^2}dx^2}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \left (-\frac {x^2}{a^4}-\frac {x^4}{2 a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^6}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (a^2 x^2-1\right )}\right )dx^2}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \left (-\frac {x^2}{a^4}-\frac {x^4}{2 a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^6}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \left (-\frac {x^2}{a^4}-\frac {x^4}{2 a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^6}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\frac {\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int \coth ^{-1}(a x)dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \left (-\frac {x^2}{a^4}-\frac {x^4}{2 a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^6}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6437 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\frac {\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)-a \int \frac {x}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \left (-\frac {x^2}{a^4}-\frac {x^4}{2 a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^6}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\frac {\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \left (-\frac {x^2}{a^4}-\frac {x^4}{2 a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^6}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6511 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \left (\frac {\frac {\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \left (-\frac {x^2}{a^4}-\frac {x^4}{2 a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^6}\right )}{a^2}\right )\) |
(x^6*ArcCoth[a*x]^2)/6 - (a*(-(((x^5*ArcCoth[a*x])/5 - (a*(-(x^2/a^4) - x^ 4/(2*a^2) - Log[1 - a^2*x^2]/a^6))/10)/a^2) + (-(((x^3*ArcCoth[a*x])/3 - ( a*(-(x^2/a^2) - Log[1 - a^2*x^2]/a^4))/6)/a^2) + (ArcCoth[a*x]^2/(2*a^3) - (x*ArcCoth[a*x] + Log[1 - a^2*x^2]/(2*a))/a^2)/a^2)/a^2))/3
3.1.12.3.1 Defintions of rubi rules used
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_)^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]
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88
| method | result | size |
| parallelrisch | \(-\frac {-30 a^{6} x^{6} \operatorname {arccoth}\left (a x \right )^{2}-12 a^{5} x^{5} \operatorname {arccoth}\left (a x \right )-16-3 a^{4} x^{4}-20 a^{3} x^{3} \operatorname {arccoth}\left (a x \right )-16 a^{2} x^{2}-60 a x \,\operatorname {arccoth}\left (a x \right )+30 \operatorname {arccoth}\left (a x \right )^{2}-92 \ln \left (a x -1\right )-92 \,\operatorname {arccoth}\left (a x \right )}{180 a^{6}}\) | \(92\) |
| parts | \(\frac {x^{6} \operatorname {arccoth}\left (a x \right )^{2}}{6}+\frac {\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )}{5}+\frac {a^{3} x^{3} \operatorname {arccoth}\left (a x \right )}{3}+a x \,\operatorname {arccoth}\left (a x \right )+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {a^{4} x^{4}}{20}+\frac {4 a^{2} x^{2}}{15}+\frac {23 \ln \left (a x -1\right )}{30}+\frac {23 \ln \left (a x +1\right )}{30}}{3 a^{6}}\) | \(166\) |
| derivativedivides | \(\frac {\frac {a^{6} x^{6} \operatorname {arccoth}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )}{15}+\frac {a^{3} x^{3} \operatorname {arccoth}\left (a x \right )}{9}+\frac {a x \,\operatorname {arccoth}\left (a x \right )}{3}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{6}-\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{6}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (a x -1\right )^{2}}{24}+\frac {\ln \left (a x +1\right )^{2}}{24}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{12}+\frac {a^{4} x^{4}}{60}+\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a x -1\right )}{90}+\frac {23 \ln \left (a x +1\right )}{90}}{a^{6}}\) | \(168\) |
| default | \(\frac {\frac {a^{6} x^{6} \operatorname {arccoth}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )}{15}+\frac {a^{3} x^{3} \operatorname {arccoth}\left (a x \right )}{9}+\frac {a x \,\operatorname {arccoth}\left (a x \right )}{3}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{6}-\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{6}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (a x -1\right )^{2}}{24}+\frac {\ln \left (a x +1\right )^{2}}{24}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{12}+\frac {a^{4} x^{4}}{60}+\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a x -1\right )}{90}+\frac {23 \ln \left (a x +1\right )}{90}}{a^{6}}\) | \(168\) |
| risch | \(\frac {\left (a^{6} x^{6}-1\right ) \ln \left (a x +1\right )^{2}}{24 a^{6}}-\frac {\left (15 \ln \left (a x -1\right ) x^{6} a^{6}-6 a^{5} x^{5}-10 a^{3} x^{3}-30 a x -15 \ln \left (a x -1\right )\right ) \ln \left (a x +1\right )}{180 a^{6}}+\frac {x^{6} \ln \left (a x -1\right )^{2}}{24}-\frac {\ln \left (a x -1\right ) x^{5}}{30 a}+\frac {x^{4}}{60 a^{2}}-\frac {\ln \left (a x -1\right ) x^{3}}{18 a^{3}}+\frac {4 x^{2}}{45 a^{4}}-\frac {\ln \left (a x -1\right ) x}{6 a^{5}}-\frac {\ln \left (a x -1\right )^{2}}{24 a^{6}}+\frac {23 \ln \left (a^{2} x^{2}-1\right )}{90 a^{6}}+\frac {16}{135 a^{6}}\) | \(180\) |
-1/180*(-30*a^6*x^6*arccoth(a*x)^2-12*a^5*x^5*arccoth(a*x)-16-3*a^4*x^4-20 *a^3*x^3*arccoth(a*x)-16*a^2*x^2-60*a*x*arccoth(a*x)+30*arccoth(a*x)^2-92* ln(a*x-1)-92*arccoth(a*x))/a^6
Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {6 \, a^{4} x^{4} + 32 \, a^{2} x^{2} + 15 \, {\left (a^{6} x^{6} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (3 \, a^{5} x^{5} + 5 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (\frac {a x + 1}{a x - 1}\right ) + 92 \, \log \left (a^{2} x^{2} - 1\right )}{360 \, a^{6}} \]
1/360*(6*a^4*x^4 + 32*a^2*x^2 + 15*(a^6*x^6 - 1)*log((a*x + 1)/(a*x - 1))^ 2 + 4*(3*a^5*x^5 + 5*a^3*x^3 + 15*a*x)*log((a*x + 1)/(a*x - 1)) + 92*log(a ^2*x^2 - 1))/a^6
Time = 0.40 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\begin {cases} \frac {x^{6} \operatorname {acoth}^{2}{\left (a x \right )}}{6} + \frac {x^{5} \operatorname {acoth}{\left (a x \right )}}{15 a} + \frac {x^{4}}{60 a^{2}} + \frac {x^{3} \operatorname {acoth}{\left (a x \right )}}{9 a^{3}} + \frac {4 x^{2}}{45 a^{4}} + \frac {x \operatorname {acoth}{\left (a x \right )}}{3 a^{5}} + \frac {23 \log {\left (a x + 1 \right )}}{45 a^{6}} - \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{6 a^{6}} - \frac {23 \operatorname {acoth}{\left (a x \right )}}{45 a^{6}} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} x^{6}}{24} & \text {otherwise} \end {cases} \]
Piecewise((x**6*acoth(a*x)**2/6 + x**5*acoth(a*x)/(15*a) + x**4/(60*a**2) + x**3*acoth(a*x)/(9*a**3) + 4*x**2/(45*a**4) + x*acoth(a*x)/(3*a**5) + 23 *log(a*x + 1)/(45*a**6) - acoth(a*x)**2/(6*a**6) - 23*acoth(a*x)/(45*a**6) , Ne(a, 0)), (-pi**2*x**6/24, True))
Time = 0.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.29 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {1}{6} \, x^{6} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{90} \, a {\left (\frac {2 \, {\left (3 \, a^{4} x^{5} + 5 \, a^{2} x^{3} + 15 \, x\right )}}{a^{6}} - \frac {15 \, \log \left (a x + 1\right )}{a^{7}} + \frac {15 \, \log \left (a x - 1\right )}{a^{7}}\right )} \operatorname {arcoth}\left (a x\right ) + \frac {6 \, a^{4} x^{4} + 32 \, a^{2} x^{2} - 2 \, {\left (15 \, \log \left (a x - 1\right ) - 46\right )} \log \left (a x + 1\right ) + 15 \, \log \left (a x + 1\right )^{2} + 15 \, \log \left (a x - 1\right )^{2} + 92 \, \log \left (a x - 1\right )}{360 \, a^{6}} \]
1/6*x^6*arccoth(a*x)^2 + 1/90*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) + 1/360*(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))/a^6
Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (89) = 178\).
Time = 0.28 (sec) , antiderivative size = 534, normalized size of antiderivative = 5.09 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {1}{90} \, {\left (\frac {15 \, {\left (\frac {3 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {10 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {3 \, {\left (a x + 1\right )}}{a x - 1}\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2}}{\frac {{\left (a x + 1\right )}^{6} a^{7}}{{\left (a x - 1\right )}^{6}} - \frac {6 \, {\left (a x + 1\right )}^{5} a^{7}}{{\left (a x - 1\right )}^{5}} + \frac {15 \, {\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} - \frac {20 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} + \frac {15 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} - \frac {6 \, {\left (a x + 1\right )} a^{7}}{a x - 1} + a^{7}} + \frac {2 \, {\left (\frac {45 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} - \frac {90 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {140 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - \frac {70 \, {\left (a x + 1\right )}}{a x - 1} + 23\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{5} a^{7}}{{\left (a x - 1\right )}^{5}} - \frac {5 \, {\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} + \frac {10 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} - \frac {10 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} + \frac {5 \, {\left (a x + 1\right )} a^{7}}{a x - 1} - a^{7}} + \frac {4 \, {\left (\frac {11 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} - \frac {16 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {11 \, {\left (a x + 1\right )}}{a x - 1}\right )}}{\frac {{\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} - \frac {4 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} - \frac {4 \, {\left (a x + 1\right )} a^{7}}{a x - 1} + a^{7}} - \frac {46 \, \log \left (\frac {a x + 1}{a x - 1} - 1\right )}{a^{7}} + \frac {46 \, \log \left (\frac {a x + 1}{a x - 1}\right )}{a^{7}}\right )} a \]
1/90*(15*(3*(a*x + 1)^5/(a*x - 1)^5 + 10*(a*x + 1)^3/(a*x - 1)^3 + 3*(a*x + 1)/(a*x - 1))*log((a*x + 1)/(a*x - 1))^2/((a*x + 1)^6*a^7/(a*x - 1)^6 - 6*(a*x + 1)^5*a^7/(a*x - 1)^5 + 15*(a*x + 1)^4*a^7/(a*x - 1)^4 - 20*(a*x + 1)^3*a^7/(a*x - 1)^3 + 15*(a*x + 1)^2*a^7/(a*x - 1)^2 - 6*(a*x + 1)*a^7/( a*x - 1) + a^7) + 2*(45*(a*x + 1)^4/(a*x - 1)^4 - 90*(a*x + 1)^3/(a*x - 1) ^3 + 140*(a*x + 1)^2/(a*x - 1)^2 - 70*(a*x + 1)/(a*x - 1) + 23)*log((a*x + 1)/(a*x - 1))/((a*x + 1)^5*a^7/(a*x - 1)^5 - 5*(a*x + 1)^4*a^7/(a*x - 1)^ 4 + 10*(a*x + 1)^3*a^7/(a*x - 1)^3 - 10*(a*x + 1)^2*a^7/(a*x - 1)^2 + 5*(a *x + 1)*a^7/(a*x - 1) - a^7) + 4*(11*(a*x + 1)^3/(a*x - 1)^3 - 16*(a*x + 1 )^2/(a*x - 1)^2 + 11*(a*x + 1)/(a*x - 1))/((a*x + 1)^4*a^7/(a*x - 1)^4 - 4 *(a*x + 1)^3*a^7/(a*x - 1)^3 + 6*(a*x + 1)^2*a^7/(a*x - 1)^2 - 4*(a*x + 1) *a^7/(a*x - 1) + a^7) - 46*log((a*x + 1)/(a*x - 1) - 1)/a^7 + 46*log((a*x + 1)/(a*x - 1))/a^7)*a
Time = 4.63 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.81 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {x^6\,{\mathrm {acoth}\left (a\,x\right )}^2}{6}+\frac {\frac {23\,\ln \left (a^2\,x^2-1\right )}{90}+\frac {4\,a^2\,x^2}{45}+\frac {a^4\,x^4}{60}-\frac {{\mathrm {acoth}\left (a\,x\right )}^2}{6}+\frac {a^3\,x^3\,\mathrm {acoth}\left (a\,x\right )}{9}+\frac {a^5\,x^5\,\mathrm {acoth}\left (a\,x\right )}{15}+\frac {a\,x\,\mathrm {acoth}\left (a\,x\right )}{3}}{a^6} \]