Integrand size = 8, antiderivative size = 95 \[ \int x \coth ^{-1}(a x)^3 \, dx=\frac {3 \coth ^{-1}(a x)^2}{2 a^2}+\frac {3 x \coth ^{-1}(a x)^2}{2 a}-\frac {\coth ^{-1}(a x)^3}{2 a^2}+\frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^2}-\frac {3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^2} \]
3/2*arccoth(a*x)^2/a^2+3/2*x*arccoth(a*x)^2/a-1/2*arccoth(a*x)^3/a^2+1/2*x ^2*arccoth(a*x)^3-3*arccoth(a*x)*ln(2/(-a*x+1))/a^2-3/2*polylog(2,1-2/(-a* x+1))/a^2
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.72 \[ \int x \coth ^{-1}(a x)^3 \, dx=\frac {\coth ^{-1}(a x) \left (3 (-1+a x) \coth ^{-1}(a x)+\left (-1+a^2 x^2\right ) \coth ^{-1}(a x)^2-6 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+3 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{2 a^2} \]
(ArcCoth[a*x]*(3*(-1 + a*x)*ArcCoth[a*x] + (-1 + a^2*x^2)*ArcCoth[a*x]^2 - 6*Log[1 - E^(-2*ArcCoth[a*x])]) + 3*PolyLog[2, E^(-2*ArcCoth[a*x])])/(2*a ^2)
Time = 0.80 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6453, 6543, 6437, 6511, 6547, 6471, 2849, 2752}
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.
| \(\displaystyle \int x \coth ^{-1}(a x)^3 \, dx\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3}{2} a \int \frac {x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3}{2} a \left (\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\int \coth ^{-1}(a x)^2dx}{a^2}\right )\) |
| \(\Big \downarrow \) 6437 |
| \(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3}{2} a \left (\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}\right )\) |
| \(\Big \downarrow \) 6511 |
| \(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3}{2} a \left (\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}\right )\) |
| \(\Big \downarrow \) 6547 |
| \(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3}{2} a \left (\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a x}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6471 |
| \(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3}{2} a \left (\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \left (\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3}{2} a \left (\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \left (\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^3-\frac {3}{2} a \left (\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \left (\frac {\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )}{a^2}\right )\) |
(x^2*ArcCoth[a*x]^3)/2 - (3*a*(ArcCoth[a*x]^3/(3*a^3) - (x*ArcCoth[a*x]^2 - 2*a*(-1/2*ArcCoth[a*x]^2/a^2 + ((ArcCoth[a*x]*Log[2/(1 - a*x)])/a + Poly Log[2, 1 - 2/(1 - a*x)]/(2*a))/a))/a^2))/2
3.1.27.3.1 Defintions of rubi rules used
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.28 (sec) , antiderivative size = 2910, normalized size of antiderivative = 30.63
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2910\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2916\) |
| default | \(\text {Expression too large to display}\) | \(2916\) |
1/2*x^2*arccoth(a*x)^3+3/2/a^2*(arccoth(a*x)^2*a*x+1/4*I*Pi*csgn(I/((a*x+1 )/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x -1)-1))*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))+1/4*I*Pi*csgn(I/((a*x+1)/(a*x -1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1) )*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))+1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1 ))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))*dil og(1/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I* (a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))*dilog(1+1/((a *x-1)/(a*x+1))^(1/2))-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(a*x)*ln(1-1/((a*x-1)/(a*x+1))^(1/2))+1/ 4*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth(a *x)*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*c sgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^2*arccoth(a*x)*ln(1-1/((a*x-1)/ (a*x+1))^(1/2))-1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1 ))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))*arccoth(a*x)^2-1/4*I*Pi*csg n(I*(a*x+1)/(a*x-1))^3*dilog(1+1/((a*x-1)/(a*x+1))^(1/2))+1/4*I*Pi*csgn(I* (a*x+1)/(a*x-1))^3*polylog(2,1/((a*x-1)/(a*x+1))^(1/2))+1/4*I*Pi*csgn(I*(a *x+1)/(a*x-1))^3*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))+1/4*I*Pi*csgn(I/((a *x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/ (a*x-1)-1))*arccoth(a*x)*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-polylog(2,1/((...
\[ \int x \coth ^{-1}(a x)^3 \, dx=\int { x \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]
\[ \int x \coth ^{-1}(a x)^3 \, dx=\int x \operatorname {acoth}^{3}{\left (a x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (82) = 164\).
Time = 0.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.26 \[ \int x \coth ^{-1}(a x)^3 \, dx=\frac {1}{2} \, x^{2} \operatorname {arcoth}\left (a x\right )^{3} + \frac {3}{4} \, a {\left (\frac {2 \, x}{a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{16} \, a {\left (\frac {\frac {3 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} - 3 \, {\left (\log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 6 \, \log \left (a x - 1\right )^{2}}{a} - \frac {24 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a}}{a^{2}} - \frac {6 \, {\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \operatorname {arcoth}\left (a x\right )}{a^{3}}\right )} \]
1/2*x^2*arccoth(a*x)^3 + 3/4*a*(2*x/a^2 - log(a*x + 1)/a^3 + log(a*x - 1)/ a^3)*arccoth(a*x)^2 + 1/16*a*(((3*(log(a*x - 1) - 2)*log(a*x + 1)^2 - log( a*x + 1)^3 + log(a*x - 1)^3 - 3*(log(a*x - 1)^2 - 4*log(a*x - 1))*log(a*x + 1) + 6*log(a*x - 1)^2)/a - 24*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(- 1/2*a*x + 1/2))/a)/a^2 - 6*(2*(log(a*x - 1) - 2)*log(a*x + 1) - log(a*x + 1)^2 - log(a*x - 1)^2 - 4*log(a*x - 1))*arccoth(a*x)/a^3)
\[ \int x \coth ^{-1}(a x)^3 \, dx=\int { x \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x \coth ^{-1}(a x)^3 \, dx=\int x\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \]