Integrand size = 10, antiderivative size = 141 \[ \int \frac {\coth ^{-1}(a x)^3}{x^5} \, dx=-\frac {a^3}{4 x}-\frac {a^2 \coth ^{-1}(a x)}{4 x^2}+a^4 \coth ^{-1}(a x)^2-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} a^4 \text {arctanh}(a x)+2 a^4 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a^4 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
-1/4*a^3/x-1/4*a^2*arccoth(a*x)/x^2+a^4*arccoth(a*x)^2-1/4*a*arccoth(a*x)^ 2/x^3-3/4*a^3*arccoth(a*x)^2/x+1/4*a^4*arccoth(a*x)^3-1/4*arccoth(a*x)^3/x ^4+1/4*a^4*arctanh(a*x)+2*a^4*arccoth(a*x)*ln(2-2/(a*x+1))-a^4*polylog(2,- 1+2/(a*x+1))
Time = 0.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84 \[ \int \frac {\coth ^{-1}(a x)^3}{x^5} \, dx=\frac {-a^3 x^3+a x \left (-1-3 a^2 x^2+4 a^3 x^3\right ) \coth ^{-1}(a x)^2+\left (-1+a^4 x^4\right ) \coth ^{-1}(a x)^3+a^2 x^2 \coth ^{-1}(a x) \left (-1+a^2 x^2+8 a^2 x^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )\right )-4 a^4 x^4 \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )}{4 x^4} \]
(-(a^3*x^3) + a*x*(-1 - 3*a^2*x^2 + 4*a^3*x^3)*ArcCoth[a*x]^2 + (-1 + a^4* x^4)*ArcCoth[a*x]^3 + a^2*x^2*ArcCoth[a*x]*(-1 + a^2*x^2 + 8*a^2*x^2*Log[1 + E^(-2*ArcCoth[a*x])]) - 4*a^4*x^4*PolyLog[2, -E^(-2*ArcCoth[a*x])])/(4* x^4)
Time = 1.44 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.35, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6453, 6545, 6453, 6545, 6453, 264, 219, 6511, 6551, 6495, 2897}
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.
| \(\displaystyle \int \frac {\coth ^{-1}(a x)^3}{x^5} \, dx\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {3}{4} a \int \frac {\coth ^{-1}(a x)^2}{x^4 \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 6545 |
| \(\displaystyle \frac {3}{4} a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )}dx+\int \frac {\coth ^{-1}(a x)^2}{x^4}dx\right )-\frac {\coth ^{-1}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {3}{4} a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )}dx+\frac {2}{3} a \int \frac {\coth ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)^2}{3 x^3}\right )-\frac {\coth ^{-1}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 6545 |
| \(\displaystyle \frac {3}{4} a \left (a^2 \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx+\int \frac {\coth ^{-1}(a x)^2}{x^2}dx\right )+\frac {2}{3} a \left (a^2 \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx+\int \frac {\coth ^{-1}(a x)}{x^3}dx\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\right )-\frac {\coth ^{-1}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {3}{4} a \left (a^2 \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx+2 a \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)^2}{x}\right )+\frac {2}{3} a \left (\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )}dx+a^2 \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\right )-\frac {\coth ^{-1}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {3}{4} a \left (a^2 \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx+2 a \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)^2}{x}\right )+\frac {2}{3} a \left (\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx-\frac {1}{x}\right )+a^2 \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\right )-\frac {\coth ^{-1}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a \left (a^2 \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {\coth ^{-1}(a x)}{2 x^2}\right )+a^2 \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx+2 a \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)^2}{x}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\right )-\frac {\coth ^{-1}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 6511 |
| \(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a \left (a^2 \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {\coth ^{-1}(a x)}{2 x^2}\right )+a^2 \left (2 a \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{3} a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2}{x}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\right )-\frac {\coth ^{-1}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 6551 |
| \(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a \left (a^2 \left (\int \frac {\coth ^{-1}(a x)}{x (a x+1)}dx+\frac {1}{2} \coth ^{-1}(a x)^2\right )+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {\coth ^{-1}(a x)}{2 x^2}\right )+a^2 \left (2 a \left (\int \frac {\coth ^{-1}(a x)}{x (a x+1)}dx+\frac {1}{2} \coth ^{-1}(a x)^2\right )+\frac {1}{3} a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2}{x}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\right )-\frac {\coth ^{-1}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 6495 |
| \(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a \left (a^2 \left (-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \coth ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)\right )+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {\coth ^{-1}(a x)}{2 x^2}\right )+a^2 \left (2 a \left (-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \coth ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)\right )+\frac {1}{3} a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2}{x}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\right )-\frac {\coth ^{-1}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a \left (a^2 \left (-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {1}{2} \coth ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)\right )+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {\coth ^{-1}(a x)}{2 x^2}\right )+a^2 \left (2 a \left (-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {1}{2} \coth ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)\right )+\frac {1}{3} a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2}{x}\right )-\frac {\coth ^{-1}(a x)^2}{3 x^3}\right )-\frac {\coth ^{-1}(a x)^3}{4 x^4}\) |
-1/4*ArcCoth[a*x]^3/x^4 + (3*a*(-1/3*ArcCoth[a*x]^2/x^3 + a^2*(-(ArcCoth[a *x]^2/x) + (a*ArcCoth[a*x]^3)/3 + 2*a*(ArcCoth[a*x]^2/2 + ArcCoth[a*x]*Log [2 - 2/(1 + a*x)] - PolyLog[2, -1 + 2/(1 + a*x)]/2)) + (2*a*(-1/2*ArcCoth[ a*x]/x^2 + (a*(-x^(-1) + a*ArcTanh[a*x]))/2 + a^2*(ArcCoth[a*x]^2/2 + ArcC oth[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog[2, -1 + 2/(1 + a*x)]/2)))/3))/4
3.1.33.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
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_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcCoth[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[2 - 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[1/d Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x ], x] - Simp[e/(d*f^2) 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] && LtQ[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*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcCoth[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.73 (sec) , antiderivative size = 657, normalized size of antiderivative = 4.66
| method | result | size |
| parts | \(-\frac {\operatorname {arccoth}\left (a x \right )^{3}}{4 x^{4}}-\frac {3 a^{4} \left (\frac {\operatorname {arccoth}\left (a x \right )^{2}}{3 a^{3} x^{3}}+\frac {\operatorname {arccoth}\left (a x \right )^{2}}{a x}-\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}-\frac {8 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1+\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{3}-\frac {8 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1-\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{3}-\frac {8 \operatorname {dilog}\left (1+\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{3}-\frac {8 \operatorname {dilog}\left (1-\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{3}-\frac {a x -1}{3 a x}-\frac {\operatorname {arccoth}\left (a x \right )^{3}}{3}+\frac {2 \,\operatorname {arccoth}\left (a x \right ) \left (a x +1\right )}{3 a x}+\frac {4 \operatorname {arccoth}\left (a x \right )^{2}}{3}+\frac {i \operatorname {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} \operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{4}-\frac {\operatorname {arccoth}\left (a x \right ) \left (a x +1\right )^{2}}{3 a^{2} x^{2}}-\frac {2 \,\operatorname {arccoth}\left (a x \right ) \left (a x -1\right ) \left (a x +1\right )}{3 a^{2} x^{2}}+\frac {i \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2} \operatorname {arccoth}\left (a x \right )^{2} \pi }{2}+\frac {i \operatorname {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} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{4}-\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}-\frac {i \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{4}-\frac {i \operatorname {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} \operatorname {arccoth}\left (a x \right )^{2} \pi }{4}-\frac {i \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3} \operatorname {arccoth}\left (a x \right )^{2} \pi }{4}-\frac {i \operatorname {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 ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{4}\right )}{4}\) | \(657\) |
| derivativedivides | \(a^{4} \left (-\frac {\operatorname {arccoth}\left (a x \right )^{3}}{4 a^{4} x^{4}}-\frac {\operatorname {arccoth}\left (a x \right )^{2}}{4 a^{3} x^{3}}-\frac {3 \operatorname {arccoth}\left (a x \right )^{2}}{4 a x}+\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{8}-\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{8}+2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1+\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1-\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \operatorname {dilog}\left (1+\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \operatorname {dilog}\left (1-\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+\frac {a x -1}{4 a x}+\frac {\operatorname {arccoth}\left (a x \right )^{3}}{4}-\frac {\operatorname {arccoth}\left (a x \right ) \left (a x +1\right )}{2 a x}-\operatorname {arccoth}\left (a x \right )^{2}-\frac {3 i \operatorname {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} \operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}+\frac {\operatorname {arccoth}\left (a x \right ) \left (a x +1\right )^{2}}{4 a^{2} x^{2}}+\frac {\operatorname {arccoth}\left (a x \right ) \left (a x -1\right ) \left (a x +1\right )}{2 a^{2} x^{2}}-\frac {3 i \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2} \operatorname {arccoth}\left (a x \right )^{2} \pi }{8}-\frac {3 i \operatorname {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} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}+\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{8}+\frac {3 i \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}+\frac {3 i \operatorname {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} \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}+\frac {3 i \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3} \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}+\frac {3 i \operatorname {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 ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}\right )\) | \(659\) |
| default | \(a^{4} \left (-\frac {\operatorname {arccoth}\left (a x \right )^{3}}{4 a^{4} x^{4}}-\frac {\operatorname {arccoth}\left (a x \right )^{2}}{4 a^{3} x^{3}}-\frac {3 \operatorname {arccoth}\left (a x \right )^{2}}{4 a x}+\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{8}-\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{8}+2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1+\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1-\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \operatorname {dilog}\left (1+\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \operatorname {dilog}\left (1-\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+\frac {a x -1}{4 a x}+\frac {\operatorname {arccoth}\left (a x \right )^{3}}{4}-\frac {\operatorname {arccoth}\left (a x \right ) \left (a x +1\right )}{2 a x}-\operatorname {arccoth}\left (a x \right )^{2}-\frac {3 i \operatorname {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} \operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}+\frac {\operatorname {arccoth}\left (a x \right ) \left (a x +1\right )^{2}}{4 a^{2} x^{2}}+\frac {\operatorname {arccoth}\left (a x \right ) \left (a x -1\right ) \left (a x +1\right )}{2 a^{2} x^{2}}-\frac {3 i \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2} \operatorname {arccoth}\left (a x \right )^{2} \pi }{8}-\frac {3 i \operatorname {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} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}+\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{8}+\frac {3 i \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}+\frac {3 i \operatorname {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} \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}+\frac {3 i \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3} \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}+\frac {3 i \operatorname {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 ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {arccoth}\left (a x \right )^{2} \pi }{16}\right )\) | \(659\) |
-1/4*arccoth(a*x)^3/x^4-3/4*a^4*(1/3/a^3/x^3*arccoth(a*x)^2+1/a/x*arccoth( a*x)^2-1/2*arccoth(a*x)^2*ln(a*x+1)+1/2*arccoth(a*x)^2*ln(a*x-1)-8/3*arcco th(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1/2))-8/3*arccoth(a*x)*ln(1-I/((a*x-1)/( a*x+1))^(1/2))-8/3*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))-8/3*dilog(1-I/((a*x- 1)/(a*x+1))^(1/2))-1/3*(a*x-1)/a/x-1/3*arccoth(a*x)^3+2/3*arccoth(a*x)*(a* x+1)/a/x+4/3*arccoth(a*x)^2+1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I/(a *x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^2*arccoth(a*x)^2-1/3*arccoth(a*x)*(a*x+ 1)^2/a^2/x^2-2/3*arccoth(a*x)*(a*x-1)*(a*x+1)/a^2/x^2+1/2*I*Pi*csgn(I/((a* x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*arccoth(a*x)^2+1/4*I*Pi*csg n(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)^2-1/2*arccoth(a*x)^2*ln((a*x-1)/(a*x+1))-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)^2-1/4*I*Pi*csgn(I/(a* x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^3*arccoth(a*x)^2-1/4*I*Pi*csgn(I*(a*x+1) /(a*x-1))^3*arccoth(a*x)^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)
\[ \int \frac {\coth ^{-1}(a x)^3}{x^5} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{5}} \,d x } \]
\[ \int \frac {\coth ^{-1}(a x)^3}{x^5} \, dx=\int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x^{5}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (126) = 252\).
Time = 0.22 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.43 \[ \int \frac {\coth ^{-1}(a x)^3}{x^5} \, dx=\frac {1}{8} \, {\left (3 \, a^{3} \log \left (a x + 1\right ) - 3 \, a^{3} \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} + 1\right )}}{x^{3}}\right )} a \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{32} \, {\left ({\left (32 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a - 32 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a + 32 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a + 4 \, a \log \left (a x + 1\right ) - 4 \, a \log \left (a x - 1\right ) + \frac {a x \log \left (a x + 1\right )^{3} - a x \log \left (a x - 1\right )^{3} - 8 \, a x \log \left (a x - 1\right )^{2} - {\left (3 \, a x \log \left (a x - 1\right ) - 8 \, a x\right )} \log \left (a x + 1\right )^{2} + {\left (3 \, a x \log \left (a x - 1\right )^{2} - 16 \, a x \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) - 8}{x}\right )} a^{2} + 2 \, {\left (32 \, a^{2} \log \left (x\right ) - \frac {3 \, a^{2} x^{2} \log \left (a x + 1\right )^{2} + 3 \, a^{2} x^{2} \log \left (a x - 1\right )^{2} + 16 \, a^{2} x^{2} \log \left (a x - 1\right ) - 2 \, {\left (3 \, a^{2} x^{2} \log \left (a x - 1\right ) - 8 \, a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 4}{x^{2}}\right )} a \operatorname {arcoth}\left (a x\right )\right )} a - \frac {\operatorname {arcoth}\left (a x\right )^{3}}{4 \, x^{4}} \]
1/8*(3*a^3*log(a*x + 1) - 3*a^3*log(a*x - 1) - 2*(3*a^2*x^2 + 1)/x^3)*a*ar ccoth(a*x)^2 + 1/32*((32*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a - 32*(log(a*x + 1)*log(x) + dilog(-a*x))*a + 32*(log(-a*x + 1)* log(x) + dilog(a*x))*a + 4*a*log(a*x + 1) - 4*a*log(a*x - 1) + (a*x*log(a* x + 1)^3 - a*x*log(a*x - 1)^3 - 8*a*x*log(a*x - 1)^2 - (3*a*x*log(a*x - 1) - 8*a*x)*log(a*x + 1)^2 + (3*a*x*log(a*x - 1)^2 - 16*a*x*log(a*x - 1))*lo g(a*x + 1) - 8)/x)*a^2 + 2*(32*a^2*log(x) - (3*a^2*x^2*log(a*x + 1)^2 + 3* a^2*x^2*log(a*x - 1)^2 + 16*a^2*x^2*log(a*x - 1) - 2*(3*a^2*x^2*log(a*x - 1) - 8*a^2*x^2)*log(a*x + 1) + 4)/x^2)*a*arccoth(a*x))*a - 1/4*arccoth(a*x )^3/x^4
\[ \int \frac {\coth ^{-1}(a x)^3}{x^5} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\coth ^{-1}(a x)^3}{x^5} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^5} \,d x \]