Integrand size = 10, antiderivative size = 95 \[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\frac {3}{2} a^2 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{2 x^2}+3 a^2 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
3/2*a^2*arccoth(a*x)^2-3/2*a*arccoth(a*x)^2/x+1/2*a^2*arccoth(a*x)^3-1/2*a rccoth(a*x)^3/x^2+3*a^2*arccoth(a*x)*ln(2-2/(a*x+1))-3/2*a^2*polylog(2,-1+ 2/(a*x+1))
Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\frac {1}{2} \left (\frac {\coth ^{-1}(a x) \left (3 a x (-1+a x) \coth ^{-1}(a x)+\left (-1+a^2 x^2\right ) \coth ^{-1}(a x)^2+6 a^2 x^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )\right )}{x^2}-3 a^2 \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )\right ) \]
((ArcCoth[a*x]*(3*a*x*(-1 + a*x)*ArcCoth[a*x] + (-1 + a^2*x^2)*ArcCoth[a*x ]^2 + 6*a^2*x^2*Log[1 + E^(-2*ArcCoth[a*x])]))/x^2 - 3*a^2*PolyLog[2, -E^( -2*ArcCoth[a*x])])/2
Time = 0.79 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6453, 6545, 6453, 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.
\(\displaystyle \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx\) |
\(\Big \downarrow \) 6453 |
\(\displaystyle \frac {3}{2} a \int \frac {\coth ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 6545 |
\(\displaystyle \frac {3}{2} a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx+\int \frac {\coth ^{-1}(a x)^2}{x^2}dx\right )-\frac {\coth ^{-1}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 6453 |
\(\displaystyle \frac {3}{2} a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx+2 a \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)^2}{x}\right )-\frac {\coth ^{-1}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 6511 |
\(\displaystyle \frac {3}{2} a \left (2 a \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{3} a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2}{x}\right )-\frac {\coth ^{-1}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 6551 |
\(\displaystyle \frac {3}{2} a \left (2 a \left (\int \frac {\coth ^{-1}(a x)}{x (a x+1)}dx+\frac {1}{2} \coth ^{-1}(a x)^2\right )+\frac {1}{3} a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2}{x}\right )-\frac {\coth ^{-1}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 6495 |
\(\displaystyle \frac {3}{2} a \left (2 a \left (-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \coth ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)\right )+\frac {1}{3} a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2}{x}\right )-\frac {\coth ^{-1}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 2897 |
\(\displaystyle \frac {3}{2} a \left (2 a \left (-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {1}{2} \coth ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)\right )+\frac {1}{3} a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2}{x}\right )-\frac {\coth ^{-1}(a x)^3}{2 x^2}\) |
-1/2*ArcCoth[a*x]^3/x^2 + (3*a*(-(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
3.1.31.3.1 Defintions of rubi rules used
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 = 3.32 (sec) , antiderivative size = 3498, normalized size of antiderivative = 36.82
method | result | size |
parts | \(\text {Expression too large to display}\) | \(3498\) |
derivativedivides | \(\text {Expression too large to display}\) | \(3502\) |
default | \(\text {Expression too large to display}\) | \(3502\) |
-1/2*arccoth(a*x)^3/x^2-3/2*a^2*(-1/2*polylog(2,-(a*x+1)/(a*x-1))+1/a/x*ar ccoth(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+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*arccot h(a*x)*ln(1-I/((a*x-1)/(a*x+1))^(1/2))-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)*ln(1+(a*x+1)/(a*x-1))+1/4*I*P i*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth(a*x)*l n(1+(a*x+1)/(a*x-1))-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)*ln(1+(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)*ln( 1+I/((a*x-1)/(a*x+1))^(1/2))-1/4*I*Pi*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*cs gn(I*(a*x+1)/(a*x-1))*arccoth(a*x)*ln(1-I/((a*x-1)/(a*x+1))^(1/2))+1/4*I*P i*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^2*ar ccoth(a*x)*ln(1+I/((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-I/((a*x- 1)/(a*x+1))^(1/2))+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)*ln(1+I/((a*x-1)/(a*x+1))^(1/2))+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)*ln(1-I/((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+I/((a*x-1)/...
\[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{3}} \,d x } \]
\[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (84) = 168\).
Time = 0.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.65 \[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\frac {3}{4} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} a \operatorname {arcoth}\left (a x\right )^{2} - \frac {1}{16} \, {\left (a^{2} {\left (\frac {3 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} - 3 \, {\left (\log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 6 \, \log \left (a x - 1\right )^{2}}{a} - \frac {24 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} + \frac {24 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} - \frac {24 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} - 6 \, {\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right ) + 8 \, \log \left (x\right )\right )} a \operatorname {arcoth}\left (a x\right )\right )} a - \frac {\operatorname {arcoth}\left (a x\right )^{3}}{2 \, x^{2}} \]
3/4*(a*log(a*x + 1) - a*log(a*x - 1) - 2/x)*a*arccoth(a*x)^2 - 1/16*(a^2*( (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 + 24*(log(a*x + 1)*log(x) + dilog(-a*x))/a - 24*(log(-a*x + 1)*log(x) + dilog(a*x))/a) - 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) + 8*log(x))*a*arccoth(a*x))*a - 1/2*arccoth(a*x)^3/x^2
\[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\coth ^{-1}(a x)^3}{x^3} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^3} \,d x \]