Integrand size = 12, antiderivative size = 233 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^2} \, dx=\sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6}-\frac {1}{3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {1}{3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {2}{3} \arctan \left (\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {\log \left (1-\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{2 \sqrt {3}}-\frac {\log \left (1+\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{2 \sqrt {3}} \]
(1+1/x)^(1/6)*((-1+x)/x)^(5/6)+2/3*arctan(((-1+x)/x)^(1/6)/(1+1/x)^(1/6))+ 1/3*arctan(2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)-3^(1/2))+1/3*arctan(2*((-1+x)/ x)^(1/6)/(1+1/x)^(1/6)+3^(1/2))+1/6*ln(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)-(( -1+x)/x)^(1/6)*3^(1/2)/(1+1/x)^(1/6))*3^(1/2)-1/6*ln(1+((-1+x)/x)^(1/3)/(1 +1/x)^(1/3)+((-1+x)/x)^(1/6)*3^(1/2)/(1+1/x)^(1/6))*3^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.17 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^2} \, dx=-2 e^{\frac {1}{3} \coth ^{-1}(x)} \left (-\frac {1}{1+e^{2 \coth ^{-1}(x)}}+\operatorname {Hypergeometric2F1}\left (\frac {1}{6},1,\frac {7}{6},-e^{2 \coth ^{-1}(x)}\right )\right ) \]
-2*E^(ArcCoth[x]/3)*(-(1 + E^(2*ArcCoth[x]))^(-1) + Hypergeometric2F1[1/6, 1, 7/6, -E^(2*ArcCoth[x])])
Time = 0.39 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.92, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6721, 60, 73, 854, 824, 27, 216, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^2} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}-\frac {1}{3} \int \frac {1}{\sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle 2 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle 2 \int \frac {1}{\left (1+\frac {1}{x^6}\right ) x^4}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle 2 \left (\frac {1}{3} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{3} \int -\frac {1-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{2 \left (-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{3} \int -\frac {\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+1}{2 \left (\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {1}{3} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 2 \left (-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 2 \left (\frac {1}{6} \left (-\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (-\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {1}{6} \left (-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\frac {1}{6} \left (\arctan \left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\frac {1}{6} \left (\arctan \left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )\right )\right )+\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}\) |
(1 - x^(-1))^(5/6)*(1 + x^(-1))^(1/6) + 2*(ArcTan[(1 - x^(-1))^(1/6)/(2 - x^(-6))^(1/6)]/3 + (-ArcTan[Sqrt[3] - (2*(1 - x^(-1))^(1/6))/(2 - x^(-6))^ (1/6)] + (Sqrt[3]*Log[1 - (Sqrt[3]*(1 - x^(-1))^(1/6))/(2 - x^(-6))^(1/6) + x^(-2)])/2)/6 + (ArcTan[Sqrt[3] + (2*(1 - x^(-1))^(1/6))/(2 - x^(-6))^(1 /6)] - (Sqrt[3]*Log[1 + (Sqrt[3]*(1 - x^(-1))^(1/6))/(2 - x^(-6))^(1/6) + x^(-2)])/2)/6)
3.2.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x /a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && !IntegerQ[n] && IntegerQ[m]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 16.72 (sec) , antiderivative size = 1487, normalized size of antiderivative = 6.38
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1487\) |
| risch | \(\text {Expression too large to display}\) | \(2991\) |
(1+x)*(-(1-x)/(1+x))^(5/6)/x-9*RootOf(81*_Z^4-9*_Z^2+1)^3*ln((-27*RootOf(8 1*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(2/3)*x-27*RootOf(81*_Z^4-9*_Z^2+1)^3*(- (1-x)/(1+x))^(2/3)+(-(1-x)/(1+x))^(5/6)*x-3*RootOf(81*_Z^4-9*_Z^2+1)*(-(1- x)/(1+x))^(2/3)*x+18*x*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/2)-27* RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(1/3)*x+(-(1-x)/(1+x))^(5/6)-3*R ootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(2/3)+18*RootOf(81*_Z^4-9*_Z^2+1)^2 *(-(1-x)/(1+x))^(1/2)-27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(1/3)-2 *(-(1-x)/(1+x))^(1/2)*x+6*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(1/3)*x- 9*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/6)*x+18*RootOf(81*_Z^4-9*_Z ^2+1)^3*x-2*(-(1-x)/(1+x))^(1/2)+6*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x)) ^(1/3)-9*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/6)-RootOf(81*_Z^4-9* _Z^2+1)*x)/x)+RootOf(81*_Z^4-9*_Z^2+1)*ln((-27*RootOf(81*_Z^4-9*_Z^2+1)^3* (-(1-x)/(1+x))^(2/3)*x-27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(2/3)+ (-(1-x)/(1+x))^(5/6)*x-3*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+1 8*x*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/2)-27*RootOf(81*_Z^4-9*_Z ^2+1)^3*(-(1-x)/(1+x))^(1/3)*x+(-(1-x)/(1+x))^(5/6)-3*RootOf(81*_Z^4-9*_Z^ 2+1)*(-(1-x)/(1+x))^(2/3)+18*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/ 2)-27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(1/3)-2*(-(1-x)/(1+x))^(1/ 2)*x+6*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(1/3)*x-9*RootOf(81*_Z^4-9* _Z^2+1)^2*(-(1-x)/(1+x))^(1/6)*x+18*RootOf(81*_Z^4-9*_Z^2+1)^3*x-2*(-(1...
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^2} \, dx=\frac {x \sqrt {2 i \, \sqrt {3} + 2} \log \left (\sqrt {2 i \, \sqrt {3} + 2} {\left (i \, \sqrt {3} - 1\right )} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - x \sqrt {2 i \, \sqrt {3} + 2} \log \left (\sqrt {2 i \, \sqrt {3} + 2} {\left (-i \, \sqrt {3} + 1\right )} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - x \sqrt {-2 i \, \sqrt {3} + 2} \log \left ({\left (i \, \sqrt {3} + 1\right )} \sqrt {-2 i \, \sqrt {3} + 2} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + x \sqrt {-2 i \, \sqrt {3} + 2} \log \left ({\left (-i \, \sqrt {3} - 1\right )} \sqrt {-2 i \, \sqrt {3} + 2} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 4 \, x \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 6 \, {\left (x + 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{6 \, x} \]
1/6*(x*sqrt(2*I*sqrt(3) + 2)*log(sqrt(2*I*sqrt(3) + 2)*(I*sqrt(3) - 1) + 4 *((x - 1)/(x + 1))^(1/6)) - x*sqrt(2*I*sqrt(3) + 2)*log(sqrt(2*I*sqrt(3) + 2)*(-I*sqrt(3) + 1) + 4*((x - 1)/(x + 1))^(1/6)) - x*sqrt(-2*I*sqrt(3) + 2)*log((I*sqrt(3) + 1)*sqrt(-2*I*sqrt(3) + 2) + 4*((x - 1)/(x + 1))^(1/6)) + x*sqrt(-2*I*sqrt(3) + 2)*log((-I*sqrt(3) - 1)*sqrt(-2*I*sqrt(3) + 2) + 4*((x - 1)/(x + 1))^(1/6)) + 4*x*arctan(((x - 1)/(x + 1))^(1/6)) + 6*(x + 1)*((x - 1)/(x + 1))^(5/6))/x
\[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^2} \, dx=\int \frac {1}{x^{2} \sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^2} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{\frac {x - 1}{x + 1} + 1} + \frac {1}{3} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{3} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {2}{3} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]
-1/6*sqrt(3)*log(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/6*sqrt(3)*log(-sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1 ))^(1/3) + 1) + 2*((x - 1)/(x + 1))^(5/6)/((x - 1)/(x + 1) + 1) + 1/3*arct an(sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 1/3*arctan(-sqrt(3) + 2*((x - 1) /(x + 1))^(1/6)) + 2/3*arctan(((x - 1)/(x + 1))^(1/6))
Time = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^2} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{\frac {x - 1}{x + 1} + 1} + \frac {1}{3} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{3} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {2}{3} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]
-1/6*sqrt(3)*log(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/6*sqrt(3)*log(-sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1 ))^(1/3) + 1) + 2*((x - 1)/(x + 1))^(5/6)/((x - 1)/(x + 1) + 1) + 1/3*arct an(sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 1/3*arctan(-sqrt(3) + 2*((x - 1) /(x + 1))^(1/6)) + 2/3*arctan(((x - 1)/(x + 1))^(1/6))
Time = 4.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.47 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^2} \, dx=\frac {2\,\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\right )}{3}+\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{\frac {x-1}{x+1}+1}-\mathrm {atan}\left (\frac {64\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{-32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )-\mathrm {atan}\left (\frac {64\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right ) \]