Integrand size = 10, antiderivative size = 258 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=\frac {1}{6} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6} x^2-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {1}{9} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )-\frac {1}{36} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{36} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}+\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \]
1/6*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)*x+1/2*(1+1/x)^(7/6)*((-1+x)/x)^(5/6)*x^ 2+1/9*arctanh((1+1/x)^(1/6)/((-1+x)/x)^(1/6))-1/36*ln(1+(1+1/x)^(1/3)/((-1 +x)/x)^(1/3)-(1+1/x)^(1/6)/((-1+x)/x)^(1/6))+1/36*ln(1+(1+1/x)^(1/3)/((-1+ x)/x)^(1/3)+(1+1/x)^(1/6)/((-1+x)/x)^(1/6))-1/18*arctan(1/3*(1-2*(1+1/x)^( 1/6)/((-1+x)/x)^(1/6))*3^(1/2))*3^(1/2)+1/18*arctan(1/3*(1+2*(1+1/x)^(1/6) /((-1+x)/x)^(1/6))*3^(1/2))*3^(1/2)
Time = 0.63 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.65 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=\frac {1}{36} \left (\frac {72 e^{\frac {1}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^2}+\frac {84 e^{\frac {1}{3} \coth ^{-1}(x)}}{-1+e^{2 \coth ^{-1}(x)}}+2 \sqrt {3} \arctan \left (\frac {-1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-2 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}\right )+2 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}\right )-\log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )+\log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )\right ) \]
((72*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x]))^2 + (84*E^(ArcCoth[x]/3))/( -1 + E^(2*ArcCoth[x])) + 2*Sqrt[3]*ArcTan[(-1 + 2*E^(ArcCoth[x]/3))/Sqrt[3 ]] + 2*Sqrt[3]*ArcTan[(1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 2*Log[1 - E^(Arc Coth[x]/3)] + 2*Log[1 + E^(ArcCoth[x]/3)] - Log[1 - E^(ArcCoth[x]/3) + E^( (2*ArcCoth[x])/3)] + Log[1 + E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)])/36
Time = 0.35 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.90, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6721, 107, 105, 104, 754, 27, 219, 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.
| \(\displaystyle \int x e^{\frac {1}{3} \coth ^{-1}(x)} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\sqrt [6]{1+\frac {1}{x}} x^3}{\sqrt [6]{1-\frac {1}{x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6} x^2-\frac {1}{6} \int \frac {\sqrt [6]{1+\frac {1}{x}} x^2}{\sqrt [6]{1-\frac {1}{x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {1}{6} \left (\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-\frac {1}{3} \int \frac {x}{\sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6} x^2\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{6} \left (\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \int \frac {1}{\frac {1}{x^6}-1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6} x^2\) |
| \(\Big \downarrow \) 754 |
| \(\displaystyle \frac {1}{6} \left (\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (-\frac {1}{3} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{2 \left (-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{2 \left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6} x^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (-\frac {1}{3} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6} x^2\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} \left (\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6} x^2\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{6} \left (\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (\frac {1}{6} \left (\frac {1}{2} \int -\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {3}{2} \int \frac {1}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6} x^2\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6} x^2\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{6} \left (\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (\frac {1}{6} \left (3 \int \frac {1}{-3-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-1\right )-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{6} \left (3 \int \frac {1}{-3-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6} x^2\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} \left (\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (\frac {1}{6} \left (-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\sqrt {3}}\right )\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6} x^2\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{6} \left (\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (\frac {1}{6} \left (\frac {1}{2} \log \left (\frac {1}{x^2}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\frac {1}{x^2}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6} x^2\) |
((1 - x^(-1))^(5/6)*(1 + x^(-1))^(7/6)*x^2)/2 + ((1 - x^(-1))^(5/6)*(1 + x ^(-1))^(1/6)*x - 2*(-1/3*ArcTanh[(1 + x^(-1))^(1/6)/(1 - x^(-1))^(1/6)] + (-(Sqrt[3]*ArcTan[(-1 + (2*(1 + x^(-1))^(1/6))/(1 - x^(-1))^(1/6))/Sqrt[3] ]) + Log[1 - (1 + x^(-1))^(1/6)/(1 - x^(-1))^(1/6) + x^(-2)]/2)/6 + (-(Sqr t[3]*ArcTan[(1 + (2*(1 + x^(-1))^(1/6))/(1 - x^(-1))^(1/6))/Sqrt[3]]) - Lo g[1 + (1 + x^(-1))^(1/6)/(1 - x^(-1))^(1/6) + x^(-2)]/2)/6))/6
3.2.14.3.1 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_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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[((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[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a /b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 *k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] / ; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
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 = 5.40 (sec) , antiderivative size = 1158, normalized size of antiderivative = 4.49
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1158\) |
| risch | \(\text {Expression too large to display}\) | \(1702\) |
1/6*(1+x)*(4+3*x)*(-(1-x)/(1+x))^(5/6)+1/18*ln(3*(-(1-x)/(1+x))^(5/6)*x+9* RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)+9*Root Of(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(2/3)*x+18*RootOf( 9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/2)*x+3*(-(1-x)/(1+x))^(2/3)+18*RootOf(9*_ Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/2)+18*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^( 1/3)*x+18*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)-3*(-(1-x)/(1+x))^(1/3 )*x+9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/6)*x-3*(-(1-x)/(1+x))^(1/3)+ 9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/6)-3*(-(1-x)/(1+x))^(1/6)*x-3*(- (1-x)/(1+x))^(1/6)+3*RootOf(9*_Z^2-3*_Z+1)-2)-1/6*ln(3*(-(1-x)/(1+x))^(5/6 )*x+9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)+ 9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(2/3)*x+18*R ootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/2)*x+3*(-(1-x)/(1+x))^(2/3)+18*Root Of(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/2)+18*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1 +x))^(1/3)*x+18*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)-3*(-(1-x)/(1+x) )^(1/3)*x+9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/6)*x-3*(-(1-x)/(1+x))^ (1/3)+9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/6)-3*(-(1-x)/(1+x))^(1/6)* x-3*(-(1-x)/(1+x))^(1/6)+3*RootOf(9*_Z^2-3*_Z+1)-2)*RootOf(9*_Z^2-3*_Z+1)+ 1/6*RootOf(9*_Z^2-3*_Z+1)*ln(3*(-(1-x)/(1+x))^(5/6)*x-9*RootOf(9*_Z^2-3*_Z +1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)-9*RootOf(9*_Z^2-3*_Z+1)* (-(1-x)/(1+x))^(2/3)+6*(-(1-x)/(1+x))^(2/3)*x-18*RootOf(9*_Z^2-3*_Z+1)*...
Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.65 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=\frac {1}{6} \, {\left (3 \, x^{2} + 7 \, x + 4\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}} - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{18} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]
1/6*(3*x^2 + 7*x + 4)*((x - 1)/(x + 1))^(5/6) - 1/18*sqrt(3)*arctan(2/3*sq rt(3)*((x - 1)/(x + 1))^(1/6) + 1/3*sqrt(3)) - 1/18*sqrt(3)*arctan(2/3*sqr t(3)*((x - 1)/(x + 1))^(1/6) - 1/3*sqrt(3)) + 1/36*log(((x - 1)/(x + 1))^( 1/3) + ((x - 1)/(x + 1))^(1/6) + 1) - 1/36*log(((x - 1)/(x + 1))^(1/3) - ( (x - 1)/(x + 1))^(1/6) + 1) + 1/18*log(((x - 1)/(x + 1))^(1/6) + 1) - 1/18 *log(((x - 1)/(x + 1))^(1/6) - 1)
\[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=\int \frac {x}{\sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.75 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=-\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) + \frac {\left (\frac {x - 1}{x + 1}\right )^{\frac {11}{6}} - 7 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{3 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x + 1} - \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1\right )}} + \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{18} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]
-1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - 1/18*s qrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) - 1)) + 1/3*(((x - 1) /(x + 1))^(11/6) - 7*((x - 1)/(x + 1))^(5/6))/(2*(x - 1)/(x + 1) - (x - 1) ^2/(x + 1)^2 - 1) + 1/36*log(((x - 1)/(x + 1))^(1/3) + ((x - 1)/(x + 1))^( 1/6) + 1) - 1/36*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 1))^(1/6) + 1 ) + 1/18*log(((x - 1)/(x + 1))^(1/6) + 1) - 1/18*log(((x - 1)/(x + 1))^(1/ 6) - 1)
Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.74 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=-\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {\frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{x + 1} - 7 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{3 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{2}} + \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{18} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \]
-1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - 1/18*s qrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) - 1)) - 1/3*((x - 1)* ((x - 1)/(x + 1))^(5/6)/(x + 1) - 7*((x - 1)/(x + 1))^(5/6))/((x - 1)/(x + 1) - 1)^2 + 1/36*log(((x - 1)/(x + 1))^(1/3) + ((x - 1)/(x + 1))^(1/6) + 1) - 1/36*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 1))^(1/6) + 1) + 1/1 8*log(((x - 1)/(x + 1))^(1/6) + 1) - 1/18*log(abs(((x - 1)/(x + 1))^(1/6) - 1))
Time = 4.58 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.55 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=\frac {\frac {7\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{3}-\frac {{\left (\frac {x-1}{x+1}\right )}^{11/6}}{3}}{\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}-\frac {2\,\left (x-1\right )}{x+1}+1}-\frac {\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{9}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,2{}\mathrm {i}}{243\,\left (-\frac {1}{243}+\frac {\sqrt {3}\,1{}\mathrm {i}}{243}\right )}\right )\,\left (\frac {\sqrt {3}}{18}-\frac {1}{18}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,2{}\mathrm {i}}{243\,\left (\frac {1}{243}+\frac {\sqrt {3}\,1{}\mathrm {i}}{243}\right )}\right )\,\left (\frac {\sqrt {3}}{18}+\frac {1}{18}{}\mathrm {i}\right ) \]
((7*((x - 1)/(x + 1))^(5/6))/3 - ((x - 1)/(x + 1))^(11/6)/3)/((x - 1)^2/(x + 1)^2 - (2*(x - 1))/(x + 1) + 1) - (atan(((x - 1)/(x + 1))^(1/6)*1i)*1i) /9 - atan((((x - 1)/(x + 1))^(1/6)*2i)/(243*((3^(1/2)*1i)/243 - 1/243)))*( 3^(1/2)/18 - 1i/18) - atan((((x - 1)/(x + 1))^(1/6)*2i)/(243*((3^(1/2)*1i) /243 + 1/243)))*(3^(1/2)/18 + 1i/18)