Integrand size = 10, antiderivative size = 224 \[ \int e^{\frac {\text {arctanh}(x)}{3}} x \, dx=-\frac {1}{6} (1-x)^{5/6} \sqrt [6]{1+x}-\frac {1}{2} (1-x)^{5/6} (1+x)^{7/6}-\frac {1}{9} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {1}{18} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {1}{18} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt {3}}+\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt {3}} \]
-1/6*(1-x)^(5/6)*(1+x)^(1/6)-1/2*(1-x)^(5/6)*(1+x)^(7/6)-1/9*arctan((1-x)^ (1/6)/(1+x)^(1/6))-1/18*arctan(2*(1-x)^(1/6)/(1+x)^(1/6)-3^(1/2))-1/18*arc tan(2*(1-x)^(1/6)/(1+x)^(1/6)+3^(1/2))-1/36*ln(1+(1-x)^(1/3)/(1+x)^(1/3)-( 1-x)^(1/6)*3^(1/2)/(1+x)^(1/6))*3^(1/2)+1/36*ln(1+(1-x)^(1/3)/(1+x)^(1/3)+ (1-x)^(1/6)*3^(1/2)/(1+x)^(1/6))*3^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.22 \[ \int e^{\frac {\text {arctanh}(x)}{3}} x \, dx=-\frac {1}{10} (1-x)^{5/6} \left (5 (1+x)^{7/6}+2 \sqrt [6]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {5}{6},\frac {11}{6},\frac {1-x}{2}\right )\right ) \]
-1/10*((1 - x)^(5/6)*(5*(1 + x)^(7/6) + 2*2^(1/6)*Hypergeometric2F1[-1/6, 5/6, 11/6, (1 - x)/2]))
Time = 0.40 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.97, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {6676, 90, 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.
| \(\displaystyle \int x e^{\frac {\text {arctanh}(x)}{3}} \, dx\) |
| \(\Big \downarrow \) 6676 |
| \(\displaystyle \int \frac {x \sqrt [6]{x+1}}{\sqrt [6]{1-x}}dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{6} \int \frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}dx-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \int \frac {1}{\sqrt [6]{1-x} (x+1)^{5/6}}dx-(1-x)^{5/6} \sqrt [6]{x+1}\right )-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{6} \left (-2 \int \frac {(1-x)^{2/3}}{(x+1)^{5/6}}d\sqrt [6]{1-x}-(1-x)^{5/6} \sqrt [6]{x+1}\right )-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {1}{6} \left (-2 \int \frac {(1-x)^{2/3}}{2-x}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}-(1-x)^{5/6} \sqrt [6]{x+1}\right )-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {1}{6} \left (-2 \left (\frac {1}{3} \int \frac {1}{\sqrt [3]{1-x}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\frac {1}{3} \int -\frac {1-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}}{2 \left (\sqrt [3]{1-x}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\frac {1}{3} \int -\frac {\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}{2 \left (\sqrt [3]{1-x}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-(1-x)^{5/6} \sqrt [6]{x+1}\right )-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (-2 \left (\frac {1}{3} \int \frac {1}{\sqrt [3]{1-x}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}}{\sqrt [3]{1-x}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}{\sqrt [3]{1-x}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-(1-x)^{5/6} \sqrt [6]{x+1}\right )-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{6} \left (-2 \left (-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}}{\sqrt [3]{1-x}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}{\sqrt [3]{1-x}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )\right )-(1-x)^{5/6} \sqrt [6]{x+1}\right )-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{6} \left (-2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-x}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}}{\sqrt [3]{1-x}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-x}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}}{\sqrt [3]{1-x}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )\right )-(1-x)^{5/6} \sqrt [6]{x+1}\right )-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (-2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-x}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}}{\sqrt [3]{1-x}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-x}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}}{\sqrt [3]{1-x}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )\right )-(1-x)^{5/6} \sqrt [6]{x+1}\right )-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{6} \left (-2 \left (\frac {1}{6} \left (-\int \frac {1}{-\sqrt [3]{1-x}-1}d\left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}-\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}}{\sqrt [3]{1-x}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {1}{6} \left (-\int \frac {1}{-\sqrt [3]{1-x}-1}d\left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}}{\sqrt [3]{1-x}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )\right )-(1-x)^{5/6} \sqrt [6]{x+1}\right )-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} \left (-2 \left (\frac {1}{6} \left (-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}}{\sqrt [3]{1-x}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )\right )+\frac {1}{6} \left (\arctan \left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}}{\sqrt [3]{1-x}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1}d\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )\right )-(1-x)^{5/6} \sqrt [6]{x+1}\right )-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{6} \left (-2 \left (\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\sqrt [3]{1-x}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )\right )+\frac {1}{6} \left (\arctan \left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\sqrt [3]{1-x}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )\right )\right )-(1-x)^{5/6} \sqrt [6]{x+1}\right )-\frac {1}{2} (1-x)^{5/6} (x+1)^{7/6}\) |
-1/2*((1 - x)^(5/6)*(1 + x)^(7/6)) + (-((1 - x)^(5/6)*(1 + x)^(1/6)) - 2*( ArcTan[(1 - x)^(1/6)/(1 + x)^(1/6)]/3 + (-ArcTan[Sqrt[3] - (2*(1 - x)^(1/6 ))/(1 + x)^(1/6)] + (Sqrt[3]*Log[1 + (1 - x)^(1/3) - (Sqrt[3]*(1 - x)^(1/6 ))/(1 + x)^(1/6)])/2)/6 + (ArcTan[Sqrt[3] + (2*(1 - x)^(1/6))/(1 + x)^(1/6 )] - (Sqrt[3]*Log[1 + (1 - x)^(1/3) + (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6 )])/2)/6))/6
3.2.22.3.1 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) ^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] && !Int egerQ[(n - 1)/2]
\[\int {\left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )}^{\frac {1}{3}} x d x\]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.11 \[ \int e^{\frac {\text {arctanh}(x)}{3}} x \, dx=\frac {1}{36} \, \sqrt {2} \sqrt {i \, \sqrt {3} + 1} \log \left (\sqrt {2} \sqrt {i \, \sqrt {3} + 1} + 2 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) - \frac {1}{36} \, \sqrt {2} \sqrt {i \, \sqrt {3} + 1} \log \left (-\sqrt {2} \sqrt {i \, \sqrt {3} + 1} + 2 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) + \frac {1}{36} \, \sqrt {2} \sqrt {-i \, \sqrt {3} + 1} \log \left (\sqrt {2} \sqrt {-i \, \sqrt {3} + 1} + 2 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) - \frac {1}{36} \, \sqrt {2} \sqrt {-i \, \sqrt {3} + 1} \log \left (-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1} + 2 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) + \frac {1}{6} \, {\left (3 \, x^{2} + x - 4\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + \frac {1}{9} \, \arctan \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) \]
1/36*sqrt(2)*sqrt(I*sqrt(3) + 1)*log(sqrt(2)*sqrt(I*sqrt(3) + 1) + 2*(-sqr t(-x^2 + 1)/(x - 1))^(1/3)) - 1/36*sqrt(2)*sqrt(I*sqrt(3) + 1)*log(-sqrt(2 )*sqrt(I*sqrt(3) + 1) + 2*(-sqrt(-x^2 + 1)/(x - 1))^(1/3)) + 1/36*sqrt(2)* sqrt(-I*sqrt(3) + 1)*log(sqrt(2)*sqrt(-I*sqrt(3) + 1) + 2*(-sqrt(-x^2 + 1) /(x - 1))^(1/3)) - 1/36*sqrt(2)*sqrt(-I*sqrt(3) + 1)*log(-sqrt(2)*sqrt(-I* sqrt(3) + 1) + 2*(-sqrt(-x^2 + 1)/(x - 1))^(1/3)) + 1/6*(3*x^2 + x - 4)*(- sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1/9*arctan((-sqrt(-x^2 + 1)/(x - 1))^(1/3) )
\[ \int e^{\frac {\text {arctanh}(x)}{3}} x \, dx=\int x \sqrt [3]{\frac {x + 1}{\sqrt {1 - x^{2}}}}\, dx \]
\[ \int e^{\frac {\text {arctanh}(x)}{3}} x \, dx=\int { x \left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {1}{3}} \,d x } \]
\[ \int e^{\frac {\text {arctanh}(x)}{3}} x \, dx=\int { x \left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {1}{3}} \,d x } \]
Timed out. \[ \int e^{\frac {\text {arctanh}(x)}{3}} x \, dx=\int x\,{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{1/3} \,d x \]