\(\int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx\) [123]

Optimal result

Integrand size = 8, antiderivative size = 175 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=\left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{1+\frac {1}{x}} x-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{1-\frac {1}{x}}}\right ) \sqrt [6]{1-\frac {1}{x}}}\right ) \] Output:

(1-1/x)^(5/6)*(1+1/x)^(1/6)*x-1/3*arctan(1/3*(1-2*(1+1/x)^(1/6)/(1-1/x)^(1 
/6))*3^(1/2))*3^(1/2)+1/3*arctan(1/3*(1+2*(1+1/x)^(1/6)/(1-1/x)^(1/6))*3^( 
1/2))*3^(1/2)+2/3*arctanh((1+1/x)^(1/6)/(1-1/x)^(1/6))+1/3*arctanh((1+1/x) 
^(1/6)/(1+(1+1/x)^(1/3)/(1-1/x)^(1/3))/(1-1/x)^(1/6))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.20 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=2 e^{\frac {1}{3} \coth ^{-1}(x)} \left (\frac {1}{-1+e^{2 \coth ^{-1}(x)}}+\operatorname {Hypergeometric2F1}\left (\frac {1}{6},1,\frac {7}{6},e^{2 \coth ^{-1}(x)}\right )\right ) \] Input:

Integrate[E^(ArcCoth[x]/3),x]
 

Output:

2*E^(ArcCoth[x]/3)*((-1 + E^(2*ArcCoth[x]))^(-1) + Hypergeometric2F1[1/6, 
1, 7/6, E^(2*ArcCoth[x])])
 

Rubi [A] (warning: unable to verify)

Time = 0.55 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {6720, 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.

\(\displaystyle \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx\)

\(\Big \downarrow \) 6720

\(\displaystyle -\int \frac {\sqrt [6]{1+\frac {1}{x}} x^2}{\sqrt [6]{1-\frac {1}{x}}}d\frac {1}{x}\)

\(\Big \downarrow \) 105

\(\displaystyle \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-\frac {1}{3} \int \frac {x}{\sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}\)

\(\Big \downarrow \) 104

\(\displaystyle \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \int \frac {1}{\frac {1}{x^6}-1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\)

\(\Big \downarrow \) 754

\(\displaystyle \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (-\frac {1}{3} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{2 \left (-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{2 \left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (-\frac {1}{3} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (\frac {1}{6} \left (\frac {1}{2} \int -\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {3}{2} \int \frac {1}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (\frac {1}{6} \left (3 \int \frac {1}{-3-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-1\right )-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{6} \left (3 \int \frac {1}{-3-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (\frac {1}{6} \left (-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\sqrt {3}}\right )\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-2 \left (\frac {1}{6} \left (\frac {1}{2} \log \left (\frac {1}{x^2}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\frac {1}{x^2}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\)

Input:

Int[E^(ArcCoth[x]/3),x]
 

Output:

(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 + (-(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)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

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]
 

rule 105

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]
 

rule 217

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])
 

rule 219

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])
 

rule 754

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]
 

rule 1083

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 6720

Int[E^(ArcCoth[(a_.)*(x_)]*(n_)), x_Symbol] :> -Subst[Int[(1 + x/a)^(n/2)/( 
x^2*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] &&  !IntegerQ[n]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 7.30 (sec) , antiderivative size = 1151, normalized size of antiderivative = 6.58

method	result	size
trager	\(\text {Expression too large to display}\)	\(1151\)
risch	\(\text {Expression too large to display}\)	\(2884\)

Input:

int(1/((x-1)/(1+x))^(1/6),x,method=_RETURNVERBOSE)
 

Output:

(1+x)*(-(1-x)/(1+x))^(5/6)+RootOf(9*_Z^2+3*_Z+1)*ln(-9*RootOf(9*_Z^2+3*_Z+ 
1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)*x-9*RootOf(9*_Z^2+3*_Z+1) 
*(-(1-x)/(1+x))^(2/3)+18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/2)*x+3*(- 
(1-x)/(1+x))^(5/6)-6*(-(1-x)/(1+x))^(2/3)*x+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-6*(-(1-x)/ 
(1+x))^(2/3)+6*(-(1-x)/(1+x))^(1/2)*x-18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+ 
x))^(1/3)+9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/6)*x+6*(-(1-x)/(1+x))^ 
(1/2)-3*(-(1-x)/(1+x))^(1/3)*x+9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/6 
)-3*(-(1-x)/(1+x))^(1/3)-3*RootOf(9*_Z^2+3*_Z+1)+1)-1/3*ln(9*RootOf(9*_Z^2 
+3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)*x+9*RootOf(9*_Z^2+3 
*_Z+1)*(-(1-x)/(1+x))^(2/3)-18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/2)* 
x+3*(-(1-x)/(1+x))^(5/6)-3*(-(1-x)/(1+x))^(2/3)*x-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-3*(- 
(1-x)/(1+x))^(2/3)+18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/3)-9*RootOf( 
9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/6)*x+3*(-(1-x)/(1+x))^(1/3)*x-9*RootOf(9* 
_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/6)+3*(-(1-x)/(1+x))^(1/3)-3*(-(1-x)/(1+x))^ 
(1/6)*x+3*RootOf(9*_Z^2+3*_Z+1)-3*(-(1-x)/(1+x))^(1/6)+2)-ln(9*RootOf(9*_Z 
^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)*x+9*RootOf(9*_Z^2 
+3*_Z+1)*(-(1-x)/(1+x))^(2/3)-18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/2 
)*x+3*(-(1-x)/(1+x))^(5/6)-3*(-(1-x)/(1+x))^(2/3)*x-18*RootOf(9*_Z^2+3*...
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.91 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx={\left (x + 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \] Input:

integrate(1/((x-1)/(1+x))^(1/6),x, algorithm="fricas")
 

Output:

(x + 1)*((x - 1)/(x + 1))^(5/6) - 1/3*sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/ 
(x + 1))^(1/6) + 1/3*sqrt(3)) - 1/3*sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x 
 + 1))^(1/6) - 1/3*sqrt(3)) + 1/6*log(((x - 1)/(x + 1))^(1/3) + ((x - 1)/( 
x + 1))^(1/6) + 1) - 1/6*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 1))^( 
1/6) + 1) + 1/3*log(((x - 1)/(x + 1))^(1/6) + 1) - 1/3*log(((x - 1)/(x + 1 
))^(1/6) - 1)
 

Sympy [F]

\[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=\int \frac {1}{\sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \] Input:

integrate(1/((x-1)/(1+x))**(1/6),x)
 

Output:

Integral(((x - 1)/(x + 1))**(-1/6), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.95 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{\frac {x - 1}{x + 1} - 1} + \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \] Input:

integrate(1/((x-1)/(1+x))^(1/6),x, algorithm="maxima")
 

Output:

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - 1/3*sqr 
t(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) - 1)) - 2*((x - 1)/(x + 
 1))^(5/6)/((x - 1)/(x + 1) - 1) + 1/6*log(((x - 1)/(x + 1))^(1/3) + ((x - 
 1)/(x + 1))^(1/6) + 1) - 1/6*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 
1))^(1/6) + 1) + 1/3*log(((x - 1)/(x + 1))^(1/6) + 1) - 1/3*log(((x - 1)/( 
x + 1))^(1/6) - 1)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.96 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{\frac {x - 1}{x + 1} - 1} + \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{3} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \] Input:

integrate(1/((x-1)/(1+x))^(1/6),x, algorithm="giac")
 

Output:

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - 1/3*sqr 
t(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) - 1)) - 2*((x - 1)/(x + 
 1))^(5/6)/((x - 1)/(x + 1) - 1) + 1/6*log(((x - 1)/(x + 1))^(1/3) + ((x - 
 1)/(x + 1))^(1/6) + 1) - 1/6*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 
1))^(1/6) + 1) + 1/3*log(((x - 1)/(x + 1))^(1/6) + 1) - 1/3*log(abs(((x - 
1)/(x + 1))^(1/6) - 1))
 

Mupad [B] (verification not implemented)

Time = 23.64 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.66 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=-\frac {\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3}-\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{\frac {x-1}{x+1}-1}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,64{}\mathrm {i}}{-32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{3}-\frac {1}{3}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,64{}\mathrm {i}}{32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{3}+\frac {1}{3}{}\mathrm {i}\right ) \] Input:

int(1/((x - 1)/(x + 1))^(1/6),x)
 

Output:

- (atan(((x - 1)/(x + 1))^(1/6)*1i)*2i)/3 - (2*((x - 1)/(x + 1))^(5/6))/(( 
x - 1)/(x + 1) - 1) - atan((((x - 1)/(x + 1))^(1/6)*64i)/(3^(1/2)*32i - 32 
))*(3^(1/2)/3 - 1i/3) - atan((((x - 1)/(x + 1))^(1/6)*64i)/(3^(1/2)*32i + 
32))*(3^(1/2)/3 + 1i/3)
 

Reduce [F]

\[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=\int \frac {\left (x +1\right )^{\frac {1}{6}}}{\left (x -1\right )^{\frac {1}{6}}}d x \] Input:

int(1/((x-1)/(1+x))^(1/6),x)
 

Output:

int((x + 1)**(1/6)/(x - 1)**(1/6),x)