\(\int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx\) [115]

Optimal result

Integrand size = 8, antiderivative size = 223 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=\sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )-\frac {1}{6} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{6} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}+\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {6305, 96, 95, 216, 648, 632, 210, 642, 212} \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}\right )+\sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x-\frac {1}{6} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )+\frac {1}{6} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right ) \]

[In]

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

[Out]

(1 + x^(-1))^(1/6)*((-1 + x)/x)^(5/6)*x - ArcTan[(1 - (2*(1 + x^(-1))^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]]/Sqrt
[3] + ArcTan[(1 + (2*(1 + x^(-1))^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]]/Sqrt[3] + (2*ArcTanh[(1 + x^(-1))^(1/6)/
((-1 + x)/x)^(1/6)])/3 - Log[1 + (1 + x^(-1))^(1/3)/((-1 + x)/x)^(1/3) - (1 + x^(-1))^(1/6)/((-1 + x)/x)^(1/6)
]/6 + Log[1 + (1 + x^(-1))^(1/3)/((-1 + x)/x)^(1/3) + (1 + x^(-1))^(1/6)/((-1 + x)/x)^(1/6)]/6

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[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] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[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 210

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 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 216

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*C
os[(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] + Dis
t[2*(r/(a*n)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 6305

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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x} x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right ) \\ & = \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x-2 \text {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \\ & = \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x+\frac {2}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \\ & = \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x+\frac {2}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{6} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \\ & = \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x+\frac {2}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{6} \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}{6} \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 )-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \\ & = \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x+\frac {\arctan \left (\frac {-1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{6} \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}{6} \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 ) \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.16 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=2 e^{\frac {1}{3} \coth ^{-1}(x)} \left (\frac {1}{-1+e^{2 \coth ^{-1}(x)}}+\operatorname {Hypergeometric2F1}\left (\frac {1}{6},1,\frac {7}{6},e^{2 \coth ^{-1}(x)}\right )\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 5.15 (sec) , antiderivative size = 1151, normalized size of antiderivative = 5.16

method	result	size
trager	\(\text {Expression too large to display}\)	\(1151\)
risch	\(\text {Expression too large to display}\)	\(1761\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.72 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx={\left (x + 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]

[In]

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

[Out]

(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]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.75 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{\frac {x - 1}{x + 1} - 1} + \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]

[In]

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

[Out]

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

none

Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.75 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{\frac {x - 1}{x + 1} - 1} + \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{3} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \]

[In]

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

[Out]

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - 1/3*sqrt(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 = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.52 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx=-\frac {\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3}-\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{\frac {x-1}{x+1}-1}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,64{}\mathrm {i}}{-32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{3}-\frac {1}{3}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,64{}\mathrm {i}}{32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{3}+\frac {1}{3}{}\mathrm {i}\right ) \]

[In]

int(1/((x - 1)/(x + 1))^(1/6),x)

[Out]

- (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)