\(\int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx\) [114]

Optimal result

Integrand size = 10, antiderivative size = 258 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=\frac {1}{6} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6} x^2-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {1}{9} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )-\frac {1}{36} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{36} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}+\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \]

[Out]

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)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6306, 98, 96, 95, 216, 648, 632, 210, 642, 212} \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {1}{9} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}\right )+\frac {1}{2} \left (\frac {1}{x}+1\right )^{7/6} \left (\frac {x-1}{x}\right )^{5/6} x^2+\frac {1}{6} \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x-\frac {1}{36} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )+\frac {1}{36} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right ) \]

[In]

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

[Out]

((1 + x^(-1))^(1/6)*((-1 + x)/x)^(5/6)*x)/6 + ((1 + x^(-1))^(7/6)*((-1 + x)/x)^(5/6)*x^2)/2 - ArcTan[(1 - (2*(
1 + x^(-1))^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]]/(6*Sqrt[3]) + ArcTan[(1 + (2*(1 + x^(-1))^(1/6))/((-1 + x)/x)^
(1/6))/Sqrt[3]]/(6*Sqrt[3]) + ArcTanh[(1 + x^(-1))^(1/6)/((-1 + x)/x)^(1/6)]/9 - Log[1 + (1 + x^(-1))^(1/3)/((
-1 + x)/x)^(1/3) - (1 + x^(-1))^(1/6)/((-1 + x)/x)^(1/6)]/36 + Log[1 + (1 + x^(-1))^(1/3)/((-1 + x)/x)^(1/3) +
(1 + x^(-1))^(1/6)/((-1 + x)/x)^(1/6)]/36

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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[(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] || Sum
SimplerQ[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 6306

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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x} x^3} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6} x^2-\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x} x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \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 {1}{18} \text {Subst}\left (\int \frac {1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right ) \\ & = \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 {1}{3} \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 ) \\ & = \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 {1}{9} \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 {1}{9} \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 {1}{9} \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 {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 {1}{9} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{36} \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}{36} \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}{12} \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}{12} \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}{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 {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 )-\frac {1}{6} \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 )-\frac {1}{6} \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 ) \\ & = \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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.65 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=\frac {1}{36} \left (\frac {72 e^{\frac {1}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^2}+\frac {84 e^{\frac {1}{3} \coth ^{-1}(x)}}{-1+e^{2 \coth ^{-1}(x)}}+2 \sqrt {3} \arctan \left (\frac {-1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-2 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}\right )+2 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}\right )-\log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )+\log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )\right ) \]

[In]

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

[Out]

((72*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x]))^2 + (84*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x])) + 2*Sqrt[3]*A
rcTan[(-1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 2*Log[1 - E^(A
rcCoth[x]/3)] + 2*Log[1 + E^(ArcCoth[x]/3)] - Log[1 - E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)] + Log[1 + E^(Ar
cCoth[x]/3) + E^((2*ArcCoth[x])/3)])/36

Maple [C] (verified)

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	\(\text {Expression too large to display}\)	\(1702\)

[In]

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

[Out]

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*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(2/3)*x+18*RootO
f(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)+1
8*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*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)*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)*(-(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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.65 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=\frac {1}{6} \, {\left (3 \, x^{2} + 7 \, x + 4\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}} - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{18} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]

[In]

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

[Out]

1/6*(3*x^2 + 7*x + 4)*((x - 1)/(x + 1))^(5/6) - 1/18*sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 1/3*
sqrt(3)) - 1/18*sqrt(3)*arctan(2/3*sqrt(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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.75 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=-\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) + \frac {\left (\frac {x - 1}{x + 1}\right )^{\frac {11}{6}} - 7 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{3 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x + 1} - \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1\right )}} + \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{18} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]

[In]

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

[Out]

-1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - 1/18*sqrt(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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.74 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=-\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {\frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{x + 1} - 7 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{3 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{2}} + \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{36} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{18} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \]

[In]

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

[Out]

-1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - 1/18*sqrt(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) + 1/18*log(((x - 1)/(x + 1))^(1/6) + 1) - 1/18*log(abs(((x - 1)/(x + 1))^(1
/6) - 1))

Mupad [B] (verification not implemented)

Time = 4.58 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.55 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x \, dx=\frac {\frac {7\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{3}-\frac {{\left (\frac {x-1}{x+1}\right )}^{11/6}}{3}}{\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}-\frac {2\,\left (x-1\right )}{x+1}+1}-\frac {\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{9}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,2{}\mathrm {i}}{243\,\left (-\frac {1}{243}+\frac {\sqrt {3}\,1{}\mathrm {i}}{243}\right )}\right )\,\left (\frac {\sqrt {3}}{18}-\frac {1}{18}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,2{}\mathrm {i}}{243\,\left (\frac {1}{243}+\frac {\sqrt {3}\,1{}\mathrm {i}}{243}\right )}\right )\,\left (\frac {\sqrt {3}}{18}+\frac {1}{18}{}\mathrm {i}\right ) \]

[In]

int(x/((x - 1)/(x + 1))^(1/6),x)

[Out]

((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/1
8)