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

Optimal. Leaf size=223 \[ \sqrt [6]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{5/6} x-\frac{1}{6} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}-\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )+\frac{1}{6} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}+\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{2}{3} \tanh ^{-1}\left (\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}\right ) \]

[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

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Rubi [A]  time = 0.179596, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.125, Rules used = {6170, 94, 93, 210, 634, 618, 204, 628, 206} \[ \sqrt [6]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{5/6} x-\frac{1}{6} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}-\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )+\frac{1}{6} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}+\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{2}{3} \tanh ^{-1}\left (\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}\right ) \]

Antiderivative was successfully verified.

[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 6170

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]

Rule 94

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[p, 1] &&  !SumSimplerQ[m, 1])

Rule 93

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 210

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*Int[1/(r^2 - s^2*x^2), x])/(a*n) +
Dist[(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 634

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 618

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int e^{\frac{1}{3} \coth ^{-1}(x)} \, dx &=-\operatorname{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} \operatorname{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 \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \tanh ^{-1}\left (\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )-\frac{1}{6} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \tanh ^{-1}\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 )-\operatorname{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 )-\operatorname{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{\tan ^{-1}\left (\frac{-1+\frac{2 \sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\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} \tanh ^{-1}\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]  time = 0.0413237, size = 35, normalized size = 0.16 \[ 2 e^{\frac{1}{3} \coth ^{-1}(x)} \left (\text{Hypergeometric2F1}\left (\frac{1}{6},1,\frac{7}{6},e^{2 \coth ^{-1}(x)}\right )+\frac{1}{e^{2 \coth ^{-1}(x)}-1}\right ) \]

Warning: Unable to verify antiderivative.

[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])])

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [6]{{\frac{-1+x}{1+x}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.53153, size = 225, normalized size = 1.01 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.70431, size = 520, normalized size = 2.33 \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [6]{\frac{x - 1}{x + 1}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20744, size = 227, normalized size = 1.02 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))