### 3.122 $$\int e^{\frac{2}{3} \coth ^{-1}(x)} \, dx$$

Optimal. Leaf size=96 $\sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{\log (x)}{3}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3}}$

[Out]

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

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Rubi [A]  time = 0.0286798, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.375, Rules used = {6170, 94, 91} $\sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{\log (x)}{3}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^((2*ArcCoth[x])/3),x]

[Out]

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

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 91

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/
3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*
x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rubi steps

\begin{align*} \int e^{\frac{2}{3} \coth ^{-1}(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^2} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt [3]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{2/3} x-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt [3]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{2/3} x-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{-\frac{1-x}{x}}}{\sqrt{3} \sqrt [3]{1+\frac{1}{x}}}\right )}{\sqrt{3}}-\log \left (\sqrt [3]{1+\frac{1}{x}}-\sqrt [3]{-\frac{1-x}{x}}\right )-\frac{\log (x)}{3}\\ \end{align*}

Mathematica [A]  time = 0.160126, size = 85, normalized size = 0.89 $\frac{1}{3} \left (\frac{6 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}-2 \log \left (1-e^{\frac{2}{3} \coth ^{-1}(x)}\right )+\log \left (e^{\frac{2}{3} \coth ^{-1}(x)}+e^{\frac{4}{3} \coth ^{-1}(x)}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 e^{\frac{2}{3} \coth ^{-1}(x)}+1}{\sqrt{3}}\right )\right )$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^((2*ArcCoth[x])/3),x]

[Out]

((6*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x])) + 2*Sqrt[3]*ArcTan[(1 + 2*E^((2*ArcCoth[x])/3))/Sqrt[3]] - 2
*Log[1 - E^((2*ArcCoth[x])/3)] + Log[1 + E^((2*ArcCoth[x])/3) + E^((4*ArcCoth[x])/3)])/3

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{{\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))^(1/3),x)

[Out]

int(1/((-1+x)/(1+x))^(1/3),x)

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Maxima [A]  time = 1.54199, size = 130, normalized size = 1.35 \begin{align*} -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right )}\right ) - \frac{2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}}{\frac{x - 1}{x + 1} - 1} + \frac{1}{3} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - \frac{2}{3} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.5966, size = 282, normalized size = 2.94 \begin{align*}{\left (x + 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + \frac{1}{3} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - \frac{2}{3} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/3),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{\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))**(1/3),x)

[Out]

Integral(((x - 1)/(x + 1))**(-1/3), x)

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Giac [A]  time = 1.1857, size = 131, normalized size = 1.36 \begin{align*} -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right )}\right ) - \frac{2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}}{\frac{x - 1}{x + 1} - 1} + \frac{1}{3} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - \frac{2}{3} \, \log \left ({\left | \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/3),x, algorithm="giac")

[Out]

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