### 3.121 $$\int e^{\frac{2}{3} \coth ^{-1}(x)} x \, dx$$

Optimal. Leaf size=130 $\frac{1}{2} \left (\frac{1}{x}+1\right )^{4/3} \left (\frac{x-1}{x}\right )^{2/3} x^2+\frac{1}{3} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\frac{1}{3} \log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{\log (x)}{9}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3}}$

[Out]

((1 + x^(-1))^(1/3)*((-1 + x)/x)^(2/3)*x)/3 + ((1 + x^(-1))^(4/3)*((-1 + x)/x)^(2/3)*x^2)/2 - (2*ArcTan[1/Sqrt
[3] + (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + x^(-1))^(1/3))])/(3*Sqrt[3]) - Log[(1 + x^(-1))^(1/3) - ((-1 + x)/x
)^(1/3)]/3 - Log[x]/9

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Rubi [A]  time = 0.0466163, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {6171, 96, 94, 91} $\frac{1}{2} \left (\frac{1}{x}+1\right )^{4/3} \left (\frac{x-1}{x}\right )^{2/3} x^2+\frac{1}{3} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\frac{1}{3} \log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{\log (x)}{9}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((1 + x^(-1))^(1/3)*((-1 + x)/x)^(2/3)*x)/3 + ((1 + x^(-1))^(4/3)*((-1 + x)/x)^(2/3)*x^2)/2 - (2*ArcTan[1/Sqrt
[3] + (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + x^(-1))^(1/3))])/(3*Sqrt[3]) - Log[(1 + x^(-1))^(1/3) - ((-1 + x)/x
)^(1/3)]/3 - Log[x]/9

Rule 6171

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]

Rule 96

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 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)} x \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \left (1+\frac{1}{x}\right )^{4/3} \left (\frac{-1+x}{x}\right )^{2/3} x^2-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3} x+\frac{1}{2} \left (1+\frac{1}{x}\right )^{4/3} \left (\frac{-1+x}{x}\right )^{2/3} x^2-\frac{2}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3} x+\frac{1}{2} \left (1+\frac{1}{x}\right )^{4/3} \left (\frac{-1+x}{x}\right )^{2/3} x^2-\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 )}{3 \sqrt{3}}-\frac{1}{3} \log \left (\sqrt [3]{1+\frac{1}{x}}-\sqrt [3]{-\frac{1-x}{x}}\right )-\frac{\log (x)}{9}\\ \end{align*}

Mathematica [A]  time = 0.429379, size = 165, normalized size = 1.27 $\frac{1}{9} \left (\frac{24 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}+\frac{18 e^{\frac{2}{3} \coth ^{-1}(x)}}{\left (e^{2 \coth ^{-1}(x)}-1\right )^2}-2 \log \left (1-e^{\frac{1}{3} \coth ^{-1}(x)}\right )-2 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}+1\right )+\log \left (-e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right )+\log \left (e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}-1}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}+1}{\sqrt{3}}\right )\right )$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((18*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x]))^2 + (24*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x])) + 2*S
qrt[3]*ArcTan[(-1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 2*Log[
1 - E^(ArcCoth[x]/3)] - 2*Log[1 + E^(ArcCoth[x]/3)] + Log[1 - E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)] + Log[1
+ E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)])/9

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{x{\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,x)

[Out]

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

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Maxima [A]  time = 1.51746, size = 166, normalized size = 1.28 \begin{align*} -\frac{2}{9} \, \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 (\left (\frac{x - 1}{x + 1}\right )^{\frac{5}{3}} - 4 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}\right )}}{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}{9} \, \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}{9} \, \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,x, algorithm="maxima")

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) + 2/3*(((x - 1)/(x + 1))^(5/3) - 4*((x - 1)/(
x + 1))^(2/3))/(2*(x - 1)/(x + 1) - (x - 1)^2/(x + 1)^2 - 1) + 1/9*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x +
1))^(1/3) + 1) - 2/9*log(((x - 1)/(x + 1))^(1/3) - 1)

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Fricas [A]  time = 1.56023, size = 301, normalized size = 2.32 \begin{align*} \frac{1}{6} \,{\left (3 \, x^{2} + 8 \, x + 5\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - \frac{2}{9} \, \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}{9} \, \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}{9} \, \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,x, algorithm="fricas")

[Out]

1/6*(3*x^2 + 8*x + 5)*((x - 1)/(x + 1))^(2/3) - 2/9*sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/3) + 1/3*s
qrt(3)) + 1/9*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1) - 2/9*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{x}{\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,x)

[Out]

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

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Giac [A]  time = 1.1516, size = 162, normalized size = 1.25 \begin{align*} -\frac{2}{9} \, \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{{\left (x - 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}}{x + 1} - 4 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}\right )}}{3 \,{\left (\frac{x - 1}{x + 1} - 1\right )}^{2}} + \frac{1}{9} \, \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}{9} \, \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,x, algorithm="giac")

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) - 2/3*((x - 1)*((x - 1)/(x + 1))^(2/3)/(x + 1
) - 4*((x - 1)/(x + 1))^(2/3))/((x - 1)/(x + 1) - 1)^2 + 1/9*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(
1/3) + 1) - 2/9*log(abs(((x - 1)/(x + 1))^(1/3) - 1))