### 3.124 $$\int \frac{e^{\frac{2}{3} \coth ^{-1}(x)}}{x^2} \, dx$$

Optimal. Leaf size=99 $\sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3}-\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+1\right )-\frac{1}{3} \log \left (\frac{1}{x}+1\right )-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}\right )}{\sqrt{3}}$

[Out]

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

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Rubi [A]  time = 0.0429347, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6171, 50, 60} $\sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3}-\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+1\right )-\frac{1}{3} \log \left (\frac{1}{x}+1\right )-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}\right )}{\sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 60

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-(d/b), 3]}, Simp[(Sq
rt[3]*q*ArcTan[1/Sqrt[3] - (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3))])/d, x] + (Simp[(3*q*Log[(q*(a + b*
x)^(1/3))/(c + d*x)^(1/3) + 1])/(2*d), x] + Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ
[b*c - a*d, 0] && NegQ[d/b]

Rubi steps

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

Mathematica [A]  time = 0.144131, size = 87, normalized size = 0.88 $\frac{2 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}+1}-\frac{2}{3} \log \left (e^{\frac{2}{3} \coth ^{-1}(x)}+1\right )+\frac{1}{3} \log \left (-e^{\frac{2}{3} \coth ^{-1}(x)}+e^{\frac{4}{3} \coth ^{-1}(x)}+1\right )-\frac{2 \tan ^{-1}\left (\frac{2 e^{\frac{2}{3} \coth ^{-1}(x)}-1}{\sqrt{3}}\right )}{\sqrt{3}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\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^2,x)

[Out]

int(1/((-1+x)/(1+x))^(1/3)/x^2,x)

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Maxima [A]  time = 1.62854, size = 132, normalized size = 1.33 \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^2,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 = 1.64112, size = 293, normalized size = 2.96 \begin{align*} \frac{2 \, \sqrt{3} x \arctan \left (\frac{2}{3} \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + x \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - 2 \, x \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) + 3 \,{\left (x + 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}}{3 \, x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/3)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \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**2,x)

[Out]

Integral(1/(x**2*((x - 1)/(x + 1))**(1/3)), x)

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Giac [A]  time = 1.13534, size = 134, 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 | \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^2,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
))