3.132 \(\int \frac{e^{\frac{2}{3} \tanh ^{-1}(x)}}{x^2} \, dx\)

Optimal. Leaf size=85 \[ -\frac{(1-x)^{2/3} \sqrt [3]{x+1}}{x}-\frac{\log (x)}{3}+\log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{x+1}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0314888, antiderivative size = 85, 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 = {6126, 94, 91} \[ -\frac{(1-x)^{2/3} \sqrt [3]{x+1}}{x}-\frac{\log (x)}{3}+\log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{x+1}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6126

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[(x^m*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x] /; Fre
eQ[{a, m, n}, x] &&  !IntegerQ[(n - 1)/2]

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

Mathematica [C]  time = 0.0109406, size = 45, normalized size = 0.53 \[ -\frac{(1-x)^{2/3} \left (x \text{Hypergeometric2F1}\left (\frac{2}{3},1,\frac{5}{3},\frac{1-x}{x+1}\right )+x+1\right )}{x (x+1)^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((1 - x)^(2/3)*(1 + x + x*Hypergeometric2F1[2/3, 1, 5/3, (1 - x)/(1 + x)]))/(x*(1 + x)^(2/3)))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ({(1+x){\frac{1}{\sqrt{-{x}^{2}+1}}}} \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{x + 1}{\sqrt{-x^{2} + 1}}\right )^{\frac{2}{3}}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((x + 1)/sqrt(-x^2 + 1))^(2/3)/x^2, x)

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Fricas [B]  time = 1.6858, size = 398, normalized size = 4.68 \begin{align*} -\frac{2 \, \sqrt{3} x \arctan \left (\frac{2}{3} \, \sqrt{3} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - 2 \, x \log \left (\left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}} - 1\right ) + x \log \left (\frac{{\left (x - 1\right )} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}} + x - \sqrt{-x^{2} + 1} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} - 1}{x - 1}\right ) - 3 \,{\left (x - 1\right )} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}}}{3 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{x + 1}{\sqrt{-x^{2} + 1}}\right )^{\frac{2}{3}}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((x + 1)/sqrt(-x^2 + 1))^(2/3)/x^2, x)