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

Optimal. Leaf size=135 \[ -\frac{\log (x)}{2}+\frac{1}{2} \log (x+1)+\frac{3}{2} \log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+1\right )+\frac{3}{2} \log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{x+1}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{x+1}}+\frac{1}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0317323, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6126, 105, 60, 91} \[ -\frac{\log (x)}{2}+\frac{1}{2} \log (x+1)+\frac{3}{2} \log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+1\right )+\frac{3}{2} \log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{x+1}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{x+1}}+\frac{1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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]

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

Mathematica [C]  time = 0.0227641, size = 74, normalized size = 0.55 \[ -\frac{3 (1-x)^{2/3} \left (\sqrt [3]{2} (x+1)^{2/3} \text{Hypergeometric2F1}\left (\frac{2}{3},\frac{2}{3},\frac{5}{3},\frac{1-x}{2}\right )+2 \text{Hypergeometric2F1}\left (\frac{2}{3},1,\frac{5}{3},\frac{1-x}{x+1}\right )\right )}{4 (x+1)^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(1 - x)^(2/3)*(2^(1/3)*(1 + x)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (1 - x)/2] + 2*Hypergeometric2F1[2/3
, 1, 5/3, (1 - x)/(1 + x)]))/(4*(1 + x)^(2/3))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \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,x)

[Out]

int(((1+x)/(-x^2+1)^(1/2))^(2/3)/x,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}\,{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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.6711, size = 405, normalized size = 3. \begin{align*} -\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (x - 1\right )} - 2 \, \sqrt{3} \sqrt{-x^{2} + 1} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}}}{3 \,{\left (x - 1\right )}}\right ) - \frac{1}{2} \, \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 ) + \log \left (-\frac{x + \sqrt{-x^{2} + 1} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} - 1}{x - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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,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}\,{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,x, algorithm="giac")

[Out]

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