3.2.31 \(\int \frac {e^{\frac {2}{3} \tanh ^{-1}(x)}}{x} \, dx\) [131]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 135, normalized size of antiderivative = 1.00, 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 = {6261, 132, 62, 93} \begin {gather*} \sqrt {3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}\right )+\sqrt {3} \text {ArcTan}\left (\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )-\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 ) \end {gather*}

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 62

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

Rule 93

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]

Rule 132

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

Rule 6261

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]

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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 74, normalized size = 0.55 \begin {gather*} -\frac {3 (1-x)^{2/3} \left (\sqrt [3]{2} (1+x)^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {1-x}{2}\right )+2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1-x}{1+x}\right )\right )}{4 (1+x)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {2}{3}}}{x}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.35, size = 158, normalized size = 1.17 \begin {gather*} -\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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {x + 1}{\sqrt {1 - x^{2}}}\right )^{\frac {2}{3}}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{2/3}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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