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

Optimal. Leaf size=116 \[ -\frac {(1-x)^{2/3} (x+1)^{4/3}}{2 x^2}-\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{3 x}-\frac {\log (x)}{9}+\frac {1}{3} \log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]

[Out]

-1/3*(1-x)^(2/3)*(1+x)^(1/3)/x-1/2*(1-x)^(2/3)*(1+x)^(4/3)/x^2-1/9*ln(x)+1/3*ln((1-x)^(1/3)-(1+x)^(1/3))+2/9*a
rctan(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 = 0.04, antiderivative size = 116, 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 = {6126, 96, 94, 91} \[ -\frac {(1-x)^{2/3} (x+1)^{4/3}}{2 x^2}-\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{3 x}-\frac {\log (x)}{9}+\frac {1}{3} \log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 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 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]

Rubi steps

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

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Mathematica [C]  time = 0.01, size = 57, normalized size = 0.49 \[ -\frac {(1-x)^{2/3} \left (2 x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1-x}{x+1}\right )+5 x^2+8 x+3\right )}{6 x^2 (x+1)^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/6*((1 - x)^(2/3)*(3 + 8*x + 5*x^2 + 2*x^2*Hypergeometric2F1[2/3, 1, 5/3, (1 - x)/(1 + x)]))/(x^2*(1 + x)^(2
/3))

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fricas [A]  time = 0.71, size = 166, normalized size = 1.43 \[ -\frac {4 \, \sqrt {3} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 4 \, x^{2} \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - 1\right ) + 2 \, x^{2} \log \left (\frac {{\left (x - 1\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + x - \sqrt {-x^{2} + 1} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - 1}{x - 1}\right ) - 3 \, {\left (5 \, x^{2} - 2 \, x - 3\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}}}{18 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(4*sqrt(3)*x^2*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1/3*sqrt(3)) - 4*x^2*log((-sqrt(-x^2
+ 1)/(x - 1))^(2/3) - 1) + 2*x^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)) - 3*(5*x^2 - 2*x - 3)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {2}{3}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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