3.125 \(\int \frac{e^{\frac{2}{3} i \tan ^{-1}(x)}}{x} \, dx\)

Optimal. Leaf size=163 \[ \frac{3}{2} \log \left (1+\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )+\frac{3}{2} \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )+\frac{1}{2} \log (1+i x)-\frac{\log (x)}{2}+\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-i x}}{\sqrt{3} \sqrt [3]{1+i x}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{1-i x}}{\sqrt{3} \sqrt [3]{1+i x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0403384, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5062, 105, 60, 91} \[ \frac{3}{2} \log \left (1+\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )+\frac{3}{2} \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )+\frac{1}{2} \log (1+i x)-\frac{\log (x)}{2}+\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-i x}}{\sqrt{3} \sqrt [3]{1+i x}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{1-i x}}{\sqrt{3} \sqrt [3]{1+i x}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5062

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[(x^m*(1 - I*a*x)^((I*n)/2))/(1 + I*a*x)^((I*n)/2
), x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[(I*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} i \tan ^{-1}(x)}}{x} \, dx &=\int \frac{\sqrt [3]{1+i x}}{\sqrt [3]{1-i x} x} \, dx\\ &=i \int \frac{1}{\sqrt [3]{1-i x} (1+i x)^{2/3}} \, dx+\int \frac{1}{\sqrt [3]{1-i x} (1+i x)^{2/3} x} \, dx\\ &=\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-i x}}{\sqrt{3} \sqrt [3]{1+i x}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{1-i x}}{\sqrt{3} \sqrt [3]{1+i x}}\right )+\frac{3}{2} \log \left (1+\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )+\frac{3}{2} \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )+\frac{1}{2} \log (1+i x)-\frac{\log (x)}{2}\\ \end{align*}

Mathematica [C]  time = 0.0297735, size = 90, normalized size = 0.55 \[ -\frac{3 (1-i x)^{2/3} \left (\sqrt [3]{2} (1+i x)^{2/3} \text{Hypergeometric2F1}\left (\frac{2}{3},\frac{2}{3},\frac{5}{3},\frac{1}{2}-\frac{i x}{2}\right )+2 \text{Hypergeometric2F1}\left (\frac{2}{3},1,\frac{5}{3},\frac{x+i}{-x+i}\right )\right )}{4 (1+i x)^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(1 - I*x)^(2/3)*(2^(1/3)*(1 + I*x)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, 1/2 - (I/2)*x] + 2*Hypergeometri
c2F1[2/3, 1, 5/3, (I + x)/(I - x)]))/(4*(1 + I*x)^(2/3))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ({(1+ix){\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+I*x)/(x^2+1)^(1/2))^(2/3)/x,x)

[Out]

int(((1+I*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{i \, 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+I*x)/(x^2+1)^(1/2))^(2/3)/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.65603, size = 423, normalized size = 2.6 \begin{align*} \frac{1}{2} \,{\left (i \, \sqrt{3} - 1\right )} \log \left (\frac{\sqrt{3}{\left (i \, x - 1\right )} + x + 2 i \, \sqrt{x^{2} + 1} \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} + i}{2 \, x + 2 i}\right ) + \frac{1}{2} \,{\left (-i \, \sqrt{3} - 1\right )} \log \left (\frac{\sqrt{3}{\left (-i \, x + 1\right )} + x + 2 i \, \sqrt{x^{2} + 1} \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} + i}{2 \, x + 2 i}\right ) + \log \left (-\frac{x - i \, \sqrt{x^{2} + 1} \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} + i}{x + i}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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+I*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{i \, 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+I*x)/(x^2+1)^(1/2))^(2/3)/x,x, algorithm="giac")

[Out]

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