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

Optimal. Leaf size=142 \[ -\frac{(1-i x)^{2/3} (1+i x)^{4/3}}{2 x^2}-\frac{i (1-i x)^{2/3} \sqrt [3]{1+i x}}{3 x}-\frac{1}{3} \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )+\frac{\log (x)}{9}-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{1-i x}}{\sqrt{3} \sqrt [3]{1+i x}}\right )}{3 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0398932, antiderivative size = 142, 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, 96, 94, 91} \[ -\frac{(1-i x)^{2/3} (1+i x)^{4/3}}{2 x^2}-\frac{i (1-i x)^{2/3} \sqrt [3]{1+i x}}{3 x}-\frac{1}{3} \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )+\frac{\log (x)}{9}-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{1-i x}}{\sqrt{3} \sqrt [3]{1+i x}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.0142606, size = 69, normalized size = 0.49 \[ \frac{(1-i x)^{2/3} \left (2 x^2 \text{Hypergeometric2F1}\left (\frac{2}{3},1,\frac{5}{3},\frac{x+i}{-x+i}\right )+5 x^2-8 i x-3\right )}{6 (1+i x)^{2/3} x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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^3,x)

[Out]

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

[Out]

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

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Fricas [A]  time = 1.68321, size = 387, normalized size = 2.73 \begin{align*} -\frac{4 \, x^{2} \log \left (\left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{2}{3}} - 1\right ) + 2 \,{\left (-i \, \sqrt{3} x^{2} - x^{2}\right )} \log \left (\left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{2}{3}} + \frac{1}{2} i \, \sqrt{3} + \frac{1}{2}\right ) + 2 \,{\left (i \, \sqrt{3} x^{2} - x^{2}\right )} \log \left (\left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{2}{3}} - \frac{1}{2} i \, \sqrt{3} + \frac{1}{2}\right ) + 3 \,{\left (5 \, x^{2} + 2 i \, x + 3\right )} \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{2}{3}}}{18 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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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**3,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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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