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

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

[Out]

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

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Rubi [A]  time = 0.0205023, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5061, 50, 60} \[ i (1-i x)^{2/3} \sqrt [3]{1+i x}-i \log \left (1+\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )-\frac{1}{3} i \log (1+i x)-\frac{2 i \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-i x}}{\sqrt{3} \sqrt [3]{1+i x}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5061

Int[E^(ArcTan[(a_.)*(x_)]*(n_.)), x_Symbol] :> Int[(1 - I*a*x)^((I*n)/2)/(1 + I*a*x)^((I*n)/2), x] /; FreeQ[{a
, n}, x] &&  !IntegerQ[(I*n - 1)/2]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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]

Rubi steps

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

Mathematica [C]  time = 0.0172518, size = 34, normalized size = 0.29 \[ -\frac{3}{2} i e^{\frac{8}{3} i \tan ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{4}{3},2,\frac{7}{3},-e^{2 i \tan ^{-1}(x)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-3*I)/2)*E^(((8*I)/3)*ArcTan[x])*Hypergeometric2F1[4/3, 2, 7/3, -E^((2*I)*ArcTan[x])]

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \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)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\frac{i \, x + 1}{\sqrt{x^{2} + 1}}\right )^{\frac{2}{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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.70289, size = 346, normalized size = 2.98 \begin{align*} \frac{1}{3} \,{\left (\sqrt{3} + i\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 ) - \frac{1}{3} \,{\left (\sqrt{3} - i\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 ) + \frac{1}{3} \,{\left (3 \, x + 3 i\right )} \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{2}{3}} - \frac{2}{3} i \, \log \left (\left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{2}{3}} + 1\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, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\frac{i \, x + 1}{\sqrt{x^{2} + 1}}\right )^{\frac{2}{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, algorithm="giac")

[Out]

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