### 3.709 $$\int \frac{1}{(2+3 i x) \sqrt [3]{4-27 x^2}} \, dx$$

Optimal. Leaf size=109 $-\frac{i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{12 \sqrt [3]{2}}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt{3}}+\frac{i \log (2+3 i x)}{12 \sqrt [3]{2}}$

[Out]

((I/6)*ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - (3*I)*x))/(Sqrt[3]*(4 - 27*x^2)^(1/3))])/(2^(1/3)*Sqrt[3]) + ((I/12)*L
og[2 + (3*I)*x])/2^(1/3) - ((I/12)*Log[-54 + (81*I)*x + 27*2^(2/3)*(4 - 27*x^2)^(1/3)])/2^(1/3)

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Rubi [A]  time = 0.0162252, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.048, Rules used = {751} $-\frac{i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{12 \sqrt [3]{2}}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt{3}}+\frac{i \log (2+3 i x)}{12 \sqrt [3]{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((2 + (3*I)*x)*(4 - 27*x^2)^(1/3)),x]

[Out]

((I/6)*ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - (3*I)*x))/(Sqrt[3]*(4 - 27*x^2)^(1/3))])/(2^(1/3)*Sqrt[3]) + ((I/12)*L
og[2 + (3*I)*x])/2^(1/3) - ((I/12)*Log[-54 + (81*I)*x + 27*2^(2/3)*(4 - 27*x^2)^(1/3)])/2^(1/3)

Rule 751

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[(6*c^2*e^2)/d^2, 3]}, -Simp
[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] + (2*c*(d - e*x))/(Sqrt[3]*d*q*(a + c*x^2)^(1/3))])/(d^2*q^2), x] + (-Simp[(3*c
*e*Log[d + e*x])/(2*d^2*q^2), x] + Simp[(3*c*e*Log[c*d - c*e*x - d*q*(a + c*x^2)^(1/3)])/(2*d^2*q^2), x])] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - 3*a*e^2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(2+3 i x) \sqrt [3]{4-27 x^2}} \, dx &=\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt{3}}+\frac{i \log (2+3 i x)}{12 \sqrt [3]{2}}-\frac{i \log \left (-54+81 i x+27\ 2^{2/3} \sqrt [3]{4-27 x^2}\right )}{12 \sqrt [3]{2}}\\ \end{align*}

Mathematica [C]  time = 0.0829706, size = 125, normalized size = 1.15 $\frac{i \sqrt [3]{\frac{2 \sqrt{3}-9 x}{-3 x+2 i}} \sqrt [3]{\frac{9 x+2 \sqrt{3}}{3 x-2 i}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 \left (3 i+\sqrt{3}\right )}{6 i-9 x},\frac{2 \left (-3 i+\sqrt{3}\right )}{9 x-6 i}\right )}{2\ 3^{2/3} \sqrt [3]{4-27 x^2}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((2 + (3*I)*x)*(4 - 27*x^2)^(1/3)),x]

[Out]

((I/2)*((2*Sqrt[3] - 9*x)/(2*I - 3*x))^(1/3)*((2*Sqrt[3] + 9*x)/(-2*I + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/
3, (2*(3*I + Sqrt[3]))/(6*I - 9*x), (2*(-3*I + Sqrt[3]))/(-6*I + 9*x)])/(3^(2/3)*(4 - 27*x^2)^(1/3))

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Maple [F]  time = 0.411, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{2+3\,ix}{\frac{1}{\sqrt [3]{-27\,{x}^{2}+4}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+3*I*x)/(-27*x^2+4)^(1/3),x)

[Out]

int(1/(2+3*I*x)/(-27*x^2+4)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 i \, x + 2\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*I*x)/(-27*x^2+4)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((-27*x^2 + 4)^(1/3)*(3*I*x + 2)), x)

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Fricas [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/(2+3*I*x)/(-27*x^2+4)^(1/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{4 - 27 x^{2}} \left (3 i x + 2\right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*I*x)/(-27*x**2+4)**(1/3),x)

[Out]

Integral(1/((4 - 27*x**2)**(1/3)*(3*I*x + 2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 i \, x + 2\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*I*x)/(-27*x^2+4)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((-27*x^2 + 4)^(1/3)*(3*I*x + 2)), x)