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

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)*Log[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 = 0.0688802, 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 \[ -\frac{i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{12 \sqrt [3]{2}}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt{3}}+\frac{i \log (2+3 i x)}{12 \sqrt [3]{2}} \]

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)*Log[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 in Sympy [A]  time = 4.91561, size = 100, normalized size = 0.92 \[ \frac{2^{\frac{2}{3}} i \log{\left (3 i x + 2 \right )}}{24} - \frac{2^{\frac{2}{3}} i \log{\left (81 i x + 27 \cdot 2^{\frac{2}{3}} \sqrt [3]{- 27 x^{2} + 4} - 54 \right )}}{24} + \frac{2^{\frac{2}{3}} \sqrt{3} i \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{\sqrt [3]{2} \sqrt{3} \left (- 3 i x + 2\right )}{3 \sqrt [3]{- 27 x^{2} + 4}} \right )}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(2/3)*I*log(3*I*x + 2)/24 - 2**(2/3)*I*log(81*I*x + 27*2**(2/3)*(-27*x**2 + 4
)**(1/3) - 54)/24 + 2**(2/3)*sqrt(3)*I*atan(sqrt(3)/3 + 2**(1/3)*sqrt(3)*(-3*I*x
+ 2)/(3*(-27*x**2 + 4)**(1/3)))/36

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Mathematica [C]  time = 0.115404, 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 + Sqr
t[3]))/(-6*I + 9*x)])/(3^(2/3)*(4 - 27*x^2)^(1/3))

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{1}{2+3\,ix}{\frac{1}{\sqrt [3]{-27\,{x}^{2}+4}}}}\, dx \]

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. \[ \int \frac{1}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 i \, x + 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-27*x^2 + 4)^(1/3)*(3*I*x + 2)),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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 i \, x + 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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