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

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

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Rubi [A]  time = 0.02, antiderivative size = 109, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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*}

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Mathematica [C]  time = 0.09, size = 125, normalized size = 1.15 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.35, size = 192, normalized size = 1.76 \begin {gather*} -\frac {i \log \left (2 \sqrt [3]{4-27 x^2}+3 i \sqrt [3]{2} x-2 \sqrt [3]{2}\right )}{18 \sqrt [3]{2}}+\frac {i \log \left (9\ 2^{2/3} x^2-4 \left (4-27 x^2\right )^{2/3}+\left (-4 \sqrt [3]{2}+6 i \sqrt [3]{2} x\right ) \sqrt [3]{4-27 x^2}+12 i 2^{2/3} x-4\ 2^{2/3}\right )}{36 \sqrt [3]{2}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{4-27 x^2}}{\sqrt [3]{4-27 x^2}-3 i \sqrt [3]{2} x+2 \sqrt [3]{2}}\right )}{6 \sqrt [3]{2} \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/6*I)*ArcTan[(Sqrt[3]*(4 - 27*x^2)^(1/3))/(2*2^(1/3) - (3*I)*2^(1/3)*x + (4 - 27*x^2)^(1/3))])/(2^(1/3)*Sq
rt[3]) - ((I/18)*Log[-2*2^(1/3) + (3*I)*2^(1/3)*x + 2*(4 - 27*x^2)^(1/3)])/2^(1/3) + ((I/36)*Log[-4*2^(2/3) +
(12*I)*2^(2/3)*x + 9*2^(2/3)*x^2 + (-4*2^(1/3) + (6*I)*2^(1/3)*x)*(4 - 27*x^2)^(1/3) - 4*(4 - 27*x^2)^(2/3)])/
2^(1/3)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}} {\left (3 i \, x + 2\right )}}\,{d x} \end {gather*}

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)

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maple [F]  time = 0.74, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 i x +2\right ) \left (-27 x^{2}+4\right )^{\frac {1}{3}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}} {\left (3 i \, x + 2\right )}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (2+x\,3{}\mathrm {i}\right )\,{\left (4-27\,x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i \int \frac {1}{3 x \sqrt [3]{4 - 27 x^{2}} - 2 i \sqrt [3]{4 - 27 x^{2}}}\, dx \end {gather*}

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]

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

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