3.8.9 \(\int \frac {1}{(2+3 i x) \sqrt [3]{4-27 x^2}} \, dx\) [709]

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

[Out]

1/24*I*ln(2+3*I*x)*2^(2/3)-1/24*I*ln(-54+81*I*x+27*2^(2/3)*(-27*x^2+4)^(1/3))*2^(2/3)-1/36*I*arctan(-1/3*3^(1/
2)-1/3*2^(1/3)*(2-3*I*x)/(-27*x^2+4)^(1/3)*3^(1/2))*2^(2/3)*3^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, 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 = {765} \begin {gather*} \frac {i \text {ArcTan}\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 \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{12 \sqrt [3]{2}}+\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 765

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 [A]
time = 0.30, size = 166, normalized size = 1.52 \begin {gather*} -\frac {i \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{4-27 x^2}}{2 \sqrt [3]{2}-3 i \sqrt [3]{2} x+\sqrt [3]{4-27 x^2}}\right )+2 \log \left (-2 \sqrt [3]{2}+3 i \sqrt [3]{2} x+2 \sqrt [3]{4-27 x^2}\right )-\log \left (-4 2^{2/3}+12 i 2^{2/3} x+9\ 2^{2/3} x^2+2 (-2+3 i x) \sqrt [3]{8-54 x^2}-4 \left (4-27 x^2\right )^{2/3}\right )\right )}{36 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/36*I)*(2*Sqrt[3]*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*Log[-2*2^(1/3) + (3*I)*2^(1/3)*x + 2*(4 - 27*x^2)^(1/3)] - Log[-4*2^(2/3) + (12*I)*2^(2/3)*x + 9*2^(2/3)*x
^2 + 2*(-2 + (3*I)*x)*(8 - 54*x^2)^(1/3) - 4*(4 - 27*x^2)^(2/3)]))/2^(1/3)

________________________________________________________________________________________

Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (3 i x +2\right ) \left (-27 x^{2}+4\right )^{\frac {1}{3}}}\, dx\]

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)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)

________________________________________________________________________________________

Fricas [F(-1)] Timed out
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

________________________________________________________________________________________

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)

________________________________________________________________________________________

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)

________________________________________________________________________________________