3.55 \(\int \frac{1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{\log \left (\frac{2 \sqrt [3]{2} \sqrt [3]{3 x^2-6 x+4}+2 x-4}{x}\right )}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{-2 \sqrt [3]{2} \sqrt [3]{3 x^2-6 x+4}+x-2}{\sqrt{3} (x-2)}\right )}{2^{2/3} \sqrt{3}} \]

[Out]

ArcTan[(-2 + x - 2*2^(1/3)*(4 - 6*x + 3*x^2)^(1/3))/(Sqrt[3]*(-2 + x))]/(2^(2/3)
*Sqrt[3]) + Log[(-4 + 2*x + 2*2^(1/3)*(4 - 6*x + 3*x^2)^(1/3))/x]/(2*2^(2/3))

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Rubi [A]  time = 0.0472327, antiderivative size = 97, normalized size of antiderivative = 1.1, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{\log \left (-3 \sqrt [3]{2} \sqrt [3]{3 x^2-6 x+4}-3 x+6\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (2-x)}{\sqrt{3} \sqrt [3]{3 x^2-6 x+4}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log (x)}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(4 - 6*x + 3*x^2)^(1/3)),x]

[Out]

-(ArcTan[1/Sqrt[3] + (2^(2/3)*(2 - x))/(Sqrt[3]*(4 - 6*x + 3*x^2)^(1/3))]/(2^(2/
3)*Sqrt[3])) - Log[x]/(2*2^(2/3)) + Log[6 - 3*x - 3*2^(1/3)*(4 - 6*x + 3*x^2)^(1
/3)]/(2*2^(2/3))

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Rubi in Sympy [A]  time = 2.41973, size = 94, normalized size = 1.07 \[ - \frac{\sqrt [3]{2} \log{\left (x \right )}}{4} + \frac{\sqrt [3]{2} \log{\left (- 3 x - 3 \sqrt [3]{2} \sqrt [3]{3 x^{2} - 6 x + 4} + 6 \right )}}{4} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \sqrt{3} \left (- 3 x + 6\right )}{9 \sqrt [3]{3 x^{2} - 6 x + 4}} + \frac{\sqrt{3}}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(3*x**2-6*x+4)**(1/3),x)

[Out]

-2**(1/3)*log(x)/4 + 2**(1/3)*log(-3*x - 3*2**(1/3)*(3*x**2 - 6*x + 4)**(1/3) +
6)/4 - 2**(1/3)*sqrt(3)*atan(2**(2/3)*sqrt(3)*(-3*x + 6)/(9*(3*x**2 - 6*x + 4)**
(1/3)) + sqrt(3)/3)/6

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Mathematica [C]  time = 0.810162, size = 273, normalized size = 3.1 \[ -\frac{15 x \left (3 x-i \sqrt{3}-3\right ) \left (3 x+i \sqrt{3}-3\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{3-i \sqrt{3}}{3 x},\frac{3+i \sqrt{3}}{3 x}\right )}{2 \left (3 x^2-6 x+4\right )^{4/3} \left (15 x F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{3-i \sqrt{3}}{3 x},\frac{3+i \sqrt{3}}{3 x}\right )+\left (3+i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{3-i \sqrt{3}}{3 x},\frac{3+i \sqrt{3}}{3 x}\right )+\left (3-i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{3-i \sqrt{3}}{3 x},\frac{3+i \sqrt{3}}{3 x}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x*(4 - 6*x + 3*x^2)^(1/3)),x]

[Out]

(-15*x*(-3 - I*Sqrt[3] + 3*x)*(-3 + I*Sqrt[3] + 3*x)*AppellF1[2/3, 1/3, 1/3, 5/3
, (3 - I*Sqrt[3])/(3*x), (3 + I*Sqrt[3])/(3*x)])/(2*(4 - 6*x + 3*x^2)^(4/3)*(15*
x*AppellF1[2/3, 1/3, 1/3, 5/3, (3 - I*Sqrt[3])/(3*x), (3 + I*Sqrt[3])/(3*x)] + (
3 + I*Sqrt[3])*AppellF1[5/3, 1/3, 4/3, 8/3, (3 - I*Sqrt[3])/(3*x), (3 + I*Sqrt[3
])/(3*x)] + (3 - I*Sqrt[3])*AppellF1[5/3, 4/3, 1/3, 8/3, (3 - I*Sqrt[3])/(3*x),
(3 + I*Sqrt[3])/(3*x)]))

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Maple [F]  time = 0.199, size = 0, normalized size = 0. \[ \int{\frac{1}{x}{\frac{1}{\sqrt [3]{3\,{x}^{2}-6\,x+4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(3*x^2-6*x+4)^(1/3),x)

[Out]

int(1/x/(3*x^2-6*x+4)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^2 - 6*x + 4)^(1/3)*x),x, algorithm="maxima")

[Out]

integrate(1/((3*x^2 - 6*x + 4)^(1/3)*x), 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/((3*x^2 - 6*x + 4)^(1/3)*x),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt [3]{3 x^{2} - 6 x + 4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(3*x**2-6*x+4)**(1/3),x)

[Out]

Integral(1/(x*(3*x**2 - 6*x + 4)**(1/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^2 - 6*x + 4)^(1/3)*x),x, algorithm="giac")

[Out]

integrate(1/((3*x^2 - 6*x + 4)^(1/3)*x), x)