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

Optimal. Leaf size=110 \[ \frac{3 \log \left (-2^{2/3} \sqrt [3]{x^2-3 x+2}-x+2\right )}{4 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2} (2-x)}{\sqrt{3} \sqrt [3]{x^2-3 x+2}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\log (2-x)}{4 \sqrt [3]{2}}-\frac{\log (x)}{2 \sqrt [3]{2}} \]

[Out]

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

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Rubi [A]  time = 0.0992248, antiderivative size = 176, normalized size of antiderivative = 1.6, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{3 \sqrt [3]{x-2} \sqrt [3]{x-1} \log \left (-\frac{(x-2)^{2/3}}{\sqrt [3]{2}}-\sqrt [3]{2} \sqrt [3]{x-1}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}}-\frac{\sqrt [3]{x-2} \sqrt [3]{x-1} \log (x)}{2 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}}-\frac{\sqrt{3} \sqrt [3]{x-2} \sqrt [3]{x-1} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{\sqrt [3]{2} (x-2)^{2/3}}{\sqrt{3} \sqrt [3]{x-1}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.67779, size = 155, normalized size = 1.41 \[ - \frac{\sqrt [3]{2 x - 4} \sqrt [3]{2 x - 2} \log{\left (x \right )}}{4 \sqrt [3]{x^{2} - 3 x + 2}} + \frac{3 \sqrt [3]{2 x - 4} \sqrt [3]{2 x - 2} \log{\left (- \frac{\left (2 x - 4\right )^{\frac{2}{3}}}{2} - \sqrt [3]{2 x - 2} \right )}}{8 \sqrt [3]{x^{2} - 3 x + 2}} + \frac{\sqrt{3} \sqrt [3]{2 x - 4} \sqrt [3]{2 x - 2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (2 x - 4\right )^{\frac{2}{3}}}{3 \sqrt [3]{2 x - 2}} - \frac{\sqrt{3}}{3} \right )}}{4 \sqrt [3]{x^{2} - 3 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x - 4)**(1/3)*(2*x - 2)**(1/3)*log(x)/(4*(x**2 - 3*x + 2)**(1/3)) + 3*(2*x -
4)**(1/3)*(2*x - 2)**(1/3)*log(-(2*x - 4)**(2/3)/2 - (2*x - 2)**(1/3))/(8*(x**2
- 3*x + 2)**(1/3)) + sqrt(3)*(2*x - 4)**(1/3)*(2*x - 2)**(1/3)*atan(sqrt(3)*(2*
x - 4)**(2/3)/(3*(2*x - 2)**(1/3)) - sqrt(3)/3)/(4*(x**2 - 3*x + 2)**(1/3))

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Mathematica [C]  time = 0.19044, size = 109, normalized size = 0.99 \[ -\frac{15 x F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{1}{x},\frac{2}{x}\right )}{2 \sqrt [3]{x^2-3 x+2} \left (5 x F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{1}{x},\frac{2}{x}\right )+2 F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{1}{x},\frac{2}{x}\right )+F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{1}{x},\frac{2}{x}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \[ \int{\frac{1}{x}{\frac{1}{\sqrt [3]{{x}^{2}-3\,x+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^2-3*x+2)^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*((x - 2)*(x - 1))**(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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