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

Optimal. Leaf size=99 \[ \frac{3 \log \left (\frac{(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{2 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}} \]

[Out]

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

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Rubi [A]  time = 0.0631019, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{3 \log \left (\frac{(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{2 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.1441, size = 85, normalized size = 0.86 \[ - \frac{2^{\frac{2}{3}} \log{\left (x \right )}}{4} + \frac{3 \cdot 2^{\frac{2}{3}} \log{\left (- \sqrt [3]{- x + 1} + \frac{\sqrt [3]{2} \left (- x + 2\right )^{\frac{2}{3}}}{2} \right )}}{8} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{\sqrt [3]{2} \sqrt{3} \left (- x + 2\right )^{\frac{2}{3}}}{3 \sqrt [3]{- x + 1}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.170194, size = 115, normalized size = 1.16 \[ \frac{15 (1-x)^{2/3} F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x-1,1-x\right )}{2 \sqrt [3]{2-x} x \left ((x-1) \left (3 F_1\left (\frac{5}{3};\frac{1}{3},2;\frac{8}{3};x-1,1-x\right )-F_1\left (\frac{5}{3};\frac{4}{3},1;\frac{8}{3};x-1,1-x\right )\right )-5 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x-1,1-x\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{1}{x}{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{2-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(1/3)/(2-x)^(1/3)/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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