3.1629 \(\int \frac{\sqrt [3]{-1+x}}{\sqrt [3]{1+x}} \, dx\)

Optimal. Leaf size=77 \[ \sqrt [3]{x-1} (x+1)^{2/3}+\frac{1}{3} \log (x-1)+\log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{x-1}}-1\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{x+1}}{\sqrt{3} \sqrt [3]{x-1}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0441209, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \sqrt [3]{x-1} (x+1)^{2/3}+\frac{1}{3} \log (x-1)+\log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{x-1}}-1\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{x+1}}{\sqrt{3} \sqrt [3]{x-1}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.65314, size = 75, normalized size = 0.97 \[ \sqrt [3]{x - 1} \left (x + 1\right )^{\frac{2}{3}} + \log{\left (-1 + \frac{\sqrt [3]{x + 1}}{\sqrt [3]{x - 1}} \right )} + \frac{\log{\left (x - 1 \right )}}{3} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{x + 1}}{3 \sqrt [3]{x - 1}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0295901, size = 50, normalized size = 0.65 \[ \sqrt [3]{\frac{x-1}{x+1}} \left (-2^{2/3} \sqrt [3]{x+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1-x}{2}\right )+x+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-1 + x)/(1 + x))^(1/3)*(1 + x - 2^(2/3)*(1 + x)^(1/3)*Hypergeometric2F1[1/3, 1
/3, 4/3, (1 - x)/2])

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{1\sqrt [3]{-1+x}{\frac{1}{\sqrt [3]{1+x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+x)^(1/3)/(1+x)^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.211761, size = 161, normalized size = 2.09 \[ \frac{1}{9} \, \sqrt{3}{\left (3 \, \sqrt{3}{\left (x + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )}^{\frac{1}{3}} - \sqrt{3} \log \left (\frac{{\left (x + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )}^{\frac{1}{3}} +{\left (x + 1\right )}^{\frac{1}{3}}{\left (x - 1\right )}^{\frac{2}{3}} + x + 1}{x + 1}\right ) + 2 \, \sqrt{3} \log \left (\frac{{\left (x + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )}^{\frac{1}{3}} - x - 1}{x + 1}\right ) - 6 \, \arctan \left (\frac{\sqrt{3}{\left (x + 1\right )} + 2 \, \sqrt{3}{\left (x + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )}^{\frac{1}{3}}}{3 \,{\left (x + 1\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*sqrt(3)*(3*sqrt(3)*(x + 1)^(2/3)*(x - 1)^(1/3) - sqrt(3)*log(((x + 1)^(2/3)*
(x - 1)^(1/3) + (x + 1)^(1/3)*(x - 1)^(2/3) + x + 1)/(x + 1)) + 2*sqrt(3)*log(((
x + 1)^(2/3)*(x - 1)^(1/3) - x - 1)/(x + 1)) - 6*arctan(1/3*(sqrt(3)*(x + 1) + 2
*sqrt(3)*(x + 1)^(2/3)*(x - 1)^(1/3))/(x + 1)))

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Sympy [A]  time = 3.68554, size = 39, normalized size = 0.51 \[ \frac{2^{\frac{2}{3}} \left (x - 1\right )^{\frac{4}{3}} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{\left (x - 1\right ) e^{i \pi }}{2}} \right )}}{2 \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(2/3)*(x - 1)**(4/3)*gamma(4/3)*hyper((1/3, 4/3), (7/3,), (x - 1)*exp_polar(I
*pi)/2)/(2*gamma(7/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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