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

Optimal. Leaf size=67 \[ -\frac{1}{2} \log (x+1)-\frac{3}{2} \log \left (1-\frac{x-1}{\sqrt [3]{(x-1)^2 (x+1)}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x-1)}{\sqrt [3]{(x-1)^2 (x+1)}}+1}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [B]  time = 0.235168, antiderivative size = 188, normalized size of antiderivative = 2.81, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{(1-x)^{2/3} \sqrt [3]{x+1} \log \left (\frac{8 (1-x)}{3}\right )}{2 \sqrt [3]{x^3-x^2-x+1}}-\frac{3 (1-x)^{2/3} \sqrt [3]{x+1} \log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+1\right )}{2 \sqrt [3]{x^3-x^2-x+1}}-\frac{\sqrt{3} (1-x)^{2/3} \sqrt [3]{x+1} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{x+1}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{\sqrt [3]{x^3-x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[3]*(1 - x)^(2/3)*(1 + x)^(1/3)*ArcTan[1/Sqrt[3] - (2*(1 + x)^(1/3))/(Sqr
t[3]*(1 - x)^(1/3))])/(1 - x - x^2 + x^3)^(1/3)) - ((1 - x)^(2/3)*(1 + x)^(1/3)*
Log[(8*(1 - x))/3])/(2*(1 - x - x^2 + x^3)^(1/3)) - (3*(1 - x)^(2/3)*(1 + x)^(1/
3)*Log[1 + (1 + x)^(1/3)/(1 - x)^(1/3)])/(2*(1 - x - x^2 + x^3)^(1/3))

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Rubi in Sympy [A]  time = 2.88529, size = 144, normalized size = 2.15 \[ - \frac{3 \left (x - 1\right )^{\frac{2}{3}} \sqrt [3]{x + 1} \log{\left (-1 + \frac{\sqrt [3]{x + 1}}{\sqrt [3]{x - 1}} \right )}}{2 \sqrt [3]{x^{3} - x^{2} - x + 1}} - \frac{\left (x - 1\right )^{\frac{2}{3}} \sqrt [3]{x + 1} \log{\left (x - 1 \right )}}{2 \sqrt [3]{x^{3} - x^{2} - x + 1}} - \frac{\sqrt{3} \left (x - 1\right )^{\frac{2}{3}} \sqrt [3]{x + 1} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{x + 1}}{3 \sqrt [3]{x - 1}} \right )}}{\sqrt [3]{x^{3} - x^{2} - x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0228561, size = 49, normalized size = 0.73 \[ \frac{3 (x-1) \sqrt [3]{x+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1-x}{2}\right )}{\sqrt [3]{2} \sqrt [3]{(x-1)^2 (x+1)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [3]{ \left ( -1+x \right ) ^{2} \left ( 1+x \right ) }}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((-1+x)^2*(1+x))^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.202478, size = 166, normalized size = 2.48 \[ -\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (x + 2 \,{\left (x^{3} - x^{2} - x + 1\right )}^{\frac{1}{3}} - 1\right )}}{3 \,{\left (x - 1\right )}}\right ) + \frac{1}{2} \, \log \left (\frac{x^{2} +{\left (x^{3} - x^{2} - x + 1\right )}^{\frac{1}{3}}{\left (x - 1\right )} - 2 \, x +{\left (x^{3} - x^{2} - x + 1\right )}^{\frac{2}{3}} + 1}{x^{2} - 2 \, x + 1}\right ) - \log \left (-\frac{x -{\left (x^{3} - x^{2} - x + 1\right )}^{\frac{1}{3}} - 1}{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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