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