Optimal. Leaf size=145 \[ \frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]
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Rubi [A] time = 0.193547, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]
Antiderivative was successfully verified.
[In] Int[x/((1 + x)*(1 - x^3)^(1/3)),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (x + 1\right ) \sqrt [3]{- x^{3} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(1+x)/(-x**3+1)**(1/3),x)
[Out]
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Mathematica [A] time = 0.0889764, size = 0, normalized size = 0. \[ \int \frac{x}{(1+x) \sqrt [3]{1-x^3}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[x/((1 + x)*(1 - x^3)^(1/3)),x]
[Out]
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Maple [F] time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{x}{1+x}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(1+x)/(-x^3+1)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((-x^3 + 1)^(1/3)*(x + 1)),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((-x^3 + 1)^(1/3)*(x + 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(1+x)/(-x**3+1)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((-x^3 + 1)^(1/3)*(x + 1)),x, algorithm="giac")
[Out]