Optimal. Leaf size=108 \[ \frac{3}{4} \log \left (-\sqrt [3]{x^3+2}+x+2\right )-\frac{1}{4} \log \left (\sqrt [3]{x^3+2}-x\right )+\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )-\frac{1}{2} \log (x+1) \]
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Rubi [A] time = 0.160376, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3}{4} \log \left (-\sqrt [3]{x^3+2}+x+2\right )-\frac{1}{4} \log \left (\sqrt [3]{x^3+2}-x\right )+\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )-\frac{1}{2} \log (x+1) \]
Antiderivative was successfully verified.
[In] Int[1/((1 + x)*(2 + x^3)^(1/3)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x + 1\right ) \sqrt [3]{x^{3} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1+x)/(x**3+2)**(1/3),x)
[Out]
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Mathematica [A] time = 0.0469073, size = 0, normalized size = 0. \[ \int \frac{1}{(1+x) \sqrt [3]{2+x^3}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[1/((1 + x)*(2 + x^3)^(1/3)),x]
[Out]
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Maple [F] time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{1}{1+x}{\frac{1}{\sqrt [3]{{x}^{3}+2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1+x)/(x^3+2)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^3 + 2)^(1/3)*(x + 1)),x, algorithm="maxima")
[Out]
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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^3 + 2)^(1/3)*(x + 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x + 1\right ) \sqrt [3]{x^{3} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1+x)/(x**3+2)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^3 + 2)^(1/3)*(x + 1)),x, algorithm="giac")
[Out]