Optimal. Leaf size=23 \[ \frac{2}{3} \tanh ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{x^3+1}}\right ) \]
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Rubi [A] time = 0.103326, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2}{3} \tanh ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{x^3+1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + x)/((2 - x)*Sqrt[1 + x^3]),x]
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Rubi in Sympy [A] time = 98.4245, size = 371, normalized size = 16.13 \[ \frac{3 \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (\frac{\sqrt{3}}{3} + 1\right ) \left (x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{- \sqrt{3} + 2} \sqrt{- \frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} + 1}}{3 \sqrt{\frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} - 4 \sqrt{3} + 7}} \right )}}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{x^{3} + 1}} - \frac{2 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{x^{3} + 1}} - \frac{12 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) \Pi \left (4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{- x - 1 + \sqrt{3}}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{x^{3} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x)/(2-x)/(x**3+1)**(1/2),x)
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Mathematica [C] time = 0.319002, size = 265, normalized size = 11.52 \[ \frac{2 \sqrt{6} \sqrt{-\frac{i (x+1)}{\sqrt{3}-3 i}} \left (2 \sqrt{3} \sqrt{2 i x+\sqrt{3}-i} \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )-i \sqrt{-2 i x+\sqrt{3}+i} \left (\left (\sqrt{3}-i\right ) x-\sqrt{3}-i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )\right )}{\left (\sqrt{3}+3 i\right ) \sqrt{2 i x+\sqrt{3}-i} \sqrt{x^3+1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 + x)/((2 - x)*Sqrt[1 + x^3]),x]
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Maple [C] time = 0.032, size = 240, normalized size = 10.4 \[ -2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }+2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},1/2-i/6\sqrt{3},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x)/(2-x)/(x^3+1)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{x + 1}{\sqrt{x^{3} + 1}{\left (x - 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x + 1)/(sqrt(x^3 + 1)*(x - 2)),x, algorithm="maxima")
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Fricas [A] time = 0.34117, size = 59, normalized size = 2.57 \[ \frac{1}{3} \, \log \left (\frac{x^{3} + 12 \, x^{2} + 6 \, \sqrt{x^{3} + 1}{\left (x + 1\right )} - 6 \, x + 10}{x^{3} - 6 \, x^{2} + 12 \, x - 8}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x + 1)/(sqrt(x^3 + 1)*(x - 2)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{x \sqrt{x^{3} + 1} - 2 \sqrt{x^{3} + 1}}\, dx - \int \frac{1}{x \sqrt{x^{3} + 1} - 2 \sqrt{x^{3} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x)/(2-x)/(x**3+1)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{x + 1}{\sqrt{x^{3} + 1}{\left (x - 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x + 1)/(sqrt(x^3 + 1)*(x - 2)),x, algorithm="giac")
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