Optimal. Leaf size=25 \[ \frac{2}{3} \tan ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{-x^3-1}}\right ) \]
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Rubi [A] time = 0.116235, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2}{3} \tan ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{-x^3-1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + x)/((2 - x)*Sqrt[-1 - x^3]),x]
[Out]
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Rubi in Sympy [A] time = 94.9076, size = 377, normalized size = 15.08 \[ \frac{3 \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \frac{\sqrt{3}}{3} + 1\right ) \left (x + 1\right ) \operatorname{atan}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 - \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2}}{3 \sqrt{4 \sqrt{3} + 7 + \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \sqrt{- x^{3} - 1}} + \frac{2 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \sqrt{- x^{3} - 1}} + \frac{12 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) \Pi \left (- 4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{- x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{4 \sqrt{3} + 7} \sqrt{- x^{3} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x)/(2-x)/(-x**3-1)**(1/2),x)
[Out]
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Mathematica [C] time = 0.321923, size = 267, normalized size = 10.68 \[ \frac{2 \sqrt{6} \sqrt{-\frac{i (x+1)}{\sqrt{3}-3 i}} \left (2 \sqrt{3} \sqrt{2 i x+\sqrt{3}-i} \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )-i \sqrt{-2 i x+\sqrt{3}+i} \left (\left (\sqrt{3}-i\right ) x-\sqrt{3}-i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )\right )}{\left (\sqrt{3}+3 i\right ) \sqrt{2 i x+\sqrt{3}-i} \sqrt{-x^3-1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 + x)/((2 - x)*Sqrt[-1 - x^3]),x]
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Maple [C] time = 0.036, size = 240, normalized size = 9.6 \[{{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}+{\frac{2\,i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x)/(2-x)/(-x^3-1)^(1/2),x)
[Out]
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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 = 0.351834, size = 43, normalized size = 1.72 \[ \frac{1}{3} \, \arctan \left (\frac{x^{3} + 12 \, x^{2} - 6 \, x + 10}{6 \, \sqrt{-x^{3} - 1}{\left (x + 1\right )}}\right ) \]
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)
[Out]
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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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