Optimal. Leaf size=156 \[ \frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{-x^3-1}}\right )}{3 \sqrt{3}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.366649, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{-x^3-1}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[x/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 144.518, size = 454, normalized size = 2.91 \[ - \frac{2^{\frac{2}{3}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 + \sqrt [3]{2}} \sqrt{1 - \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2}}{3 \sqrt{-1 + \sqrt [3]{2}} \sqrt{4 \sqrt{3} + 7 + \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{-1 + \sqrt [3]{2}} \left (1 + \sqrt [3]{2}\right )^{\frac{3}{2}} \sqrt{- x^{3} - 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 1\right ) \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1} \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} + \frac{4 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) \Pi \left (\frac{\left (-1 + 2^{\frac{2}{3}} + \sqrt{3}\right )^{2}}{\left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{- x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{4 \sqrt{3} + 7} \sqrt{- x^{3} - 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right ) \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(2**(2/3)+x)/(-x**3-1)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.526008, size = 209, normalized size = 1.34 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (-\frac{\left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac{i 2^{2/3} \sqrt{x^2-x+1} \Pi \left (\frac{i \sqrt{3}}{\sqrt [3]{-1}+2^{2/3}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1}+2^{2/3}}\right )}{\sqrt{-x^3-1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.029, size = 249, normalized size = 1.6 \[{-{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}+{\frac{{\frac{2\,i}{3}}{2}^{{\frac{2}{3}}}\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{2}^{{\frac{2}{3}}}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{2}^{{\frac{2}{3}}}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(2^(2/3)+x)/(-x^3-1)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{-x^{3} - 1}{\left (x + 2^{\frac{2}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(-x^3 - 1)*(x + 2^(2/3))),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{\sqrt{-x^{3} - 1}{\left (x + 2^{\frac{2}{3}}\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(-x^3 - 1)*(x + 2^(2/3))),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 2^{\frac{2}{3}}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(2**(2/3)+x)/(-x**3-1)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{-x^{3} - 1}{\left (x + 2^{\frac{2}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(-x^3 - 1)*(x + 2^(2/3))),x, algorithm="giac")
[Out]