Optimal. Leaf size=147 \[ \frac{2 \sqrt{\frac{7}{6}+\frac{2}{\sqrt{3}}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{x^3+1}}\right )}{3^{3/4}} \]
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Rubi [A] time = 0.461211, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{\sqrt{2 \left (7+4 \sqrt{3}\right )} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{x^3+1}}\right )}{3^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[x/((1 - Sqrt[3] + x)*Sqrt[1 + x^3]),x]
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Rubi in Sympy [A] time = 41.6472, size = 240, normalized size = 1.63 \[ \frac{\sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (- \sqrt{3} + 1\right ) \left (x + 1\right ) \operatorname{atanh}{\left (\frac{\left (- \sqrt{3} + 2\right ) \sqrt{- \frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} + 1}}{\sqrt{\frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} - 4 \sqrt{3} + 7}} \right )}}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt{x^{3} + 1}} + \frac{\sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (1 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(1+x-3**(1/2))/(x**3+1)**(1/2),x)
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Mathematica [C] time = 0.97551, size = 225, normalized size = 1.53 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (\frac{\sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \left (\left ((1+2 i) \sqrt{3}-3 i\right ) x-(2+i) \sqrt{3}+3\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}-2 \left (\sqrt{3}-1\right ) \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )\right )}{\left ((1+2 i) \sqrt{3}-3 i\right ) \sqrt{x^3+1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x/((1 - Sqrt[3] + x)*Sqrt[1 + x^3]),x]
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Maple [B] time = 0.033, size = 255, normalized size = 1.7 \[{\frac{ \left ( -2\,\sqrt{3}+2 \right ) \left ({\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{3}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},-{\frac{ \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{3}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}+1}}}}+2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(1+x-3^(1/2))/(x^3+1)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{3} + 1}{\left (x - \sqrt{3} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(x^3 + 1)*(x - sqrt(3) + 1)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{\sqrt{x^{3} + 1}{\left (x - \sqrt{3} + 1\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(x^3 + 1)*(x - sqrt(3) + 1)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x - \sqrt{3} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(1+x-3**(1/2))/(x**3+1)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{3} + 1}{\left (x - \sqrt{3} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(x^3 + 1)*(x - sqrt(3) + 1)),x, algorithm="giac")
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