Optimal. Leaf size=180 \[ -\frac{1}{2} \log \left (-\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-x+3}{x^2}\right )+\frac{1}{14} \left (7+\sqrt{7}\right ) \log \left (-\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{7}+\sqrt{3}+1\right )+\frac{1}{14} \left (7-\sqrt{7}\right ) \log \left (-\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}+\sqrt{7}+\sqrt{3}+1\right )+\tan ^{-1}\left (\frac{\sqrt{3}-\sqrt{-x^2-2 x+3}}{x}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.41385, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{2} \log \left (-\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-x+3}{x^2}\right )+\frac{1}{14} \left (7+\sqrt{7}\right ) \log \left (-\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{7}+\sqrt{3}+1\right )+\frac{1}{14} \left (7-\sqrt{7}\right ) \log \left (-\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}+\sqrt{7}+\sqrt{3}+1\right )+\tan ^{-1}\left (\frac{\sqrt{3}-\sqrt{-x^2-2 x+3}}{x}\right ) \]
Antiderivative was successfully verified.
[In] Int[(x + Sqrt[3 - 2*x - x^2])^(-1),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 84.3252, size = 180, normalized size = 1. \[ - \frac{\log{\left (1 + \frac{\left (\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}\right )^{2}}{x^{2}} \right )}}{2} + \frac{\log{\left (-6 + 4 \sqrt{3} + \frac{\left (4 \sqrt{3} + 12\right ) \left (\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}\right )}{x} + \frac{\left (\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}\right )^{2}}{x^{2}} \right )}}{2} - \operatorname{atan}{\left (\frac{\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}}{x} \right )} + \frac{\sqrt{2} \left (- 2 \sqrt{3} + 5\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt{3} + 3 + \frac{\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}}{2 x}\right )}{\sqrt{10 \sqrt{3} + 27}} \right )}}{2 \sqrt{10 \sqrt{3} + 27}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x+(-x**2-2*x+3)**(1/2)),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.89127, size = 250, normalized size = 1.39 \[ \frac{1}{28} \left (-\sqrt{14 \left (4+\sqrt{7}\right )} \log \left (\sqrt{14 \left (4+\sqrt{7}\right )} \sqrt{-x^2-2 x+3}-\sqrt{7} x+7 x+7 \sqrt{7}+7\right )-\frac{1}{3} \left (\sqrt{7}-4\right ) \sqrt{14 \left (4+\sqrt{7}\right )} \log \left (-\sqrt{14} \sqrt{\left (\sqrt{7}-4\right ) \left (x^2+2 x-3\right )}+\left (7+\sqrt{7}\right ) x-7 \sqrt{7}+7\right )-\left (\sqrt{7}-7\right ) \log \left (-2 x+\sqrt{7}-1\right )+\frac{1}{3} \left (\sqrt{7}-4\right ) \sqrt{14 \left (4+\sqrt{7}\right )} \log \left (2 x-\sqrt{7}+1\right )+\left (7+\sqrt{7}\right ) \log \left (2 x+\sqrt{7}+1\right )+\sqrt{14 \left (4+\sqrt{7}\right )} \log \left (2 x+\sqrt{7}+1\right )+14 \sin ^{-1}\left (\frac{x+1}{2}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[(x + Sqrt[3 - 2*x - x^2])^(-1),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.084, size = 551, normalized size = 3.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x+(-x^2-2*x+3)^(1/2)),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x + \sqrt{-x^{2} - 2 \, x + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x + sqrt(-x^2 - 2*x + 3)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.291535, size = 379, normalized size = 2.11 \[ \frac{1}{56} \, \sqrt{7}{\left (4 \, \sqrt{7} \arctan \left (\frac{x + 1}{\sqrt{-x^{2} - 2 \, x + 3}}\right ) + 2 \, \sqrt{7} \log \left (2 \, x^{2} + 2 \, x - 3\right ) - \sqrt{7} \log \left (\frac{2 \, \sqrt{-x^{2} - 2 \, x + 3} x + 2 \, x - 3}{x^{2}}\right ) + \sqrt{7} \log \left (-\frac{2 \, \sqrt{-x^{2} - 2 \, x + 3} x - 2 \, x + 3}{x^{2}}\right ) + 2 \, \log \left (\frac{2 \, \sqrt{7}{\left (x^{2} + x + 2\right )} + 14 \, x + 7}{2 \, x^{2} + 2 \, x - 3}\right ) + \log \left (\frac{28 \, x^{2} + \sqrt{7}{\left (7 \, x^{2} - 30 \, x + 45\right )} + 3 \, \sqrt{-x^{2} - 2 \, x + 3}{\left (4 \, \sqrt{7} x + 7 \, x - 21\right )} - 84 \, x}{2 \, \sqrt{-x^{2} - 2 \, x + 3} x + 2 \, x - 3}\right ) + \log \left (\frac{28 \, x^{2} - \sqrt{7}{\left (7 \, x^{2} - 30 \, x + 45\right )} + 3 \, \sqrt{-x^{2} - 2 \, x + 3}{\left (4 \, \sqrt{7} x - 7 \, x + 21\right )} - 84 \, x}{2 \, \sqrt{-x^{2} - 2 \, x + 3} x - 2 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x + sqrt(-x^2 - 2*x + 3)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x + \sqrt{- x^{2} - 2 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x+(-x**2-2*x+3)**(1/2)),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.323625, size = 387, normalized size = 2.15 \[ -\frac{1}{28} \, \sqrt{7}{\rm ln}\left (\frac{{\left | 4 \, x - 2 \, \sqrt{7} + 2 \right |}}{{\left | 4 \, x + 2 \, \sqrt{7} + 2 \right |}}\right ) + \frac{1}{28} \, \sqrt{7}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{7} + \frac{6 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}{{\left | 2 \, \sqrt{7} + \frac{6 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}\right ) - \frac{1}{28} \, \sqrt{7}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{7} + \frac{2 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}{{\left | 2 \, \sqrt{7} + \frac{2 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}\right ) + \frac{1}{2} \, \arcsin \left (\frac{1}{2} \, x + \frac{1}{2}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | 2 \, x^{2} + 2 \, x - 3 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | \frac{4 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac{3 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | -\frac{4 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac{{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x + sqrt(-x^2 - 2*x + 3)),x, algorithm="giac")
[Out]