Optimal. Leaf size=17 \[ x-2 \sqrt{3} \tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]
[Out]
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Rubi [A] time = 0.0222769, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ x-2 \sqrt{3} \tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(3 + x^2)/(-3 + x^2),x]
[Out]
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Rubi in Sympy [A] time = 4.40597, size = 17, normalized size = 1. \[ x - 2 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} x}{3} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+3)/(x**2-3),x)
[Out]
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Mathematica [A] time = 0.0165828, size = 33, normalized size = 1.94 \[ x+\sqrt{3} \log \left (\sqrt{3}-x\right )-\sqrt{3} \log \left (x+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(3 + x^2)/(-3 + x^2),x]
[Out]
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Maple [A] time = 0.003, size = 15, normalized size = 0.9 \[ x-2\,{\it Artanh} \left ( 1/3\,x\sqrt{3} \right ) \sqrt{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+3)/(x^2-3),x)
[Out]
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Maxima [A] time = 1.48273, size = 30, normalized size = 1.76 \[ \sqrt{3} \log \left (\frac{x - \sqrt{3}}{x + \sqrt{3}}\right ) + x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 3)/(x^2 - 3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226644, size = 35, normalized size = 2.06 \[ \sqrt{3} \log \left (\frac{x^{2} - 2 \, \sqrt{3} x + 3}{x^{2} - 3}\right ) + x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 3)/(x^2 - 3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.076731, size = 27, normalized size = 1.59 \[ x + \sqrt{3} \log{\left (x - \sqrt{3} \right )} - \sqrt{3} \log{\left (x + \sqrt{3} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+3)/(x**2-3),x)
[Out]
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GIAC/XCAS [A] time = 0.213023, size = 41, normalized size = 2.41 \[ \sqrt{3}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} \right |}}\right ) + x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 3)/(x^2 - 3),x, algorithm="giac")
[Out]