3.45 \(\int \frac{3+x^2}{-3+x^2} \, dx\)

Optimal. Leaf size=17 \[ x-2 \sqrt{3} \tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

[Out]

x - 2*Sqrt[3]*ArcTanh[x/Sqrt[3]]

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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]

x - 2*Sqrt[3]*ArcTanh[x/Sqrt[3]]

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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]

x - 2*sqrt(3)*atanh(sqrt(3)*x/3)

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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]

x + Sqrt[3]*Log[Sqrt[3] - x] - Sqrt[3]*Log[Sqrt[3] + x]

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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]

x-2*arctanh(1/3*x*3^(1/2))*3^(1/2)

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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]

sqrt(3)*log((x - sqrt(3))/(x + sqrt(3))) + x

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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]

sqrt(3)*log((x^2 - 2*sqrt(3)*x + 3)/(x^2 - 3)) + x

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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]

x + sqrt(3)*log(x - sqrt(3)) - sqrt(3)*log(x + sqrt(3))

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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]

sqrt(3)*ln(abs(2*x - 2*sqrt(3))/abs(2*x + 2*sqrt(3))) + x