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 = 0.0057999, 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, Rules used = {388, 207} \[ x-2 \sqrt{3} \tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(3 + x^2)/(-3 + x^2),x]

[Out]

x - 2*Sqrt[3]*ArcTanh[x/Sqrt[3]]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{3+x^2}{-3+x^2} \, dx &=x+6 \int \frac{1}{-3+x^2} \, dx\\ &=x-2 \sqrt{3} \tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0074821, 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 = 0.002, size = 15, normalized size = 0.9 \begin{align*} x-2\,{\it Artanh} \left ( 1/3\,x\sqrt{3} \right ) \sqrt{3} \end{align*}

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 = 1.45037, size = 30, normalized size = 1.76 \begin{align*} \sqrt{3} \log \left (\frac{x - \sqrt{3}}{x + \sqrt{3}}\right ) + x \end{align*}

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 = 1.41783, size = 70, normalized size = 4.12 \begin{align*} \sqrt{3} \log \left (\frac{x^{2} - 2 \, \sqrt{3} x + 3}{x^{2} - 3}\right ) + x \end{align*}

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 = 0.082351, size = 27, normalized size = 1.59 \begin{align*} x + \sqrt{3} \log{\left (x - \sqrt{3} \right )} - \sqrt{3} \log{\left (x + \sqrt{3} \right )} \end{align*}

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 [B]  time = 1.13001, size = 41, normalized size = 2.41 \begin{align*} \sqrt{3} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} \right |}}\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+3)/(x^2-3),x, algorithm="giac")

[Out]

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