3.421 \(\int \frac{2}{-1+4 x^2} \, dx\)

Optimal. Leaf size=6 \[ -\tanh ^{-1}(2 x) \]

[Out]

-ArcTanh[2*x]

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Rubi [A]  time = 0.0024724, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {12, 207} \[ -\tanh ^{-1}(2 x) \]

Antiderivative was successfully verified.

[In]

Int[2/(-1 + 4*x^2),x]

[Out]

-ArcTanh[2*x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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{2}{-1+4 x^2} \, dx &=2 \int \frac{1}{-1+4 x^2} \, dx\\ &=-\tanh ^{-1}(2 x)\\ \end{align*}

Mathematica [B]  time = 0.0029948, size = 23, normalized size = 3.83 \[ 2 \left (\frac{1}{4} \log (1-2 x)-\frac{1}{4} \log (2 x+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[2/(-1 + 4*x^2),x]

[Out]

2*(Log[1 - 2*x]/4 - Log[1 + 2*x]/4)

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Maple [B]  time = 0.004, size = 18, normalized size = 3. \begin{align*}{\frac{\ln \left ( 2\,x-1 \right ) }{2}}-{\frac{\ln \left ( 1+2\,x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(4*x^2-1),x)

[Out]

1/2*ln(2*x-1)-1/2*ln(1+2*x)

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Maxima [B]  time = 0.957874, size = 23, normalized size = 3.83 \begin{align*} -\frac{1}{2} \, \log \left (2 \, x + 1\right ) + \frac{1}{2} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(4*x^2-1),x, algorithm="maxima")

[Out]

-1/2*log(2*x + 1) + 1/2*log(2*x - 1)

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Fricas [B]  time = 0.96732, size = 51, normalized size = 8.5 \begin{align*} -\frac{1}{2} \, \log \left (2 \, x + 1\right ) + \frac{1}{2} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(4*x^2-1),x, algorithm="fricas")

[Out]

-1/2*log(2*x + 1) + 1/2*log(2*x - 1)

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Sympy [B]  time = 0.0836, size = 15, normalized size = 2.5 \begin{align*} \frac{\log{\left (x - \frac{1}{2} \right )}}{2} - \frac{\log{\left (x + \frac{1}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(4*x**2-1),x)

[Out]

log(x - 1/2)/2 - log(x + 1/2)/2

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Giac [B]  time = 1.13055, size = 20, normalized size = 3.33 \begin{align*} -\frac{1}{2} \, \log \left ({\left | x + \frac{1}{2} \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x - \frac{1}{2} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(4*x^2-1),x, algorithm="giac")

[Out]

-1/2*log(abs(x + 1/2)) + 1/2*log(abs(x - 1/2))