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 = 0.00, antiderivative size = 6, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.00, 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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fricas [B]  time = 0.67, size = 17, normalized size = 2.83 \[ -\frac {1}{2} \, \log \left (2 \, x + 1\right ) + \frac {1}{2} \, \log \left (2 \, x - 1\right ) \]

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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giac [B]  time = 0.31, size = 15, normalized size = 2.50 \[ -\frac {1}{2} \, \log \left ({\left | x + \frac {1}{2} \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x - \frac {1}{2} \right |}\right ) \]

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))

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maple [B]  time = 0.00, size = 18, normalized size = 3.00 \[ \frac {\ln \left (2 x -1\right )}{2}-\frac {\ln \left (2 x +1\right )}{2} \]

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(2*x+1)

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maxima [B]  time = 1.06, size = 17, normalized size = 2.83 \[ -\frac {1}{2} \, \log \left (2 \, x + 1\right ) + \frac {1}{2} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 2.27, size = 6, normalized size = 1.00 \[ -\mathrm {atanh}\left (2\,x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(4*x^2 - 1),x)

[Out]

-atanh(2*x)

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sympy [B]  time = 0.09, size = 15, normalized size = 2.50 \[ \frac {\log {\left (x - \frac {1}{2} \right )}}{2} - \frac {\log {\left (x + \frac {1}{2} \right )}}{2} \]

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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