3.422 \(\int (\frac{1}{-1+2 x}-\frac{1}{1+2 x}) \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{2} \log (1-2 x)-\frac{1}{2} \log (2 x+1) \]

[Out]

Log[1 - 2*x]/2 - Log[1 + 2*x]/2

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Rubi [A]  time = 0.0035305, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \frac{1}{2} \log (1-2 x)-\frac{1}{2} \log (2 x+1) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 2*x)^(-1) - (1 + 2*x)^(-1),x]

[Out]

Log[1 - 2*x]/2 - Log[1 + 2*x]/2

Rubi steps

\begin{align*} \int \left (\frac{1}{-1+2 x}-\frac{1}{1+2 x}\right ) \, dx &=\frac{1}{2} \log (1-2 x)-\frac{1}{2} \log (1+2 x)\\ \end{align*}

Mathematica [A]  time = 0.0024301, size = 23, normalized size = 1.1 \[ 2 \left (\frac{1}{4} \log (1-2 x)-\frac{1}{4} \log (2 x+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 2*x)^(-1) - (1 + 2*x)^(-1),x]

[Out]

2*(Log[1 - 2*x]/4 - Log[1 + 2*x]/4)

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Maple [A]  time = 0.003, size = 18, normalized size = 0.9 \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(1/(2*x-1)-1/(1+2*x),x)

[Out]

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

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Maxima [A]  time = 0.964697, size = 23, normalized size = 1.1 \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(1/(-1+2*x)-1/(1+2*x),x, algorithm="maxima")

[Out]

-1/2*log(2*x + 1) + 1/2*log(2*x - 1)

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Fricas [A]  time = 0.964978, size = 51, normalized size = 2.43 \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(1/(-1+2*x)-1/(1+2*x),x, algorithm="fricas")

[Out]

-1/2*log(2*x + 1) + 1/2*log(2*x - 1)

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Sympy [A]  time = 0.089191, size = 15, normalized size = 0.71 \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(1/(-1+2*x)-1/(1+2*x),x)

[Out]

log(x - 1/2)/2 - log(x + 1/2)/2

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Giac [A]  time = 1.11539, size = 26, normalized size = 1.24 \begin{align*} -\frac{1}{2} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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