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

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

_______________________________________________________________________________________

Rubi [A]  time = 0.012799, 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. \[ \frac{1}{2} \log (1-2 x)-\frac{1}{2} \log (2 x+1) \]

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 in Sympy [A]  time = 1.93035, size = 15, normalized size = 0.71 \[ \frac{\log{\left (- 2 x + 1 \right )}}{2} - \frac{\log{\left (2 x + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-1+2*x)-1/(1+2*x),x)

[Out]

log(-2*x + 1)/2 - log(2*x + 1)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00370188, 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)

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 18, normalized size = 0.9 \[{\frac{\ln \left ( 2\,x-1 \right ) }{2}}-{\frac{\ln \left ( 1+2\,x \right ) }{2}} \]

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)

_______________________________________________________________________________________

Maxima [A]  time = 0.84113, size = 23, normalized size = 1.1 \[ -\frac{1}{2} \, \log \left (2 \, x + 1\right ) + \frac{1}{2} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(2*x + 1) + 1/(2*x - 1),x, algorithm="maxima")

[Out]

-1/2*log(2*x + 1) + 1/2*log(2*x - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.269485, size = 23, normalized size = 1.1 \[ -\frac{1}{2} \, \log \left (2 \, x + 1\right ) + \frac{1}{2} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(2*x + 1) + 1/(2*x - 1),x, algorithm="fricas")

[Out]

-1/2*log(2*x + 1) + 1/2*log(2*x - 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.177171, size = 15, normalized size = 0.71 \[ \frac{\log{\left (x - \frac{1}{2} \right )}}{2} - \frac{\log{\left (x + \frac{1}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-1+2*x)-1/(1+2*x),x)

[Out]

log(x - 1/2)/2 - log(x + 1/2)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.263533, size = 26, normalized size = 1.24 \[ -\frac{1}{2} \,{\rm ln}\left ({\left | 2 \, x + 1 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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