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

Optimal. Leaf size=31 \[ \frac{1}{4} \log \left (2 x^2+2 x+1\right )-\frac{1}{4} \log \left (2 x^2-2 x+1\right ) \]

[Out]

-Log[1 - 2*x + 2*x^2]/4 + Log[1 + 2*x + 2*x^2]/4

_______________________________________________________________________________________

Rubi [A]  time = 0.0293277, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{1}{4} \log \left (2 x^2+2 x+1\right )-\frac{1}{4} \log \left (2 x^2-2 x+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x^2)/(1 + 4*x^4),x]

[Out]

-Log[1 - 2*x + 2*x^2]/4 + Log[1 + 2*x + 2*x^2]/4

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.23885, size = 26, normalized size = 0.84 \[ - \frac{\log{\left (2 x^{2} - 2 x + 1 \right )}}{4} + \frac{\log{\left (2 x^{2} + 2 x + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2*x**2+1)/(4*x**4+1),x)

[Out]

-log(2*x**2 - 2*x + 1)/4 + log(2*x**2 + 2*x + 1)/4

_______________________________________________________________________________________

Mathematica [A]  time = 0.00748568, size = 31, normalized size = 1. \[ \frac{1}{4} \log \left (2 x^2+2 x+1\right )-\frac{1}{4} \log \left (2 x^2-2 x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x^2)/(1 + 4*x^4),x]

[Out]

-Log[1 - 2*x + 2*x^2]/4 + Log[1 + 2*x + 2*x^2]/4

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 28, normalized size = 0.9 \[ -{\frac{\ln \left ( 2\,{x}^{2}-2\,x+1 \right ) }{4}}+{\frac{\ln \left ( 2\,{x}^{2}+2\,x+1 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-2*x^2+1)/(4*x^4+1),x)

[Out]

-1/4*ln(2*x^2-2*x+1)+1/4*ln(2*x^2+2*x+1)

_______________________________________________________________________________________

Maxima [A]  time = 0.741158, size = 36, normalized size = 1.16 \[ \frac{1}{4} \, \log \left (2 \, x^{2} + 2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x^{2} - 2 \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*log(2*x^2 + 2*x + 1) - 1/4*log(2*x^2 - 2*x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.279919, size = 36, normalized size = 1.16 \[ \frac{1}{4} \, \log \left (2 \, x^{2} + 2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x^{2} - 2 \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*log(2*x^2 + 2*x + 1) - 1/4*log(2*x^2 - 2*x + 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.191743, size = 22, normalized size = 0.71 \[ - \frac{\log{\left (x^{2} - x + \frac{1}{2} \right )}}{4} + \frac{\log{\left (x^{2} + x + \frac{1}{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x**2+1)/(4*x**4+1),x)

[Out]

-log(x**2 - x + 1/2)/4 + log(x**2 + x + 1/2)/4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.275379, size = 46, normalized size = 1.48 \[ \frac{1}{4} \,{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{1}{4}\right )^{\frac{1}{4}} x + \frac{1}{2}\right ) - \frac{1}{4} \,{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{1}{4}\right )^{\frac{1}{4}} x + \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*ln(x^2 + sqrt(2)*(1/4)^(1/4)*x + 1/2) - 1/4*ln(x^2 - sqrt(2)*(1/4)^(1/4)*x +
1/2)