3.196 \(\int \frac{-x+2 x^3}{1-x^2+x^4} \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{2} \log \left (x^4-x^2+1\right ) \]

[Out]

Log[1 - x^2 + x^4]/2

_______________________________________________________________________________________

Rubi [A]  time = 0.00774903, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{1}{2} \log \left (x^4-x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-x + 2*x^3)/(1 - x^2 + x^4),x]

[Out]

Log[1 - x^2 + x^4]/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.16588, size = 10, normalized size = 0.67 \[ \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**3-x)/(x**4-x**2+1),x)

[Out]

log(x**4 - x**2 + 1)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00636606, size = 15, normalized size = 1. \[ \frac{1}{2} \log \left (x^4-x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-x + 2*x^3)/(1 - x^2 + x^4),x]

[Out]

Log[1 - x^2 + x^4]/2

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 14, normalized size = 0.9 \[{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^3-x)/(x^4-x^2+1),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.33633, size = 18, normalized size = 1.2 \[ \frac{1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^3 - x)/(x^4 - x^2 + 1),x, algorithm="maxima")

[Out]

1/2*log(x^4 - x^2 + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.194308, size = 18, normalized size = 1.2 \[ \frac{1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^3 - x)/(x^4 - x^2 + 1),x, algorithm="fricas")

[Out]

1/2*log(x^4 - x^2 + 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.089786, size = 10, normalized size = 0.67 \[ \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**3-x)/(x**4-x**2+1),x)

[Out]

log(x**4 - x**2 + 1)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207107, size = 18, normalized size = 1.2 \[ \frac{1}{2} \,{\rm ln}\left (x^{4} - x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^3 - x)/(x^4 - x^2 + 1),x, algorithm="giac")

[Out]

1/2*ln(x^4 - x^2 + 1)