3.60 \(\int \frac{1+2 x^2}{1-5 x^2+4 x^4} \, dx\)

Optimal. Leaf size=39 \[ -\frac{1}{2} \log (1-2 x)+\frac{1}{2} \log (1-x)-\frac{1}{2} \log (x+1)+\frac{1}{2} \log (2 x+1) \]

[Out]

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

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Rubi [A]  time = 0.0409623, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{1}{2} \log (1-2 x)+\frac{1}{2} \log (1-x)-\frac{1}{2} \log (x+1)+\frac{1}{2} \log (2 x+1) \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x^2)/(1 - 5*x^2 + 4*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.97766, size = 29, normalized size = 0.74 \[ - \frac{\log{\left (- 2 x + 1 \right )}}{2} + \frac{\log{\left (- x + 1 \right )}}{2} - \frac{\log{\left (x + 1 \right )}}{2} + \frac{\log{\left (2 x + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**2+1)/(4*x**4-5*x**2+1),x)

[Out]

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

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Mathematica [A]  time = 0.00995051, size = 29, normalized size = 0.74 \[ \frac{1}{2} \log \left (-2 x^2+x+1\right )-\frac{1}{2} \log \left (-2 x^2-x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x^2)/(1 - 5*x^2 + 4*x^4),x]

[Out]

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

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Maple [A]  time = 0.013, size = 30, normalized size = 0.8 \[{\frac{\ln \left ( -1+x \right ) }{2}}-{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ( 2\,x-1 \right ) }{2}}+{\frac{\ln \left ( 1+2\,x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+1)/(4*x^4-5*x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.738028, size = 39, normalized size = 1. \[ \frac{1}{2} \, \log \left (2 \, x + 1\right ) - \frac{1}{2} \, \log \left (2 \, x - 1\right ) - \frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 1)/(4*x^4 - 5*x^2 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.281927, size = 34, normalized size = 0.87 \[ -\frac{1}{2} \, \log \left (2 \, x^{2} + x - 1\right ) + \frac{1}{2} \, \log \left (2 \, x^{2} - x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 1)/(4*x^4 - 5*x^2 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.206458, size = 26, normalized size = 0.67 \[ \frac{\log{\left (x^{2} - \frac{x}{2} - \frac{1}{2} \right )}}{2} - \frac{\log{\left (x^{2} + \frac{x}{2} - \frac{1}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**2+1)/(4*x**4-5*x**2+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.271918, size = 45, normalized size = 1.15 \[ \frac{1}{2} \,{\rm ln}\left ({\left | 2 \, x + 1 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 1)/(4*x^4 - 5*x^2 + 1),x, algorithm="giac")

[Out]

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